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Reciprocal classes of p-valently spirallike and p-valently Robertson functions

Neslihan Uyanik1, Hitoshi Shiraishi2, Shigeyoshi Owa2* and Yasar Polatoglu3

Author Affiliations

1 Department of Mathematics, Kazim Karabekir Faculty of Education, Ataturk University, Erzuram 25240, Turkey

2 Department of Mathematics, Kinki University, Higashi-Osaka, 577-8502 Osaka, Japan

3 Department of Mathematics and Computer Sciences, Faculty of Science and Letters, Istanbul Kultur University, Istanbul, Turkey

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Journal of Inequalities and Applications 2011, 2011:61  doi:10.1186/1029-242X-2011-61

The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2011/1/61


Received:10 April 2011
Accepted:18 September 2011
Published:18 September 2011

© 2011 Uyanik et al; 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

For p-valently spirallike and p-valently Robertson functions in the open unit disk <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M1">View MathML</a>, reciprocal classes <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M2">View MathML</a>, and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M3">View MathML</a> are introduced. The object of the present paper is to discuss some interesting properties for functions f(z) belonging to the classes <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M2">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M3">View MathML</a>.

2010 Mathematics Subject Classification

Primary 30C45

Keywords:
Reciprocal class; Subordination; Schwarz function; Robertson function; Miller and Mocanu lemma

1 Introduction

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M4">View MathML</a> be the class of functions f(z) of the form

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M5">View MathML</a>

(1.1)

which are analytic in the open unit disk <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M6">View MathML</a>.

For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M7">View MathML</a>, we say that f(z) belongs to the class <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M2">View MathML</a> if it satisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M8">View MathML</a>

(1.2)

for some real <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M9">View MathML</a> and β (β > p cos α).

When α = 0, the class <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M10">View MathML</a> was studied by Polatoglu et al. [1], and the classes <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M11">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M12">View MathML</a> were introduced by Owa and Nishiwaki [2].

Further, let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M3">View MathML</a> denote the subclass of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M4">View MathML</a> consisting of functions f(z), which satisfy

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M13">View MathML</a>

(1.3)

for some real <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M9">View MathML</a> and β (β > p cos α).

We note that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M14">View MathML</a> if and only if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M15">View MathML</a>, and that, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M16">View MathML</a> if and only if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M17">View MathML</a>.

Remark 1 If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M18">View MathML</a> satisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M19">View MathML</a>

then we say that f(z) is p-valently spirallike in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M1">View MathML</a> (cf. [1]). Also, if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M7">View MathML</a> satisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M20">View MathML</a>

then f(z) is said to be p-valently Robertson function in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M1">View MathML</a> (cf. [3,4]). Therefore, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M2">View MathML</a> defined by (1.2) is the reciprocal class of p-valently spirallike functions in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M1">View MathML</a>, and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M3">View MathML</a> defined by (1.3) is the reciprocal class of p-valently Robertson functions in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M1">View MathML</a>.

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M21">View MathML</a> be the class of functions p(z) of the form

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M22">View MathML</a>

(1.4)

that are analytic in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M1">View MathML</a> and satisfy <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M23">View MathML</a>. A function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M24">View MathML</a> is called the Carathéodory function and satisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M25">View MathML</a>

(1.5)

with the equality for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M26">View MathML</a> (cf. [5]).

For analytic functions g(z) and h(z) in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M1">View MathML</a>, we say that g(z) is subordinate to h(z) if there exists an analytic function w(z) in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M1">View MathML</a> with w(0) = 0 and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M27">View MathML</a>, and such that g(z) = h(w(z)). We denote this subordination by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M28">View MathML</a>

(1.6)

If h(z) is univalent in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M1">View MathML</a>, then this subordination (1.6) is equivalent to g(0) = h(0) and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M29">View MathML</a> (cf. [5]).

2 Subordinations for classes

We consider subordination properties of function f(z) in the classes <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M2">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M3">View MathML</a>.

Theorem 1 A function f(z) belongs to the class <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M2">View MathML</a>if and only if

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M30">View MathML</a>

(2.1)

for some real <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M9">View MathML</a>and β (β > p cos α).

The result is sharp for f(z) given by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M31">View MathML</a>

(2.2)

Proof. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M16">View MathML</a>. If we define the function w(z) by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M32">View MathML</a>

(2.3)

then we know that w(z) is analytic in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M1">View MathML</a>, w(0) = 0, and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M33">View MathML</a>

(2.4)

Therefore, we have that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M27">View MathML</a>. If follows from (2.3) that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M34">View MathML</a>

(2.5)

which is equivalent to the subordination (2.1).

Conversely, we suppose that the subordination (2.1) holds true. Then, we have that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M35">View MathML</a>

(2.6)

for some Shwarz function w(z), which is analytic in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M1">View MathML</a>, w(0) = 0, and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M27">View MathML</a>. It is easy to see that the equality (2.6) is equivalent to the equality (2.3). Since

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M36">View MathML</a>

(2.7)

we conclude that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M37">View MathML</a>

(2.8)

which shows that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M16">View MathML</a>.

Finally, we consider the function f(z) given by (2.2). Then, f(z) satisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M38">View MathML</a>

(2.9)

This completes the proof of the theorem.   □

Noting that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M14">View MathML</a> if and only if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M39">View MathML</a>, we also have

Corollary 1 A function f(z) belongs to the class <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M3">View MathML</a>if and only if

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M40">View MathML</a>

(2.10)

for some real <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M9">View MathML</a>and β (β > p cos α).

The result is sharp for f(z) given by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M41">View MathML</a>

(2.11)

3 Coefficient inequalities

Applying the properties for Carathéodory functions, we discuss the coefficient inequalities for f(z) in the classes <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M2">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M3">View MathML</a>.

Theorem 2 If f(z) belongs to the class <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M2">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M42">View MathML</a>

(3.1)

The result is sharp for

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M43">View MathML</a>

(3.2)

for α = 0.

Proof. In view of Theorem 1, we can consider the function w(z) given by (2.3) for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M16">View MathML</a>. Since w(z) is the Schwarz function, the function q(z) defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M44">View MathML</a>

(3.3)

is the Carathéodory function. If we write that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M45">View MathML</a>

(3.4)

then we see that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M46">View MathML</a>

and the equality holds true for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M47">View MathML</a> and its rotation. It is to be noted that the equation (3.3) is equivalent to

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M48">View MathML</a>

(3.5)

This gives us that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M49">View MathML</a>

(3.6)

which implies that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M50">View MathML</a>

(3.7)

It follows from (3.7) that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M51">View MathML</a>

(3.8)

If n = p + 1, then we have that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M52">View MathML</a>

(3.9)

If n = p + 2, then we also have that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M53">View MathML</a>

(3.10)

Thus, the coefficient inequality (3.1) is true for n = p + 1 and n = p + 2. Next, we suppose that (3.1) holds true for n = p + 1, p + 2, p + 3, ..., p + k - 1. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M54">View MathML</a>

(3.11)

This means that the inequality (3.1) holds true for n = p + k. Therefore, by the mathematical induction, we prove the coefficient inequality (3.1).

Finally, let us consider the function f(z) given by (3.2). Then, f(z) can be written by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M55">View MathML</a>

(3.12)

Thus, this function f(z) satisfies the equality in (3.1).   □

Corollary 2 If f(z) belongs to the class <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M3">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M56">View MathML</a>

(3.13)

The result is sharp for f(z) defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M57">View MathML</a>

(3.14)

for α = 0.

Remark 2 We know that the extremal functions for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M16">View MathML</a> is f(z) given by (2.2) and for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M14">View MathML</a> is f(z) given by (2.11). But, we see that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M58">View MathML</a>

(3.15)

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M59">View MathML</a>

(3.16)

for such functions.

Therefore, the extremal functions for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M16">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M14">View MathML</a> do not satisfy the equalities in (3.1) and (3.13), respectively.

Furthermore, if we consider α = 0 in Theorem 2, then we obtain the corresponding result due to Polatoglu et al. [1].

4 Inequalities for the real parts

We discuss some problems of inequalities for the real parts of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M60">View MathML</a>.

Theorem 3 If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M16">View MathML</a>, then we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M61">View MathML</a>

(4.1)

for | z | = r < 1. The equalities hold true for f(z) given by (2.2).

Proof. By virtue of Theorem 1, we consider the function g(z) defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M62">View MathML</a>

(4.2)

Letting z = re(0 ≦ r < 1), we see that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M63">View MathML</a>

(4.3)

Let us define

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M64">View MathML</a>

(4.4)

Then, we know that h'(t) ≧ 0. This implies that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M65">View MathML</a>

(4.5)

which is equivalent to

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M66">View MathML</a>

(4.6)

Noting that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M67">View MathML</a> by Theorem 1 and g(z) is univalent in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M1">View MathML</a>, we prove the inequality (4.1). Since the subordination (2.1) is sharp for f(z) given by (2.2), we say that the equalities in (4.1) are attained by the function f(z) given by (2.2).   □

Taking α = 0 in Theorem 3, we have

Corollary 3 If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M68">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M69">View MathML</a>

(4.7)

for | z | = r < 1. The equalities hold true for

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M70">View MathML</a>

(4.8)

Corollary 4 If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M14">View MathML</a>, then we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M71">View MathML</a>

(4.9)

for | z | = r < 1. The equalities hold true for f(z) defined by (2.11).

Corollary 5 If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M72">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M73">View MathML</a>

(4.10)

for | z | = r < 1. The equalities hold true for f(z) defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M74">View MathML</a>

(4.11)

5 Sufficient conditions

We consider some sufficient conditions for f(z) to be in the classes <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M75">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M76">View MathML</a>.

To discuss our sufficient conditions, we have to recall here the following lemma by Miller and Mocanu [6] (also due to Jack [7]).

Lemma 1 Let w(z) be analytic in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M1">View MathML</a>with w(0) = 0. If there exists a point <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M77">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M78">View MathML</a>

(5.1)

then we can write

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M79">View MathML</a>

(5.2)

where k is real and k ≧ 1.

Applying Lemma 1, we derive

Theorem 4 If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M80">View MathML</a> satisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M81">View MathML</a>

(5.3)

for some real β > p, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M68">View MathML</a>.

Proof. Let us define the function w(z) by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M82">View MathML</a>

(5.4)

Then we see that w(z) is analytic in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M1">View MathML</a> and w(0) = 0.

It follows from (5.4) that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M83">View MathML</a>

(5.5)

We suppose that there exists a point <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M84">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M85">View MathML</a>

Then, Lemma 1 gives us that w(z0) = eand z0w'(z0) = ke. For such a point z0, we have that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M86">View MathML</a>

(5.6)

This contradicts our condition (5.3). Therefore, there is no <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M84">View MathML</a> such that |w (z0) | = 1. This implies that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M27">View MathML</a>, that is, that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M87">View MathML</a>

(5.7)

Thus, we observe that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M68">View MathML</a>.   □

Further, we derive

Theorem 5 If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M68">View MathML</a>for some real <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M88">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M89">View MathML</a>

(5.8)

Proof. We consider the function w(z) such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M90">View MathML</a>

(5.9)

for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M91">View MathML</a> and for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M68">View MathML</a>.

Then, we know that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M92">View MathML</a>

(5.10)

for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M93">View MathML</a>.

Since w(z) is analytic in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M1">View MathML</a> and w(0) = 0, we suppose that there exists a point <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M84">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M94">View MathML</a>

Then, applying Lemma 1, we can write that w(z0) = eand z0w'(z0) = ke(k ≧ 1). This gives us that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M95">View MathML</a>

(5.11)

which contradicts the inequality (5.10). Thus, there is no point <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M84">View MathML</a> such that |w (z0) | = 1. This means that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M27">View MathML</a>, and that,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M96">View MathML</a>

This completes the proof of the theorem.   □

Letting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M97">View MathML</a> instead of f(z) in Theorem 5, we have

Corollary 6 If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M14">View MathML</a>for some <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M98">View MathML</a>, Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M99">View MathML</a>

(5.12)

Finally, we consider the coefficient estimates for functions f(z) to be in the classes <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M2">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M3">View MathML</a>.

Theorem 6 If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M18">View MathML</a>satisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M100">View MathML</a>

(5.13)

for some real <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M9">View MathML</a>, β (β > p cos α), and k (0 ≦ k p cos α), then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M16">View MathML</a>

Proof. It is to be noted that if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M18">View MathML</a> satisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M101">View MathML</a>

(5.14)

Then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M16">View MathML</a>. It follows that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M102">View MathML</a>

Therefore, if f(z) satisfies the coefficient estimate (5.13), then we know that f(z) satisfies the inequality (5.14). This completes the proof of the theorem.   □

Letting α = 0 and k = p in Theorem 6, we have

Corollary 7 If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M18">View MathML</a> satisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M103">View MathML</a>

for some real <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M104">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M68">View MathML</a>.

Further, we have

Theorem 7 If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M18">View MathML</a>satisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M105">View MathML</a>

for some real <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M9">View MathML</a>, β (β > p cos α) and k (0 ≦ k p cos α), then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M14">View MathML</a>

Corollary 8 If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M18">View MathML</a>satisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M106">View MathML</a>

for some real <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M107">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/61/mathml/M14">View MathML</a>.

Acknowledgements

This paper was completed when the first author was visiting Department of Mathematics, Kinki University, Higashi-Osaka, Osaka 577-8502, Japan, between February 17 and February 26, 2011.

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