# Applications of Jack's lemma for analytic functions involving $\alpha$-convex functions

Volume 13, Issue 2, pp 113--118
Publication Date: October 19, 2019 Submission Date: March 06, 2018 Revision Date: July 01, 2019 Accteptance Date: July 23, 2019
• 121 Views

### Authors

Syed Zakir Hussain Bukhari - Department of Mathematics, Mirpur University of Science and Technology (MUST), Mirpur-10250 (AJK), Pakistan. Maryam Nazir - Department of Mathematics, Mirpur University of Science and Technology (MUST), Mirpur-10250 (AJK), Pakistan.

### Abstract

In this paper, we introduce two new subclasses of $\alpha$-convex functions and study the applications of Jack's Lemma in the characterization of functions in these classes.

### Keywords

• Analytic functions
• $\alpha$-convex functions
• Jack's Lemma

•  30C45
•  30C80

### References

• [1] H. Al-Amiri, P. T. Mocanu, Some simple criteria of starlikeness and convexity for meromorphic functions, Mathematica, 37 (1995), 11--21

• [2] M. Arif, I. Ahmad, M. Raza, K. Khan, Sufficient condition of a subclass of analytic functions defined by Hadamard product, Life Sci. J., 9 (2012), 2487--2489

• [3] J. Dziok, Applications of the Jack lemma, Acta Math. Hungar., 105 (2004), 93--102

• [4] B. A. Frasin, Some sufficient conditions for certain integral operators, J. Math. Inequal., 2 (2008), 527--535

• [5] A. W. Goodman, Univalent Functions, Vol: I & II, Mariner Publishing Co., Tampa (1983)

• [6] I. S. Jack, Functions starlike and convex of order $\alpha$, J. London Math. Soc. (2), 3 (1971), 469--474

• [7] S. S. Miller, P. T. Mocanu, Second order differential inequalities in the complex plane, Math. Anal. Appl., 65 (1978), 289--305

• [8] P. T. Mocanu, Some starlikeness conditions for analytic functions, Rev. Roumaine Math. Pures Appl., 33 (1988), 117--124

• [9] M. Nunokawa, S. Owa, Y. Polatoglu, M. Çağlar, E. Yavuz Duman, Some sufficient conditions for starlikeness and convexity, Turkish J. Math., 34 (2010), 333--337

• [10] K. Ochiai, S. Owa, M. Acu, Applications of Jack's lemma for certain subclasses of analytic functions, Gen. Math., 13 (2005), 73--82

• [11] J. A. Pfaltzgraff, M. O. Reade, T. Umezawa, Sufficient conditions for univalence, Ann. Fac. Sci. Univ. Nat. Zaïre (Kinshasa) Sect. Math.-Phys., 2 (1976), 211--218

• [12] H. Shiraishi, S. Owa, Starlikeness and convexity for analytic functions concerned with Jack's Lemma, Int. J. Open Probl. Comput. Sci. Math., 2 (2009), 37--47

• [13] R. Singh, S. Singh, Some sufficient conditions for univalence and starlikeness, Colloq. Math., 47 (1982), 309--314