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Aspects of univalent holomorphic functions involving multiplier transformation and Ruscheweyh derivative
Advances in Difference Equations volume 2014, Article number: 117 (2014)
Abstract
Making use multiplier transformation and Ruscheweyh derivative,we introduce a new class of analytic functions defined on the open unit disc, and investigate its various characteristics. Further we obtain distortion bounds, extreme points and radii of close-to-convexity, starlikeness and convexity and neighborhood property for functions belonging to the class .
MSC:30C45, 30A20, 34A40.
1 Introduction
Let denote the class of functions of the form , which are analytic and univalent in the open unit disc . is a subclass of consisting of the functions of the form . For functions given by , , we define the Hadamard product (or convolution) of f and g by , .
Definition 1.1 (Ruscheweyh [1])
For , , the operator is defined by ,
Remark 1.1 If , , then , .
If , , then , .
For , , , the operator is defined by the following infinite series:
Remark 1.2 [4]
It follows from the above definition that
Remark 1.3 The operator is the generalized Sălăgean operator introduced by Al-Oboudi [5], and is the Sălăgean differential operator [6].
Let , . Denote by the operator given by ,
Remark 1.4 If , , then
If , , then
Remark 1.5 The operator which was introduced in [9] and the operator which was introduced in [10].
Following the work of Najafzadeh and Pezeshki [11] we can define the class as follows.
Definition 1.4 For , and , let be the subclass of consisting of functions that satisfying the inequality
where
.
Remark 1.6 If , , then
Remark 1.7 The class defined and studied in [12] and defined and studied in [13].
2 Coefficient bounds
In this section we obtain coefficient bounds and extreme points for functions in .
Theorem 2.1 Let the function . Then if and only if
The result is sharp for the function defined by
Proof Suppose f satisfies (2.1). Then for , we have
Hence, by using the maximum modulus Theorem and (1.1), . Conversely, assume that
Since for all , we have
By choosing choose values of z on the real axis so that is real and letting through real values, we obtain the desired inequality (2.1). □
Corollary 2.2 If is in , then
with equality only for functions of the form .
Theorem 2.3 Let and
for , , and . Then is in the class if and only if it can be expressed in the form
where and .
Proof Suppose can be written as in (2.5). Then
Now,
Thus .
Conversely, let . Then by using (2.3), setting
and , we have . This completes the proof of Theorem 2.3. □
3 Distortion bounds
In this section we obtain distortion bounds for the class .
Theorem 3.1 If , then
holds if the sequence is non-decreasing, and
holds if the sequence is non-decreasing, where
The bounds in (3.1) and (3.2) are sharp, for given by
Proof In view of Theorem 2.1, we have
We obtain
Thus
Hence (3.1) follows from (3.5). Further,
Hence (3.2) follows from
□
4 Radius of starlikeness and convexity
The radii of close-to-convexity, starlikeness, and convexity for the class are given in this section.
Theorem 4.1 Let the function belong to the class , Then is close-to-convex of order δ, , in the disc , where
The result is sharp, with extremal function given by (2.3).
Proof For given we must show that
By a simple calculation we have
The last expression is less than if
We use the fact that if and only if
Equation (4.2) holds true if
Or, equivalently,
which completes the proof. □
Theorem 4.2 Let . Then
-
1.
f is starlike of order δ, , in the disc , where
-
2.
f is convex of order δ, , in the disc where,
Each of these results is sharp for the extremal function given by (2.5).
Proof 1. For we need to show that
We have
The last expression is less than if
We use the fact that if and only if
Equation (4.3) holds true if
Or, equivalently,
which yields the starlikeness of the family.
-
2.
Using the fact that f is convex if and only is starlike, we can prove (2) with a similar way of the proof of (1). The function f is convex if and only if
(4.4)
We have
We use the fact that if and only if
Equation (4.4) holds true if
or, equivalently,
which yields the convexity of the family. □
5 Neighborhood property
In this section we study neighborhood property for functions in the class .
Definition 5.1 For functions f belong to of the form and , we define -neighborhood of f by
where η is a fixed positive integer.
By using the following lemmas we will investigate the -neighborhood of function in .
Lemma 5.1 Let , if satisfies
then .
Proof By using of Theorem 2.1, it is sufficient to show that
But
Therefore it is enough to prove that
the result follows because the last inequality holds for all . □
Lemma 5.2 Let , , , and . If , then
where either or . The result is sharp with the extremal function
Proof Letting we have
In view of Theorem 2.3, where , ,
and
So we obtain
Since and , it follows that
Since whenever or we conclude that
is a decreasing function of j, the result will follow. The proof is complete. □
Theorem 5.1 Let or and suppose and
and , then the -neighborhood of f is the subset of , where
The result is sharp.
Proof For , let be in . So by Lemma 5.2, we have
By using Lemma 5.1, if
that is, , and the proof is complete. □
Author’s contributions
The author drafted the manuscript, read and approved the final manuscript.
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Alb Lupaş, A. Aspects of univalent holomorphic functions involving multiplier transformation and Ruscheweyh derivative. Adv Differ Equ 2014, 117 (2014). https://doi.org/10.1186/1687-1847-2014-117
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DOI: https://doi.org/10.1186/1687-1847-2014-117