On some first-order differential subordination

Nunokawa, M.; Duman, E. Yavuz; Sokoł, J.; Cho, N.E.; Owae, S.

On some first-order differential subordination

Document Type : Original Article

Authors

¹ University of Gunma, Hoshikuki-cho 798-8, Chuou-Ward, Chiba 260-0808, Japan

² Department of Mathematics and Computer Science, I_stanbul Ku¨ltu¨r University, I_stanbul, Turkey

³ Department of Mathematics, Rzeszo´w University of Technology, Al. Powstan´co´w Warszawy 12, 35-959 Rzeszo´w, Poland

⁴ Department of Applied Mathematics, College of Natural Sciences, Pukyong National University, Pusan 608-737, Republic of Korea

⁵ e Department of Mathematics, Kinki University, Higashi-Osaka, Osaka 577-8502, Japan

Abstract

Let A denote the class of functions f that are analytic in the unit disc D and normalized
by f(0) = f 0(0) 1=0. In this paper, we investigate the class of functions such that
Reff 0ðzÞ þ zf 00ðzÞ bg > a in D. We determine conditions for a and b under which the function
f is univalent, close-to-convex, and convex. To obtain this, we first estimate OEArg{f 0(z)}OE which
improves the earlier results.

Keywords