Open Access Research

New subclasses of analytic functions

Basem Aref Frasin

Author Affiliations

Faculty of Science, Department of Mathematics, Al al-Bayt University, P.O. Box 130095, Mafraq, Jordan

Journal of Inequalities and Applications 2012, 2012:24  doi:10.1186/1029-242X-2012-24


The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2012/1/24


Received:13 July 2011
Accepted:9 February 2012
Published:9 February 2012

© 2012 Frasin; 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 analytic functions f (z) in the open unit disk <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M1">View MathML</a>, subclasses <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M2">View MathML</a>, and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M3">View MathML</a> are introduced. The object of the present article is to discuss some interesting properties of functions f (z) associated with classes <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M2">View MathML</a>, and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M3">View MathML</a>.

Mathematics Subject Classification (2010): 30C45.

Keywords:
analytic; univalent functions; Cauchy-Schwarz inequality

1. Introduction and Definitions

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M4">View MathML</a> denotes the class of the normalized functions of the form

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M5">View MathML</a>

(1.1)

which are analytic in the open unit disk <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M6">View MathML</a>. Also, a function f (z) belonging to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M4">View MathML</a> is said to be convex of order α if it satisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M7">View MathML</a>

(1.2)

for some α(0 ≤ α < 1). We denote by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M8">View MathML</a> the subclass of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M4">View MathML</a> consisting of functions which are convex of order α in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M1">View MathML</a> (see, [1,2]). Further, a function f (z) belonging to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M4">View MathML</a> is said to be in the class <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M9">View MathML</a> iff

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M10">View MathML</a>

(1.3)

for some α(0 ≤ α < 1).

For analytic functions f (z), Uyanik and Owa [3], obtained some interesting properties for analytic functions in the subclass <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M11">View MathML</a> defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M12">View MathML</a>

associated with close-to-convex functions and starlike functions of order α.

In this article, we define the following subclass of analytic functions.

Definition 1.1. A function f (z) belonging to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M4">View MathML</a> is said to be in the class <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M13">View MathML</a>, if it satisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M14">View MathML</a>

(1.4)

for some complex numbers β1, β2, β3, and for some real λ > 0.

Example 1.2. Let us consider the function fγ (z), γ ∈ ℝ, given by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M15">View MathML</a>

Then, we observe that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M16">View MathML</a>

where

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M17">View MathML</a>

Therefore, if γ = 1, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M18">View MathML</a>

This implies that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M19">View MathML</a> for λ ≥ 2 |β1|. If γ = 2, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M20">View MathML</a>

Therefore, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M21">View MathML</a> for λ ≥ 10 |β1| + 6 |β2|. Further, if γ = 3; then we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M22">View MathML</a>

Thus, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M23">View MathML</a> for λ ≥ 36 |β1| + 42 |β2| + 24 |β3|.

Now, let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M24">View MathML</a> denotes the subclass of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M4">View MathML</a> consisting of functions f (z) with

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M25">View MathML</a>

Also, we introduce the subclasses <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M26">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M27">View MathML</a> of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M24">View MathML</a> as follows:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M28">View MathML</a>

2. Properties of the class <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M13">View MathML</a>

We first prove

Theorem 2.1. If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M29">View MathML</a>satisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M30">View MathML</a>

(2.1)

for some complex numbers β1, β2, β3 and for some real λ > 0, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M31">View MathML</a>.

Proof. We observe that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M32">View MathML</a>

Therefore, if f (z) satisfies the inequality (2.1), then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M31">View MathML</a>.

Next, we prove

Theorem 2.2. if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M31">View MathML</a>with arg β1 = arg β2 = arg β3 = ϕ and an = |an|ei((n-1)θ-ϕ)(n = 2, 3,...), then we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M33">View MathML</a>

Proof. For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M31">View MathML</a>, we see that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M34">View MathML</a>

for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M35">View MathML</a>. Let us consider a point <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M35">View MathML</a> such that z = |z| e-.

Then we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M36">View MathML</a>

Letting |z| → 1-, we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M37">View MathML</a>

Corollary 2.3. If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M31">View MathML</a>with arg β1 = arg β2 = arg β3 = ϕ and an = |an| ei((n-1)θ-ϕ) (n = 2, 3,...), then we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M38">View MathML</a>

Example 2.4. Let us consider the function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M31">View MathML</a>with arg β1 = arg β2 = arg β3 = ϕ and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M39">View MathML</a>

Then, we see that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M40">View MathML</a>

Corollary 2.5. If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M31">View MathML</a>with arg β1 = arg β2 = arg β3 = ϕ and an = |an| ei((n-1)θ-ϕ) (n = 2, 3,...), then we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M41">View MathML</a>

with

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M42">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M43">View MathML</a>

with

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M44">View MathML</a>

Proof. In view of Theorem 2.1, we know that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M45">View MathML</a>

Further, we note that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M46">View MathML</a>

which is equivalent to

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M47">View MathML</a>

Thus, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M48">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M49">View MathML</a>

Next, we observe that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M50">View MathML</a>

which implies that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M51">View MathML</a>

Therefore, we obtain that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M52">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M53">View MathML</a>

3. Radius problem for the class <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M26">View MathML</a>

To obtain the radius problem for the class <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M26">View MathML</a>, we need the following lemma.

Lemma 3.1. If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M54">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M55">View MathML</a>

(3.1)

Proof. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M54">View MathML</a>. Then, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M56">View MathML</a>

for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M35">View MathML</a>. Let us consider a point <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M35">View MathML</a> such that z = |z| e-.

Then we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M57">View MathML</a>

Letting |z| → 1-, we obtain the inequality (3.1).

Corollary 3.2. If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M54">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M58">View MathML</a>

Remark 3.3. By Lemma 3.1, we observe that if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M54">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M59">View MathML</a>

Applying Theorem 2.1 and Lemma 3.1, we derive

Theorem 3.4. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M54">View MathML</a>, and δ ∈ ℂ (0 < |δ| < 1). Then the function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M60">View MathML</a>for (0 < |δ| ≤ |δ0(λ)|, where |δ0(λ)| is the smallest positive root of the equation

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M61">View MathML</a>

(3.2)

in 0 < |δ| < 1.

Proof. For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M54">View MathML</a>, we see that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M62">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M63">View MathML</a>

Thus, to show that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M60">View MathML</a>, from Theorem 2.1, it is sufficient to prove that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M64">View MathML</a>

Applying Cauchy-Schwarz inequality, we note that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M65">View MathML</a>

(3.3)

We note that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M66">View MathML</a>

thus, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M67">View MathML</a>

(3.4)

Since

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M68">View MathML</a>

we see that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M69">View MathML</a>

and thus, we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M70">View MathML</a>

(3.5)

Furthermore, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M71">View MathML</a>

but

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M72">View MathML</a>

thus, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M73">View MathML</a>

which yields

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M74">View MathML</a>

(3.6)

Therefore, from (3.3)-(3.6) with |δ|2 = x, we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M75">View MathML</a>

Now, let us consider the complex number δ (0 < |δ| < 1) such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M76">View MathML</a>

If we define the function h(|δ|) by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M77">View MathML</a>

then we have h(0) = -λ < 0 and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M78">View MathML</a>. This means that there exists some δ0 such that h(|δ0|) = 0 (0 < |δ0| < 1). This completes the proof of the theorem.

4. Radius problem for the class <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M3">View MathML</a>

For the class <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M3">View MathML</a>, we prove the following lemma.

Lemma 4.1. If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M79">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M80">View MathML</a>

(4.1)

Proof. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M79">View MathML</a>. Then, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M81">View MathML</a>

for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M35">View MathML</a>. Let us consider a point <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M35">View MathML</a>. such that z = |z|e-.

Then we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M82">View MathML</a>

Letting |z| → 1-, we obtain the inequality (4.1).

Corollary 4.2. If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M79">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M83">View MathML</a>

Remark 4.3. If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M79">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M84">View MathML</a>

Applying Theorem 2.1, Lemma 4.1 and using the same technique as in the proof of Theorem 3.4, we derive

Theorem 4.4. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M79">View MathML</a>, and δ ∈ ℂ (0 < |δ| < 1). Then the function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M60">View MathML</a>for (0 < |δ| ≤ |δ0(λ)|, where |δ0(λ)| is the smallest positive root of the equation

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/24/mathml/M85">View MathML</a>

(4.2)

in 0 < |δ| < 1.

Competing interests

The author declares that they have no competing interests.

References

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  2. Goodman, AW: Univalent Functions. Mariner, Tampa (1983)

  3. Uyanik, N, Owa, S: New extensions for classes of analytic functions associated with close-to-convex and starlike of order α. Math Comput Model. 54, 359–366 (2011). Publisher Full Text OpenURL