Hokkaido Mathematical Journal

OVESEA-TUDOR Horiana, OWA Shigeyoshi,

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

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




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.

Uncontrolled Keywordsunivalent function, L\"{o}wner chain, Nehari criterion, Ozaki criterion