# Hokkaido Mathematical Journal

## Hokkaido Mathematical Journal, 34 (2005) pp.533-539

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### 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 {\rm (Bull. Amer. Math. Soc.\,{\bf 55}(1949))} and S.Ozaki and M.Nunokawa {\rm (Proc. Amer. Math. Sco.\,{\bf 33}(1972))}. Likewise we find other new sufficient conditions.

MSC(Primary) 30C45 univalent function, L\"{o}wner chain, Nehari criterion, Ozaki criterion