Journal of Mathematics

Volume 2013 (2013), Article ID 923167, 4 pages

http://dx.doi.org/10.1155/2013/923167

Research Article

On a Class of Analytic Functions Defined by an Integral Operator

Pravati Sahoo¹ and Saumya Singh²

¹Department of Mathematics, Banaras Hindu University, Banaras 221 005, India

²Department of Mathematics, O.P. Jindal Institute of Technology, Raigarh 496001, India

Received 18 November 2012; Accepted 16 January 2013

Academic Editor: Nan-Jing Huang

Copyright © 2013 Pravati Sahoo and Saumya Singh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We define a new subclass by using an integral operator . We find a coefficient inequality and using that we derive many sharp results. These results generalize many results which are existing in the literature.

1. Introduction and Preliminaries

Let denote the class of functions of the form

which are analytic in the unit disc . For and for , , the integral operator is defined by

where is the Pochhammer symbol given by

For , , is defined by Komatu in [1, 2]. Here,

The operator is the Bernardi operator [3, 4]. In fact the operator is related rather closely to the Beta or Euler transformation. Moreover, for , the operator was used by Owa and Srivastava [5–8].

For , , we define a class of all analytic functions involving the integral operator, , by

The aim of this paper is to study the class and find the similar type of results proved by Frasin in [9], where the author has defined similar type of class involving the operator

Also in [10], the authors studied the similar type class involving the well-known Salagean operator.

2. Definitions and Lemmas

Definition 1. Let be analytic and univalent in . If is analytic in , , and , then we say that the function is subordinate to in , and we write .

Definition 2 (subordinating factor sequence). A sequence of complex numbers is called a subordinating sequence if, whenever is analytic, univalent, and convex in , we have the subordination given by

Lemma 3 (see [11]). A sequence is a subordinating factor sequence if and only if

Lemma 4. If

where , and

then .

Proof. It is sufficient to show that

Now, we have

The above expansion is bounded by if

hence the proof follows from (10).

Let denote the class of functions whose coefficients satisfy the condition (10). So .

3. Main Results

By using the technique used earlier by Attiya [12] and Singh [13], we state and prove the following theorem.

Theorem 5. Let the function defined by (1) be in the class , where . Also let denote the class of functions which are convex and univalent in . Then,

The constant is the best estimate.

Proof. and let Then,

Thus by the Definition 2 and (14) will hold if the sequence

is a subordinating factor sequence, with , in view of Lemma 3, this will be the case if and only if

Now

Case I (). From (19), we obtain

Since is an increasing function of , so

Case II (). From (19), we obtain

Since is decreasing function of , so

Thus, (18) holds true in . This prove the inequality (14). The inequality (15) follows by taking the convex function in (14). To prove the sharpness of the constant, we suppose that the function given by

from (14), we have

After a simple calculation, we get

Corollary 6. Let the function defined by (1) be in the class and satisfy the condition

then

The constant is the best estimate.

Corollary 7. Let the function defined by (1) be in the class and satisfy the condition

then

The constant is the best estimate.

Corollary 8. Let the function defined by (1) be in the class and satisfy the condition

then

The constant is the best estimate.

References

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Y. Komatu, “On a one-parameter additive family of operators defined on analytic functions regular in the unit disk,” in Analytic Functions, J. Lawrynowicz, Ed., pp. 292–300, Springer, New York, NY, USA, 1980.

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