Convolution of functions with free normalization (Q2725237)

Let \(\mathcal H\) denote the class of functions which are regular in the unit disc \(\Delta =\{z: |z|<1\}\). For two functions \(f(z)=\sum _{n=0}^{\infty}a_nz^n\), \(g(z)=\sum _{n=0}^{\infty}b_nz^n\) in \(\mathcal H\), the convolution \(f*g\) is defined by \((f*g)(z)=\sum _{n=0}^{\infty}a_nb_nz^n\). Suppose that \(f(\Delta)\subset D_1\) and \(g(\Delta)\subset D_2\), where \(D_1\) and \(D_2\) are the given circular (disc, halfplane) or angularly domains. In this paper the authors determine \(D_3\) so that \((f*g)(\Delta)\subset D_3\). The authors motivate the paper with earlier known results.