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Open Access Research

Further results on common zeros of the solutions of two differential equations

Asim Asiri

Author Affiliations

Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

Journal of Inequalities and Applications 2012, 2012:222  doi:10.1186/1029-242X-2012-222


The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2012/1/222


Received:8 May 2012
Accepted:17 September 2012
Published:4 October 2012

© 2012 Asiri; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Purpose

Two problems are discussed in this paper. In the first problem, we consider one homogeneous and one non-homogeneous differential equations and study when the solutions of these differential equations can have (nearly) the same zeros. In the second problem, we consider two linear second-order differential equations and investigate when the solutions of these differential equations can take the value 0 and a non-zero value at (nearly) the same points.

Method

We apply the Nevanlinna theory and properties of entire solutions of linear differential equations.

Conclusion

In the first problem, the results determine all pairs of such equations having solutions with the same zeros or nearly the same zeros. Regarding the second problem, the results also show all pairs of such equations having solutions taking the value 0 and a non-zero value at (nearly) the same points.

Keywords:
Nevanlinna theory; differential equations

1 Introduction

There has been much research [1-8] on zeros of solutions of linear differential equations with entire coefficients. The principal paper [9] that was published in 1982 by Bank and Laine has stimulated many studies on this kind of problems. The reader is referred to [10-12] for background on some applications of the Nevanlinna theory. We use the standard notation of the Nevanlinna theory from [13].

In 1955, Wittich [12] proved the following theorem.

Theorem 1.1Iffis a non-trivial solution of<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M1">View MathML</a>, i.e., <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M2">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M3">View MathML</a>is entire, then we have:

(i) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M4">View MathML</a>.

(ii) Iffhas finite order, thenAis a polynomial.

(iii) Ifais a non-zero complex number, thenftakes the valueainfinitely often, and in fact, outside a set of finite measure,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M5">View MathML</a>

The following facts follow from the asymptotic representation for solutions of the equation

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M6">View MathML</a>

(1)

Theorem 1.2[9,11]

LetPbe a polynomial of degreen, and letwbe a non-trivial solution of the equation (1). Then, whas order of growth equal to<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M7">View MathML</a>. Moreover, ifwis a solution of (1) which has infinitely many zeros, then we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M8">View MathML</a>

(2)

Our previous paper [14] studied homogeneous linear differential equations having solutions with nearly the same zeros and proved several results, including the following.

Theorem 1.3[14]

Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M9">View MathML</a>be a polynomial of degreen. Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M10">View MathML</a>be a solution of (1). Assume thatwhas infinitely many zeros. Suppose that we have a solution<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M11">View MathML</a>of the differential equation

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M12">View MathML</a>

(3)

such thatAis an entire function and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M13">View MathML</a>counts zeros ofvwhich are not zeros ofwand zeros ofwwhich are not zeros ofv. Assume that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M14">View MathML</a>

Then<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M15">View MathML</a>is a constant and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M16">View MathML</a>.

The paper [14] includes further results for homogeneous linear differential equations, and the corresponding problem where P is a transcendental entire function of finite order is studied in [15].

Recently, the same problem, but with non-homogeneous first-order differential equations, has been studied in [16], including the following result.

Theorem 1.4[16]

Assume that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M17">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M18">View MathML</a>, whereA, B, CandDare entire functions of order less than 1 andv, ware transcendental functions. Assume that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M19">View MathML</a>, whereLhas finitely many zeros and poles, and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M20">View MathML</a>

(4)

Then the following conclusions hold.

(I) IfLis a rational function, then<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M21">View MathML</a>, Lis a constant and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M22">View MathML</a>.

(II) IfLis a transcendental function, then one of the following cases holds: If, in addition, Lhas finite order in case (ii), thenA, B, C, Dare polynomials and so is<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M31">View MathML</a>.

(i) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M23">View MathML</a>andv, whave no zeros.

(ii) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M24">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M25">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M26">View MathML</a>are non-zero constants, and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M27">View MathML</a>

where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M28">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M29">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M30">View MathML</a>.

In this paper, our first result (Theorem 2.1) looks at the same problem but with one homogeneous and one non-homogeneous differential equations. In particular, we consider the first equation to be homogeneous of the second-order with a polynomial coefficient and the second equation to be non-homogeneous of the first-order with entire coefficients.

A further result (Theorem 2.2) studies the case where the solutions of two second-order homogeneous differential equations can take the value 0 and a non-zero value at (nearly) the same points.

2 Our results

Our first result is the following theorem.

Theorem 2.1Suppose that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M9">View MathML</a>is a polynomial of degreen, andwsolves (1), and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M10">View MathML</a>has infinitely many zeros. Suppose that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M11">View MathML</a>solves

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M35">View MathML</a>

(5)

whereA, Bare entire and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M36">View MathML</a>, and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M37">View MathML</a>

(6)

Suppose that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M38">View MathML</a>has finitely many zeros and poles (i.e., wandvhave the same zeros with finitely many exceptions).

ThenAis a polynomial and there exists a polynomialQsuch that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M39">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M40">View MathML</a>

where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M29">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M42">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M43">View MathML</a>are constants.

Example 2.1 Take Q to be a polynomial. Let

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M44">View MathML</a>

Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M45">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M46">View MathML</a>

So, we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M47">View MathML</a>.

Now, let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M31">View MathML</a> be another polynomial, and let

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M49">View MathML</a>

Note that v has the same zeros as w. Now, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M50">View MathML</a>

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M51">View MathML</a>.

We choose <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M31">View MathML</a> so that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M53">View MathML</a>

For example, let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M54">View MathML</a>.

We now state our second result.

Theorem 2.2Suppose that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M9">View MathML</a>is a polynomial of degreen, andAis an entire function, and suppose thatwsolves (1) andvsolves (3), and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M56">View MathML</a>. Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M57">View MathML</a>andwhave, with finitely many exceptions, the same zeros and the same multiplicities. Then one of the following holds.

(A) whas finitely many zeros andvis a polynomial and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M58">View MathML</a>.

(B) whas infinitely many zeros andP, Aare non-zero constants and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M59">View MathML</a>is non-constant and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M60">View MathML</a>

(7)

whereσ, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M61">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M62">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M63">View MathML</a>are non-zero constants.

Example 2.2 If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M64">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M65">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M66">View MathML</a>

Hence, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M57">View MathML</a> has the same zeros as w. Here <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M68">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M69">View MathML</a>.

Example 2.3 We give an example to show that the zeros of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M57">View MathML</a> and w must necessarily have the same multiplicities. To show this, let

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M71">View MathML</a>

Then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M72">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M73">View MathML</a>.

Also <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M74">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M73">View MathML</a>.

Therefore, w and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M57">View MathML</a> have the same zeros but the zeros are simple for w, double for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M57">View MathML</a>. Here, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M78">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M79">View MathML</a>.

3 Proof of Theorem 2.1

Proof We have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M80">View MathML</a>

(8)

So,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M81">View MathML</a>

(9)

but

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M82">View MathML</a>

(10)

We also have, using (1), (5), (8), (9) and (10),

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M83">View MathML</a>

(11)

Let

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M84">View MathML</a>

(12)

Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M85">View MathML</a>

(13)

We divide (11) by L, and by using (12) and (13), we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M86">View MathML</a>

(14)

The next step is to estimate the growth of M.

We know that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M87">View MathML</a> from [9]. Therefore,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M88">View MathML</a>

(15)

Claim 3.1We claim that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M89">View MathML</a>.

To show this, we know that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M90">View MathML</a> since M has finitely many poles.

Write (5) as

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M91">View MathML</a>

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M29">View MathML</a>.

Then, there exists a constant c such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M93">View MathML</a>

Also, using (6), we can write

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M94">View MathML</a>

Also,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M95">View MathML</a>

So,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M96">View MathML</a>

Therefore, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M97">View MathML</a>

We use Lemma 2.3 in [[13], p.38] with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M98">View MathML</a> to get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M99">View MathML</a>

Now, we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M100">View MathML</a>. So

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M101">View MathML</a>

We also have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M102">View MathML</a>.

Hence,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M103">View MathML</a>

This completes the proof of Claim 3.1.

Using Claim 3.1 and (14), we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M104">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M105">View MathML</a>

Also, by Theorem 1.2, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M106">View MathML</a>

Therefore, we must have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M107">View MathML</a>

(16)

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M108">View MathML</a>

(17)

because otherwise we can write <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M109">View MathML</a> to get a contradiction.

We now divide (17) by B to get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M110">View MathML</a>

So, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M111">View MathML</a> has finitely many poles, and so B has finitely many zeros. Then we can write B in the form

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M112">View MathML</a>

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M113">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M114">View MathML</a> are polynomials.

But then, we can write

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M115">View MathML</a>

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M116">View MathML</a> is rational.

Then we also can write

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M117">View MathML</a>

(18)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M118">View MathML</a> is rational and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M119">View MathML</a>.

Substitute (18) in (16), we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M120">View MathML</a>

(19)

Now, let

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M121">View MathML</a>

Also, let

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M122">View MathML</a>

(20)

So, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M123">View MathML</a>

(21)

Substituting (21) in (19), we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M124">View MathML</a>

Thus, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M125">View MathML</a>, H is entire, and B has no zeros.

Then,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M126">View MathML</a>

Therefore, A is a polynomial.

Since B has no zeros, from (20) we can write

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M127">View MathML</a>

(22)

where Q is a polynomial.

Since w and H solve the same equation and are linearly independent (because w has zeros but H does not), we can write

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M128">View MathML</a>

Therefore,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M129">View MathML</a>

(23)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M42">View MathML</a> is a constant and Q is a polynomial.

Now, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M131">View MathML</a>

So, we can write

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M132">View MathML</a>

Therefore, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M133">View MathML</a>

Hence, using (22), we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M134">View MathML</a>

(24)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M43">View MathML</a> is a constant and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M29">View MathML</a>.

Now, from (23) and (24), we notice that w and v have the same zeros.

Also, differentiating (24), using (22), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M137">View MathML</a>

Comparing this with (5), we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M138">View MathML</a>

Moreover, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M139">View MathML</a> solves <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M140">View MathML</a>, and so

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M141">View MathML</a>

This completes the proof of Theorem 2.1.  □

4 A lemma needed to prove Theorem 2.2

In order to prove Theorem 2.2, we must state and prove the following lemma. We include a proof for completeness.

Lemma 4.1Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M142">View MathML</a>be distinct, and let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M143">View MathML</a>be rational functions such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M144">View MathML</a>

(25)

Then there exists<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M145">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M146">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M147">View MathML</a>, and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M148">View MathML</a>for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M149">View MathML</a>.

Proof The proof is by induction. It is obvious that the lemma is true when <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M150">View MathML</a>.

Assume that the lemma is true for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M151">View MathML</a>. Differentiating (25), we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M152">View MathML</a>

Now, we have two cases to consider.

Case (1): Suppose there exists k such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M153">View MathML</a>. Without loss of generality, let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M154">View MathML</a>, then we can write

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M155">View MathML</a>

Since we assumed the lemma is true for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M151">View MathML</a>, there exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M157">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M158">View MathML</a>. But this contradicts our assumption that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M159">View MathML</a> are distinct.

Case (2): Suppose that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M160">View MathML</a> for each j, i.e.,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M161">View MathML</a>

If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M162">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M163">View MathML</a> because otherwise we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M164">View MathML</a>

But this contradicts the fact that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M162">View MathML</a>.

So, we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M163">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M162">View MathML</a>. Thus, (25) becomes (for some k)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M168">View MathML</a>

and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M146">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M147">View MathML</a>. □

5 Proof of Theorem 2.2

We first note that, outside a set of finite measure, by Theorem 1.1,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M171">View MathML</a>

(26)

In particular, if w has finitely many zeros, then v is a polynomial, which gives <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M58">View MathML</a>. This completes the proof of part (A) in the conclusion.

Assume henceforth that w has infinitely many zeros. Then (26) implies that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M173">View MathML</a>, and so A is a polynomial of degree at most n by the Wiman-Valiron theory [17]. Also, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M3">View MathML</a> since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M57">View MathML</a> has infinitely many zeros.

Now, two cases have to be considered.

Case (I). Assume that P is a non-zero constant; then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M176">View MathML</a> and A is constant. Therefore, we can write

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M177">View MathML</a>

(27)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M178">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M179">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M180">View MathML</a> (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M181">View MathML</a>).

Since w and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M57">View MathML</a> have the same zeros with finitely many exceptions, we can write

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M183">View MathML</a>

(28)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M116">View MathML</a> is a rational function and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M113">View MathML</a> is a polynomial. We know that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M186">View MathML</a> because <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M187">View MathML</a>. We can now write

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M188">View MathML</a>

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M189">View MathML</a>, and so

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M190">View MathML</a>

Now, by using Lemma 4.1, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M116">View MathML</a> is constant and so we can write (28) as

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M192">View MathML</a>

(29)

where δ is constant.

Therefore,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M193">View MathML</a>

(30)

Now, by using Lemma 4.1, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M194">View MathML</a>

and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M195">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M196">View MathML</a>, β, −β, 0 cannot all be different.

We must now try six cases:

I(a): If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M197">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M198">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M199">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M200">View MathML</a>. But this contradicts (30). Thus, this case cannot happen.

I(b): If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M197">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M202">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M203">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M204">View MathML</a>. Substituting these in (30) gives

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M205">View MathML</a>

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M206">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M207">View MathML</a> are constants, which yields <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M208">View MathML</a>. Putting this in (27) gives (7) with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M209">View MathML</a>.

There are four more cases:

I(c): <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M210">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M211">View MathML</a>.

I(d): <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M210">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M202">View MathML</a>.

I(e): <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M214">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M211">View MathML</a>.

I(f): <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M214">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M198">View MathML</a>.

It is easy to check that case I(c) is impossible and cases I(d), I(e), I(f) all lead to (7) with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M218">View MathML</a>.

From these cases, we find that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M219">View MathML</a>, and so <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M220">View MathML</a> is non-constant. Also, we have (7) and case (B) of the conclusion.

Case (II). Suppose that P is non-constant. We will show that this leads to a contradiction. Let

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M221">View MathML</a>

(31)

where L is a rational function and Q is an entire function.

From (26), we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M222">View MathML</a>, and so Q is a polynomial.

Also, from (31), we have

So,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M224">View MathML</a>

(32)

Now, we have two cases to consider.

Case (i): If M is constant, then either <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M16">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M58">View MathML</a>, so that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M227">View MathML</a>, which is a contradiction, or

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M228">View MathML</a>

is a rational function, which is a contradiction since w has infinitely many zeros.

Case (ii): If M is non-constant, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M229">View MathML</a>. Therefore,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M230">View MathML</a>

(33)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M231">View MathML</a> is rational because <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M232">View MathML</a> is rational and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M233">View MathML</a> is rational, and so <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M234">View MathML</a> is rational.

Also,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M235">View MathML</a>

Then we can write (33) as

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M236">View MathML</a>

(34)

where

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M237">View MathML</a>

are rational functions and Q is a polynomial.

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M238">View MathML</a>, then we can write (34) as

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M239">View MathML</a>

(35)

Now, we have two cases to consider:

Case ii(a): If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M240">View MathML</a> in (35), then (34) gives

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M241">View MathML</a>

which is a contradiction since w has infinitely many zeros.

Case ii(b): Assume that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M242">View MathML</a> in (35); then (35) gives

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M243">View MathML</a>

Now, (1) and (35) give

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M244">View MathML</a>

and so

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M245">View MathML</a>

Therefore,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M246">View MathML</a>

(36)

because if not, w has finitely many zeros, a contradiction. Also,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M247">View MathML</a>

(37)

Put

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M248">View MathML</a>

(38)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M249">View MathML</a> is a rational function.

Then,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M250">View MathML</a>

From (36), we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M251">View MathML</a>

So, G solves (1), and since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M9">View MathML</a> and is a polynomial of degree n, we see that G is a transcendental entire function with finitely many zeros and has order <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M253">View MathML</a>.

Since w and G solve the same equation but w has infinitely many zeros and G has finitely many zeros, w and G are linearly independent, and we can write

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M254">View MathML</a>

where c is a non-zero constant. So,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M255">View MathML</a>

By integrating, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M256">View MathML</a>

(39)

Also, using (31), (38) and (39),

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M257">View MathML</a>

(40)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M258">View MathML</a> is a rational function.

Now, we can assume that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M259">View MathML</a> because if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M260">View MathML</a>, we can multiply w by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M261">View MathML</a>.

We differentiate (40) to get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M262">View MathML</a>

(41)

where

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M263">View MathML</a>

(42)

is a rational function.

So, from (3) and (41), we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M264">View MathML</a>

and so

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M265">View MathML</a>

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M266">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M267">View MathML</a>.

Since v is transcendental and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M268">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M269">View MathML</a> are rational functions, we must have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M270">View MathML</a>

(43)

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M271">View MathML</a>

(44)

Claim 5.1We claim that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M272">View MathML</a>.

To show this, let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M273">View MathML</a>.

From (44), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M274">View MathML</a>

From (42), we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M275">View MathML</a>

We integrate to get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M276">View MathML</a>

where a is a constant. But this contradicts the fact that H and K are rational functions and G is a transcendental function. This completes the proof of Claim 5.1.

Once we have Claim 5.1, (41) gives

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M277">View MathML</a>

and so

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M278">View MathML</a>

(45)

By (45), v has finitely many zeros, so we can write

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M279">View MathML</a>

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M113">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M281">View MathML</a> are polynomials, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M282">View MathML</a>, and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M281">View MathML</a> is non-constant because v is transcendental.

Therefore,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M284">View MathML</a>

Now, we can write this as

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M285">View MathML</a>

(46)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M286">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M287">View MathML</a> are rational functions and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M288">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M289">View MathML</a> are polynomials.

Here, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M290">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M291">View MathML</a> are linearly independent because <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M281">View MathML</a> is non-constant. Now, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M293">View MathML</a>

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M294">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M295">View MathML</a> are rational and satisfy

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M296">View MathML</a>

Therefore, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M297">View MathML</a> because otherwise <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M298">View MathML</a> or <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M299">View MathML</a>. Thus, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M290">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M291">View MathML</a> both solve <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/222/mathml/M302">View MathML</a> and have finitely many zeros, and they are linearly independent.

Hence, P is constant by [9], which contradicts our assumption in Case (II) that P is non-constant. □

Competing interests

The author declares that he has no competing interests.

Acknowledgements

The author would like to thank his supervisor Prof. Jim Langley for his support and guidance. Also, he would like to thank King Abdulaziz University for financial support for his PhD study.

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