An improvement of Ozaki's P-valent conditions

The old result due to [Ozaki, S.: On the theory of multivalent functions II. Sci. Rep. Tokyo Bunrika Daigaku Sect. A, 45.87 (1941)], says that if f(z)=zp+..n=p+1anzn anzn is analytic in a convex domain D and for some real . we have Re{exp(i.)f(p)(z) >} 0 in D, then f(z) is at most p-valent in D. In this paper, we consider similar problems in the unit disc D = {z . C: |z| < 1}.