Conic regions and k-uniform convexity

Let Ωk⊂C denote a domain, such that 1∈Ωk and ∂Ωk is a conic section, with eccentricity equal to 1/k. In this paper authors introduce the class of k-uniformly convex functions k-UCV, with the property that the values of the expression 1+zf″(z)/f′(z) lie inside the domain Ωk. Necessary and sufficient conditions for membership in k-UCV, as well as sharp growth and distortion theorems for k-uniformly convex functions are given. The obtained results generalize the concept of uniform convexity due to A.W. Goodman (Ann. Polon. Math. 56 (1991) 87–92).