On Ozaki's condition for p-valency

作者：Nunokawa Mamoru^*; Sokol Janusz; Thomas Derek K

来源：Comptes Rendus Mathematique, 2018, 356(4): 382-386.

DOI：10.1016/j.crma.2018.02.007

Let f be an analytic function in a convex domain D subset of C. A well-known theorem of Ozaki states that if f is analytic in D, and is given by f(z) = z(p) + Sigma(infinity)(n=p+1) a(n)z(n) for z is an element of D, and
Re{e(i alpha) f (p)(z)} > 0, ( z is an element of D),
for some real alpha, then f is at most p-valent in D. Ozaki's condition is a generalization of the well-known Noshiro-Warschawski univalence condition. The purpose of this paper is to provide some related sufficient conditions for functions analytic in the unit disk D ={z is an element of C : vertical bar z vertical bar < 1} to be p-valent in D, and to give an improvement to Ozaki's sufficient condition for p-valence when z is an element of D.

• 出版日期2018-4