Research Article | Open Access
An Application of Differential Subordination
We apply the general theory of differential subordination to obtain certian interesting criteria for -valent starlikeness and strong starlikeness. Some applications of these results are also discussed.
Let be the class of functions of the form which are analytic in the open unit disk .
Let be the class of functions of the form which are analytic in . If satisfies , then we say that is a Carathéodory function.
With a view to recalling the principle of subordination between analytic functions, let the functions and be analytic in . Then we say that the function is subordinate to if there exists a Schwarz function , analytic in with such that We denote this subordination by In particular, if the function is univalent in , the above subordination is equivalent to
For and , a function is said to be in the class if it satisfies Also, we write , the class of strongly starlike -valent functions of order in . , the class of Janowski starlike -valent function, , the class of -valent starlike function, and , the class of -valent starlike function of order .
For Carathéodory functions, Miller  obtained certain sufficient conditions applying the differential inequalities. Recently, Nunokawa et al.  have given some improvement of result by Miller . Recently Ravichandran and Jayamala  studied some subordination results for Carathéodory functions. In this paper by extending the result of Ravichandran and Jayamala , we find sufficient conditions for the subordination to hold for given and criteria for -valent starlikeness. Our results include results obtained by Nunokawa et al. . We also give some criteria for -valently starlikeness and strong starlikeness.
To prove our result we need the following lemma due to Miller and Mocanu .
Lemma 1.1 (see [4, Theorem 3.4h, page 132]). Let be analytic and univalent in the unit disk and and let be analytic in a domain containing with when . Set Suppose that(i) is starlike univalent in ,(ii) for .If is analytic in with, , , and then and is the best dominant.
2. Application of Differential Subordination
By making use of Lemma 1.1, we first prove the following theorem.
Theorem 2.1. Let and be a positive real number. Let be convex univalent in and . If satisfies where then and is the best dominant of (2.1).
Then clearly and are analytic in and . Also let
Since is convex univalent, is starlike univalent. Therefore is starlike univalent in , and
From (2.1)–(2.6) we see that
Therefore, by applying Lemma 1.1, we conclude that and is the best dominant of (2.1). The proof of the theorem is complete.
By taking as real and in Theorem 2.1, we get the following corollary.
Corollary 2.2. Let , , , be real number such that and . If satisfies where then and is the best dominant of (2.8).
Corollary 2.3. Let , . If satisfies in and where then
Corollary 2.4. Let and then
Theorem 2.5. Let and . Let be convex univalent in and satisfy If satisfies then and is the best dominant of (2.19)
Proof. By setting and it can be easily observed that and are analytic in and that .
Also, by letting
we find that is starlike univalent in and that The differential subordination becomes Now, the result follows as an application of Lemma 1.1.
Theorem 2.6. Let , and be complex numbers, . Let be univalent in and satisfy the following conditions for : (1)let be starlike,(2).If satisfies then and is the best dominant.
Proof. The proof of this theorem is much akin to the proof of Theorem 2.5 and hence can be omitted.
Corollary 2.9. Let , and . If satisfies in , then implies Also, implies
Proof. By taking , , , , and in Theorem 2.6, we get the first part.
Proof of the second part follows, by setting , , , and .
For , , , and , in Theorem 2.5, we have the following result.
By taking in Corollary 2.10, we obtain the following.
Corollary 2.11. If satisfies , , and then Region has been shown shaded in Figure 1.
Letting , , , and in Theorem 2.5, we get the following.
Corollary 2.12. If satisfies , , and where for some , then
For the univalent function given by (2.38), One now finds the image of the unit disk .
Let , where and are real. One has Elimination of yields Therefore, one concludes that which properly contains the half plane .
Corollary 2.13. Let and . If satisfies in and where then
Setting and in Corollary 2.13, we get the following corollay.
Corollary 2.14. Let and . If satisfies in and where then
Remark 2.15. For the function given by (2.48), we have which properly contains the half plane , where
Corollary 2.16. If and then
Corollary 2.18. Let and be convex univalent in with and satisfy (2.18).
Let and If then and is the best dominant.
Proof. By taking in Theorem 2.5, we have the above corollary.
Corollary 2.19. Let and be convex univalent in with and satisfy (i)If satisfies then (ii)If satisfies then and is the best dominant.
Corollary 2.20. Let , then for a positive integer , one has that implies and is the best dominant.
Corollary 2.22. Let and then and is the best dominant.
We remark here that is univalent if and only if .
Corollary 2.24. If satisfies then and is the best dominant.
Proof. By taking and , and in Theorem 2.5, we get the previous corollary.
- S. S. Miller, “Differential inequalities and Carathéodory functions,” Bulletin of the American Mathematical Society, vol. 81, pp. 79–81, 1975.
- M. Nunokawa, S. Owa, N. Takahashi, and H. Saitoh, “Sufficient conditions for Carathéodory functions,” Indian Journal of Pure and Applied Mathematics, vol. 33, no. 9, pp. 1385–1390, 2002.
- V. Ravichandran and M. Jayamala, “On sufficient conditions for Carathéodory functions,” Far East Journal of Mathematical Sciences, vol. 12, no. 2, pp. 191–201, 2004.
- S. S. Miller and P. T. Mocanu, Differential Subordinations: Theory and Applications, vol. 225 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2000.
- K. S. Padmanabhan, “On sufficient conditions for starlikeness,” Indian Journal of Pure and Applied Mathematics, vol. 32, no. 4, pp. 543–550, 2001.
- S. Singh and S. Gupta, “A differential subordination and starlikeness of analytic functions,” Applied Mathematics Letters, vol. 19, no. 7, pp. 618–627, 2006.
- N. Tuneski, “On certain sufficient conditions for starlikeness,” International Journal of Mathematics and Mathematical Sciences, vol. 23, no. 8, pp. 521–527, 2000.
- V. Singh, “Remarks on a paper by Silverman,” International Journal of Mathematics and Mathematical Sciences, vol. 27, no. 2, pp. 65–68, 2001.
- H. Silverman, “Convex and starlike criteria,” International Journal of Mathematics and Mathematical Sciences, vol. 22, no. 1, pp. 75–79, 1999.
- S. Ponnusamy, “On Bazilevic functions,” Annales Universitatis Mariae Curie-Skłodowska. Sectio A, vol. 42, no. 14, pp. 115–127, 1988.
- S. Ponnusamy and O. P. Juneja, “Some applications of first order differential subordinations,” Glasnik Matematički. Serija III, vol. 25, no. 45, pp. 287–296, 1990.
- S. Owa and M. Obradović, “An application of differential subordinations and some criteria for univalency,” Bulletin of the Australian Mathematical Society, vol. 41, no. 3, pp. 487–494, 1990.
- H. M. Srivastava and A. Y. Lashin, “Some applications of the Briot-Bouquet differential subordination,” Journal of Inequalities in Pure and Applied Mathematics, vol. 6, no. 2, pp. 1–7, 2005.
Copyright © 2009 T. N. Shanmugam and M. P. Jeyaraman. 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.