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International Journal of Mathematics and Mathematical Sciences
Volume 2008 (2008), Article ID 159029, 11 pages
http://dx.doi.org/10.1155/2008/159029
Research Article

Starlike and Convex Properties for Hypergeometric Functions

1Department of Mathematics, Kyungsung University, Busan 608-736, South Korea
2Department of Applied Mathematics, Pukyong National University, Busan 608-737, South Korea

Received 13 February 2008; Accepted 18 June 2008

Academic Editor: Linda Sons

Copyright © 2008 Oh Sang Kwon and Nak Eun Cho. 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

The purpose of the present paper is to give some characterizations for a (Gaussian) hypergeometric function to be in various subclasses of starlike and convex functions. We also consider an integral operator related to the hypergeometric function.

1. Introduction

Let be the class consisting of functions of the form that are analytic and univalent in the open unit disk . Let and denote the subclasses of consisting of starlike and convex functions of order , respectively [1].

Recently, Bharati et al. [2] introduced the following subclasses of starlike and convex functions.

Definition 1.1. A function of the form (1.1) is in if it satisfies the condition and if and only if .

Definition 1.2. A function of the form (1.1) is in if it satisfies the condition and if and only if .

Bharati et al. [2] showed that , and . In particular, we note that is the class of uniformly convex functions given by Goodman [3] (also see [46]).

Let be the (Gaussian) hypergeometric function defined by where , and is the Pochhammer symbol defined by We note that converges for and is related to the Gamma function by Silverman [7] gave necessary and sufficient conditions for to be in and , and also examined a linear operator acting on hypergeometric functions. For the other interesting developments for in connection with various subclasses of univalent functions, the readers can refer to the works of Carlson and Shaffer [8], Merkes and Scott [9], and Ruscheweyh and Singh [10].

In the present paper, we determine necessary and sufficient conditions for to be in and . Furthermore, we consider an integral operator related to the hypergeometric function.

2. Results

To establish our main results, we need the following lemmas due to Bharati et al. [2].

Lemma 2.1. (i) A function of the form (1.1) is in if and only if it satisfies
(ii) A function of the form (1.1) is in if and only if it satisfies

Lemma 2.2. (i) A function of the form (1.1) is in if and only if it satisfies
(ii) A function of the form (1.1) is in if and only if it satisfies

Theorem 2.3. (i) If and , then is in if and only if
(ii) If and , then is in if and only if

Proof. (i) Since according to (i) of Lemma 2.1, we must show that Noting that and then applying (1.6), we have Hence, (2.8) is equivalent to Thus, (2.10) is valid if and only if or, equivalently, .
(ii) Since by (i) of Lemma 2.1, we need only to show that Now, But this last expression is bounded above by if and only if (2.6) holds.

Theorem 2.4. (i) If and , then is in if and only if
(ii) If and , then is in if and only if

Proof. (i) Since has the form (2.7), we see from (ii) of Lemma 2.1 that our conclusion is equivalent to Writing , we see that This last expression is bounded above by if and only if which is equivalent to (2.14).
(ii) In view of (ii) of Lemma 2.1, we need only to show that Now, Writing , we have Substituting (2.21) into the right-hand side of (2.20), we obtain Since , we write (2.22) as By simplification, we see that the last expression is bounded above by if and only if (2.15) holds.

Theorem 2.5. (i) If and , then is in if and only if
(ii) If and , then is in if and only if

Proof. (i) Since according to (i) of Lemma 2.2, we must show that Noting that and then applying (1.6), we have Hence, (2.27) is equivalent to Thus, (2.29) is valid if and only if or, equivalently, .
(ii) Since by (i) of Lemma 2.2, we need only to show that Now, But this last expression is bounded above by if and only if (2.25) holds.

Theorem 2.6. (i) If and , then is in if and only if
(ii) If and , then is in if and only if

Proof. (i) Since has the form (2.26), we see from (ii) of Lemma 2.2 that our conclusion is equivalent to Writing , we see that This last expression is bounded above by if and only if , which is equivalent to (2.33).
(ii) In view of (ii) of Lemma 2.2, we need only to show that Now, Substituting (2.21) into the right-hand side of (2.38), we obtain Since , we may write (2.39) as By simplification, we see that the last expression is bounded above by if and only if (2.34) holds.

3. An Integral Operator

In the next theorems, we obtain similar-type results in connection with a particular integral operator acting on as follows:

Theorem 3.1. Let and . Then,
(i) defined by (3.1) is in if and only if (ii) defined by (3.1) is in if and only if

Proof. (i) Since by (i) of Lemma 2.1, we need only to show that Now, which is equivalent to (3.2).
(ii) According to (i) of Lemma 2.2, it is sufficient to show that Now, which is equivalent to (3.3).
Now, we observe that if and only if . Thus, any result of functions belonging to the class about leads to that of functions belonging to the class . Hence, we obtain the following analogues to Theorems 2.3 and 2.5.

Theorem 3.2. Let and . Then,
(i) defined by (3.1) is in if and only if (ii) defined by (3.1) is in if and only if

References

  1. H. Silverman, “Univalent functions with negative coefficients,” Proceedings of the American Mathematical Society, vol. 51, no. 1, pp. 109–116, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. R. Bharati, R. Parvatham, and A. Swaminathan, “On subclasses of uniformly convex functions and corresponding class of starlike functions,” Tamkang Journal of Mathematics, vol. 28, no. 1, pp. 17–32, 1997. View at Zentralblatt MATH · View at MathSciNet
  3. A. W. Goodman, “On uniformly convex functions,” Annales Polonici Mathematici, vol. 56, no. 1, pp. 87–92, 1991. View at Zentralblatt MATH · View at MathSciNet
  4. N. E. Cho, S. Y. Woo, and S. Owa, “Uniform convexity properties for hypergeometric functions,” Fractional Calculus & Applied Analysis, vol. 5, no. 3, pp. 303–313, 2002. View at Zentralblatt MATH · View at MathSciNet
  5. W. C. Ma and D. Minda, “Uniformly convex functions,” Annales Polonici Mathematici, vol. 57, no. 2, pp. 165–175, 1992. View at Zentralblatt MATH · View at MathSciNet
  6. F. Rønning, “Uniformly convex functions and a corresponding class of starlike functions,” Proceedings of the American Mathematical Society, vol. 118, no. 1, pp. 189–196, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. H. Silverman, “Starlike and convexity properties for hypergeometric functions,” Journal of Mathematical Analysis and Applications, vol. 172, no. 2, pp. 574–581, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. B. C. Carlson and D. B. Shaffer, “Starlike and prestarlike hypergeometric functions,” SIAM Journal on Mathematical Analysis, vol. 15, no. 4, pp. 737–745, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. E. P. Merkes and W. T. Scott, “Starlike hypergeometric functions,” Proceedings of the American Mathematical Society, vol. 12, no. 6, pp. 885–888, 1961. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. St. Ruscheweyh and V. Singh, “On the order of starlikeness of hypergeometric functions,” Journal of Mathematical Analysis and Applications, vol. 113, no. 1, pp. 1–11, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet