On Fully-Convex harmonic functions and their extension

Abstract

‎Uniformly convex univalent functions that introduced by Goodman‎, ‎maps every circular arc contained in the open unit disk with center in it into a convex curve‎. ‎On the other hand‎, ‎a fully-convex harmonic function‎, ‎maps each subdisk $|z|=r<1$ onto a convex curve‎. ‎Here we synthesis these two ideas and introduce a family of univalent harmonic functions which are fully-convex and uniformly convex also‎. ‎In the following we will mention some examples of this subclass and obtain a necessary and sufficient conditions and finally a coefficient condition will attain with convolution‎.