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

Abstract

In this paper, we obtain a sufficient condition for the univalence of analytic functions in the open unit disk $\mathbb{U}$. This condition involves two arbitrary functions $ g(z) $ and $ h(z) $ analytic in $\mathbb{U} $. Replacing $g(z)$ and $h(z)$ by some particular functions, we find the well-known conditions for univalency established by Z.Nehari (Bull. Amer. Math. Soc. 55(1949)) and S.Ozaki and M.Nunokawa (Proc. Amer. Math. Sco., 33(1972))}. Likewise we find other new sufficient conditions.