Sufficient Conditions and Applications for Carathéodory Functions

Abstract EN

Let P be the class of functions of the form p(z)=1+∑n=1∞pnzn which are analytic in U.

By using the method of differential subordinations, we give some sufficient conditions for Carathéodory function, that is, if p(z)∈P satisfies p(z)≠0 (0<|z|<1) and p(z)m+λ(zp'(z)/(p(z))μ)≺h(z), where h(z)=((1+az)/(1+bz))m+λ(a-b)z/(1+bz)2-μ(1+az)μ, -1≤b0, m, μ∈{0,1,2}, m+μ≤2, then p(z)≺(1+az)/(1+bz).

Some useful consequences of this result are also given.