Journal of Mathematics

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

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

## On a Class of Analytic Functions Defined by an Integral Operator

^{1}Department of Mathematics, Banaras Hindu University, Banaras 221 005, India^{2}Department 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

- Y. Komatu, “On a one-parameter additive family of operators defined on analytic functions regular in the unit disk,”
*Bulletin of the Faculty of Science and Engineering*, vol. 22, pp. 1–22, 1979. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Google Scholar - S. D. Bernardi, “Convex and starlike univalent functions,”
*Transactions of the American Mathematical Society*, vol. 135, pp. 429–446, 1969. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. D. Bernardi, “The radius of univalence of certain analytic functions,”
*Proceedings of the American Mathematical Society*, vol. 24, pp. 312–318, 1970. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Owa and H. M. Srivastava, “Some applications of the generalized Libera integral operator,”
*Proceedings of the Japan Academy A*, vol. 62, no. 4, pp. 125–128, 1986. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. M. Srivastava and S. Owa, “New characterizations of certain starlike and convex generalized hypergeometric functions,”
*Journal of National Academy of Mathematics*, vol. 3, pp. 198–202, 1985. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. M. Srivastava and S. Owa, “A certain one-parameter additive family of operators defined on analytic functions,”
*Journal of Mathematical Analysis and Applications*, vol. 118, no. 1, pp. 80–87, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. M. Srivastava and S. Owa, “Some characterization and distortion theorems involving fractional calculus, generalized hypergeometric functions, Hadamard products, linear operators, and certain subclasses of analytic functions,”
*Nagoya Mathematical Journal*, vol. 106, pp. 1–28, 1987. View at Google Scholar · View at MathSciNet - B. A. Frasin, “Subordination results for a class of analytic functions defined by a linear operator,”
*Journal of Inequalities in Pure and Applied Mathematics*, vol. 7, no. 4, article 134, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. M. Srivastava and S. S. Eker, “Some applications of a subordination theorem for a class of analytic functions,”
*Applied Mathematics Letters*, vol. 21, no. 4, pp. 394–399, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. S. Wilf, “Subordinating factor sequences for convex maps of the unit circle,”
*Proceedings of the American Mathematical Society*, vol. 12, pp. 689–693, 1961. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. A. Attiya, “On some applications of a subordination theorem,”
*Journal of Mathematical Analysis and Applications*, vol. 311, no. 2, pp. 489–494, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Singh, “A subordination theorem for spirallike functions,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 24, no. 7, pp. 433–435, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet