Abstract and Applied Analysis

Volume 2011 (2011), Article ID 901235, 10 pages

http://dx.doi.org/10.1155/2011/901235

## A Third-Order Differential Equation and Starlikeness of a Double Integral Operator

^{1}School of Mathematical Sciences, Universiti Sains Malaysia (USM), Penang 11800, Malaysia^{2}Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247 667, India

Received 19 August 2010; Revised 2 January 2011; Accepted 10 January 2011

Academic Editor: Jean Pierre Gossez

Copyright © 2011 Rosihan M. Ali et al. 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

Functions that are analytic in the unit disk and satisfy the differential equation are considered, where is subordinated to a normalized convex univalent function . These functions are given by a double integral operator of the form with subordinated to . The best dominant to all solutions of the differential equation is obtained. Starlikeness properties and various sharp estimates of these solutions are investigated for particular cases of the convex function .

#### 1. Introduction

Let denote the class of all analytic functions defined in the open unit disk and normalized by the conditions , . Further, let be the subclass of consisting of univalent functions, and let be its subclass of starlike functions. A starlike function is characterized analytically by the condition in , that is, the domain is starlike with respect to origin. For two functions and in , the Hadamard product (or convolution) of and is the function defined by

For and in , a function is subordinate to , written as , if there is an analytic function satisfying and , such that , . When is univalent in , then is subordinated to which is equivalent to and .

In a recent paper, Miller and Mocanu [1] investigated starlikeness properties of functions defined by double integral operators of the form

In this paper, conditions on a different kernel are investigated from the perspective of starlikeness. Specifically, we consider functions given by the double integral operator of the form In this case, it follows that

where . Further, the function satisfies a third-order differential equation of the form

for appropriate parameters and . The investigation of such functions can be seen as an extension to the study of the class

The class or its variations for an appropriate function have been investigated in several works; see, for example, [2–10] and more recently [11, 12].

#### 2. Results on Differential Subordination

We first recall the definition of best dominant solution of a differential subordination.

*Definition 2.1 ((dominant and best dominant) [13]). *Let , and let be univalent in *. *If is analytic in and satisfies the differential subordination
then is called a solution of the differential subordination. A univalent function is called a dominant if for all satisfying (2.1). A dominant that satisfies for all dominants of (2.1) is said to be the best dominant of (2.1).

In the following sequel, we will assume that is an analytic convex function in with . For , consider the third-order differential equation We will denote the class consisting of all solutions as , that is,

Let The discriminant is denoted by . Note that and .

We will rewrite the solution of

in its equivalent integral form

It follows from relations (2.4) that

Thus,

Making the substitution in the above integral and integrating again, a change of variables yields

We will use the notation for From [14] it is known that is convex in provided .

Theorem 2.2. *Let and be given by (2.4), and
**
Then the function is convex. If , then
**
and is the best dominant.*

*Proof. *It follows from (2.10) that
Thus,
Since the convolution of two convex functions is convex [15], the function is convex. Let
Then,
It is known from [16] that
Similarly,
implies

The differential chain
shows that . Since , the function
is a solution of the differential subordination , and thus for all dominants . Hence, is the best dominant.

*Remark 2.3. *
(1) When , then and , and the above subordination reduces to the result of [16], that is,

(2) The above proof also reveals that
that is, .

Theorem 2.4. *Let , , and be as given in Theorem 2.2. If , then
*

*Proof. *Let . Then
With given by (2.10), this subordination implies

In this paper, starlikeness properties will be investigated for functions given by a double integral operator of the form (1.3).

#### 3. Applications

First, we consider a class of convex univalent functions so that is symmetric with respect to the real axis. Denote by the class

where , and let . When and , let . The class of is of particular significance, and we will simply denote it by

Equivalently,

The following result is an immediate consequence of Theorems 2.2 and 2.4.

Theorem 3.1. *Under the assumptions of Theorem 2.2, if
**
then
**
where
**
is the best dominant. Further,
**
if , and
**
if .*

#### 4. Starlikeness Property

Starlikeness properties of functions given by a double integral operator are investigated in this section. The following result will be required.

Lemma 4.1 (see [5]). *If satisfies
**
for , then . This result is sharp.*

Theorem 4.2. *Let and be given by (2.4) with and . If
**
where , and satisfies
**
then .*

*Proof. *The function satisfies
and thus
Now, also implies that , and so
It follows from the proof of Theorem 2.2 that
where
Since
an application of Lemma 4.1 yields the result.

Corollary 4.3. *Let and
**
If satisfies
**
then .*

*Proof. *In this case, , , and the result now follows from Theorem 4.2.

*Example 4.4. *If
and satisfies
then .

Theorem 4.5. *Let and let and be given by (2.4) with . If satisfies
**
then .*

*Proof. *Clearly,
Since , substituting and in (3.7) gives
Hence, it follows that

#### Acknowledgments

The authors are greatly thankful to the referee for the many suggestions that helped improve the presentation of this paper. The work presented here was supported in part by a Research University grant from Universiti Sains Malaysia.

#### References

- S. S. Miller and P. T. Mocanu, “Double integral starlike operators,”
*Integral Transforms and Special Functions*, vol. 19, no. 7-8, pp. 591–597, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. M. Ali, “On a subclass of starlike functions,”
*The Rocky Mountain Journal of Mathematics*, vol. 24, no. 2, pp. 447–451, 1994. View at Publisher · View at Google Scholar - R. M. Ali and V. Singh, “Convexity and starlikeness of functions defined by a class of integral operators,”
*Complex Variables. Theory and Application*, vol. 26, no. 4, pp. 299–309, 1995. View at Google Scholar · View at Zentralblatt MATH - P. N. Chichra, “New subclasses of the class of close-to-convex functions,”
*Proceedings of the American Mathematical Society*, vol. 62, no. 1, pp. 37–43, 1977. View at Google Scholar - R. Fournier and S. Ruscheweyh, “On two extremal problems related to univalent functions,”
*The Rocky Mountain Journal of Mathematics*, vol. 24, no. 2, pp. 529–538, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y. C. Kim and H. M. Srivastava, “Some applications of a differential subordination,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 22, no. 3, pp. 649–654, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. Krzyz, “A counter example concerning univalent functions,”
*Folia Societatis Scientiarum Lublinensis*, vol. 2, pp. 57–58, 1962. View at Google Scholar - S. Ponnusamy, “Neighborhoods and Carathéodory functions,”
*Journal of Analysis*, vol. 4, pp. 41–51, 1996. View at Google Scholar · View at Zentralblatt MATH - H. Silverman, “A class of bounded starlike functions,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 17, no. 2, pp. 249–252, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. Singh and S. Singh, “Starlikeness and convexity of certain integrals,”
*Annales Universitatis Mariae Curie-Skłodowska. Sectio A. Mathematica*, vol. 35, pp. 145–148, 1981. View at Google Scholar · View at Zentralblatt MATH - R. Szász, “The sharp version of a criterion for starlikeness related to the operator of Alexander,”
*Annales Polonici Mathematici*, vol. 94, no. 1, pp. 1–14, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - D.-G. Yang and J.-L. Liu, “On a class of analytic functions with missing coefficients,”
*Applied Mathematics and Computation*, vol. 215, no. 9, pp. 3473–3481, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. S. Miller and P. T. Mocanu,
*Differential Subordinations*, vol. 225 of*Monographs and Textbooks in Pure and Applied Mathematics*, Marcel Dekker, New York, NY, USA, 2000. - S. Ruscheweyh, “New criteria for univalent functions,”
*Proceedings of the American Mathematical Society*, vol. 49, pp. 109–115, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. Ruscheweyh and T. Sheil-Small, “Hadamard products of Schlicht functions and the Pólya-Schoenberg conjecture,”
*Commentarii Mathematici Helvetici*, vol. 48, pp. 119–135, 1973. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - D. J. Hallenbeck and S. Ruscheweyh, “Subordination by convex functions,”
*Proceedings of the American Mathematical Society*, vol. 52, pp. 191–195, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH