A class of univalent functions with varying arguments (Q1196607)

The authors give various characterizations for two subclasses \(V_ n(A,B)\) and \(K_ n(A,B)\) of normalized univalent functions together with sharp growth and distortion theorems and coefficient inequalities for these families. The classes are subsets of the family \(V\) of univalent functions with varying arguments introduced by \textit{H. Silverman} [Houston J. Math. 7, 283-287 (1981; Zbl 0472.30018)]. Precisely, let \(D^ n f(z)\) be the Hadamard product \([z/(1-z)^{n+1}]*f(z)\); this quantity is called the \(n\)th Ruscheweyh derivative of \(f\). For a nonnegative integer \(n\) and \(-1\leq A<B\leq 1\), let \(V_ n(A,B)\) denote the set of \(f\in V\) such that \(D^{n+1}f(z)/D^ n f(z)\) is subordinate to \((1+Az)/(1+Bz)\). \(K_ n(A,B)\) is the class of \(f\in V\) such that \(zf'(z)\in V_ n(A,B)\).