Applications of Ruscheweyh derivatives and Hadamard product to analytic functions

Abstract

For given analytic functions ϕ(z)=z+∑m=2∞ λm zm,ψ(z)=z+∑m=2∞ μm zm in U={z||z|<1} with λm≥0,μm≥0 and λm≥μm, let En(ϕ,ψ;A,B) be the class of analytic functions f(z)=z+∑m=2∞am zm in U such that (f*Ψ)(z)≠0 and Dn+1(f*ϕ)(z)Dn(f*Ψ)(z)≪1+Az1+Bz,      −1≤A<B≤1,  z∈U, where Dnh(z)=z(zn−1h(z))(n)/n!,   n∈N0={0,1,2,…} is the nth Ruscheweyh derivative; ≪ and * denote subordination and the Hadamard product, respectively. Let T be the class of analytic functions in U of the form f(z)=z−∑m=2∞am zm,  am≥0, and let En[ϕ,ψ;A,B]=En(ϕ,ψ;A,B)∩T. Coefficient estimates, extreme points, distortion theorems and radius of starlikeness and convexity are determined for functions in the class En[ϕ,ψ;A,B]. We also consider the quasi-Hadamard product of functions in En[z/(1−z),z/(1−z);A,B] and En[z/(1−z)2,z/(1−z)2;A,B].