Sufficient conditions for univalence and starlikeness

Document Type : Research Paper


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

2 Department of Mathematics‎, ‎Rzeszow University of Technology‎, ‎Al‎. ‎Powstancow Warszawy 12‎, ‎35-959 Rzeszow‎, ‎Poland.

3 Department of Applied Mathematics‎, ‎College of Natural Sciences‎, ‎Pukyong National University‎, ‎Busan 608-737‎, ‎Korea.


It is known that the condition $mathfrak {Re} left{zf'(z)/f(z)right}>0$, $|z|<1$ is a sufficient condition for $f$, $f(0)=f'(0)-1$ to be starlike in $|z|<1$. The purpose of this
work is to present some new sufficient conditions for univalence and starlikeness.


Main Subjects

S. N. Kudryashov, On some criteria of schlichtness of analytic functions (Russian), Mat. Zametki 13 (1973) 359--366.
S. S. Miller and P. T. Mocanu, Differential subordinations and univalent functions, Michigan Math. J. 32 (1985) 185--195.
R. Nevanlinna, Uber die konforme Abbildund Sterngebieten, Oversikt av Finska-Vetenskaps Societen Forhandlingar, 63 (A) (1921), no. 6, 48--403.
M. Nunokawa, On properties of non-Caratheodory functions, Proc. Japan Acad. Ser. A 68 (1992), no. 6, 152--153.
M. Nunokawa, On the order of strongly starlikeness of strongly convex functions, Proc. Japan Acad. Ser. A Math. Sci. 69 (1993), no. 7, 234--237.
S. Ozaki, On the theory of multivalent functions II, Sci. Rep. Tokyo Bunrika Daigaku A 4 (1941) 45--87.
J. A. Pfaltzgraff, M. O. Reade and T. Umezawa, Sufficient Conditions for Univalence, Ann. Fac. Sci. de Kinshasa, Zare; Sec. Math. Phys. 2 (1976), no. 2, 94--100.
R. Singh and R. Singh, Some sufficient conditions for univalence and starlikeness, Colloq. Math. 47 (1982) no. 2, 309--314.
J. Sokol, On sufficient condition to be in a certain subclass of starlike functions defined by subordination, Appl. Math. Comput. 190 (2007), no. 1, 237--241.
J. Sokol, Starlikeness of Hadamard product of certain analytic functions, Appl. Math. Comp. 190 (2007), no. 2, 1157--1160.
J. Sokol, On a condition for alpha-starlikeness, J. Math. Anal. Appl. 352 (2009), 696--701.
J. Sokol, M. Nunokawa, On some sufficient condition for starlikeness, J. Inequal. Appl. 2012, 2012:282.
E. Study, Konforme Abbildung Einfachzusammenhangender Bereiche, B. C. Teubner, Leipzig und Berlin, 1913.
T. Umezawa, On the theory of univalent functions, Tohoku Math. J. (2) 7 (1955) 212--228.