An extension of the univalence criteria of Nehari and Ozaki and Nunokawa

Abstract

Abstract. In this paper, we obtain a sufficient condition for the univalence of analyticfunctions in the open unit disk U. This condition involves two arbitrary functions g ( z )and h ( z ) analytic in U. Replacing g ( z ) and h ( z ) by some particular functions, we findthe well-known conditions for univalency established by Z. Nehari ( Bull. Amer. Math.Soc. 55 (1949)) and S. Ozaki and M. Nunokawa ( Proc. Amer. Math. Soc. 33 (1972)).Likewise we find other new sufficient conditions. Key words : univalent function, L¨owner chain, Nehari criterion, Ozaki criterion. 1. IntroductionWe denote by U r = {z ∈ C: |z| < r} the disk of z -plane, where r ∈ (0 , 1] , U 1 = Uand I = [0 , ∞ ). Let A be the class of functions f ( z ) whichare analytic in Uwith the normalizations f (0) = 0 and f 0 (0) = 1. Inthe present paper, we consider the following conditions for univalency offunctions f ( z ) belonging to the class A .Theorem 1.1 ([1]) Let f ( z ) ∈ A. If, for all z ∈ U , f ( z ) satisfies flfl {f ; z}