On a subclass of the generalized Janowski type functions of complex order

Abstract

In this paper, we introduce the class $\mathcal {JR}^{\lambda}_{b}\left(\alpha,\beta, \delta, A,B\right)$ of generalized Janowski type functions of complex order defined by using the Ruscheweyh derivative operator in the open unit disc $ \mathbb D=\left \{z\in \mathbb C: \left \vert z\right \vert <1\right \}$. The bound for the n-th coefficient and subordination relation are obtained for the functions belonging to this class. Some consequences of our main theorems are same as the results obtained in the earlier studies.

Keywords

• Analytic function
• Subordination
• $ \lambda$-spirallike function
• $ \lambda$-Robertson function
• $ \lambda$-close-to-spirallike function
• $ \lambda$-close-to-Robertson function
• Ruscheweyh derivative operator

References

1. [1] R.M. Goel and B.C. Mehrok, A subclass of univalent functions, Houston J. Math. 8, 343-357, 1982.
2. [2] A.W. Goodman, On close-to-convex functions of higher order, Ann. Univ. Sci. Bu- dapest Eötvös Sect. Math. 15, 17–30, 1972.
3. [3] A.W. Goodman, Univalent Functions, Vol II. Somerset, NJ, USA Mariner, 1983.
4. [4] M.M. Haidan and F.M. Al-Oboudi, Spirallike functions of complex order, J. Natural Geom. 19, 53–72, 2000.
5. [5] W. Janowski, Some extremal problems for certain families of analytic functions, Ann. Polon. Math. 28, 297–326, 1973.
6. [6] W. Kaplan , Close-to-convex schlicht functions, Michigan Math. J. 1, 169–185, 1952.
7. [7] Ö.Ö. Kılıç, Coefficient Inequalities for Janowski type close-to-convex functions asso- ciated with Ruscheweyh Derivative Operator, Sakarya Uni. J. Sci. 23 (5), 714-717, 2019.
8. [8] Y. Polatoğlu, M. Bolcal, A. Şen and E. Yavuz, A study on the generalization of Janowski functions in the unit disc, Acta Math. Aca. Paed. 22, 27–31, 2006.

9. [9] M.O. Reade, On close-to-convex univalent functions, Michigan Math. J. 3, 59–62, 1955.
10. [10] S. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math Soc. 49 (1), 109–115, 1975.
11. [11] L. Špaček, Prispevek k teorii funcki prostych, Casopis Pest. Mat. a Fys. 62, 12-19, 1932.