Complex convexity and fixed point theorems in Orlicz modular spaces

In this paper, we prove that every Orlicz modular function space $L_{\Phi ,\rho }$ is complex midpoint locally uniformly convex. As a corollary, $L_{\Phi ,\rho }$ is also complex strictly convex. Furthermore, we introduce the notions of mean nonexpansive mappings in the modular sense and prove a fixed point theorem in Orlicz modular function spaces.