Convexity constant of a domain and applications

In the present paper we introduce a new characterization of the convexity of a planar domain, based on the convexity constant K(D) of a domain D⊂C. We show that in the class of simply connected planar domains, K(D)=1 characterizes the convexity of the domain D, and we derive the value of the convexity constant for some classes of doubly connected domains of the form DΩ=D−Ω‾, for certain choices of the domains D and Ω. Using the convexity constant of a domain, we derive an extension of the well-known Ozaki-Nunokawa-Krzyz univalence criterion for the case of non-convex domains, and we present some examples, which show that our condition is sharp.