The aim of this paper is to prove the theorem which generates many examples of functions belonging to a geometrically defined class of uniformly starlike functions introduced by Goodman in 1991. Only a very few explicit uniformly starlike functions were known until now. Next we obtain inclusion relations between some subclasses of convex functions and the class of uniformly starlike functions.

We investigate classes of k-uniformly convex functions which generalize the class UCV {\mathrm{UCV}} introduced by Goodman in 1991. We solve the problem of finding the largest α ≥ 0 {\alpha\geq 0} such that the class of k-uniformly convex functions is contained in the class of starlike functions of order α.

Let $Y$ be a closed 2-dimensional disk or a 2-sphere. We consider a simple, $d$-sheeted branched covering $\pi :X\to Y$. We fix a base point $A_0$ in $Y$ ($A_0\in \partial Y$ if $Y$ is a disk). We consider the homeomorphisms $h$ of $Y$ which fix $\partial

We examine the differential subordination related to the geometric and harmonic means. In particular, the discussed differential subordination generalizes the well-known Briot–Bouquet differential subordination. The main results are applicable to construct nontrivial subclasses of the class of standardly normalized holomorphic functions in the unit disk. In addition, a new type of ordinary differential equation is proposed for study.