Some Remarks on Ozaki, Ono and Umezawa’s Results

Recall that in On a general second order derivative, Sci. Rep. Tokyo Kyoiku Daigaku A, 5(124–127)(1956), 111–114, Ozaki, Ono and Umezawa proved a result that if f(z) is analytic and satisfies |f″(z)|<1 in the unit disc D=z:z<1, then |f′(z)−1|<1 and so, f(z) is univalent in D, because Re{f′(z)}>0 in D implies univalence by the Noshiro–Warschawski Theorem. In this paper, we obtain another sufficient condition for univalence of f(z) by applying a hypothesis for modulus of arg{f″(z)}.