The Hahn-Banach type theorems for generalized convex functions

In the present paper we consider $$\omega $$ -convex functions which generalized many well-known and intensively investigated classes of functions. Among other results, we prove the Jensen inequality for $$\omega $$ -convex functions, we examine operations preserving the $$\omega $$ -convexity and give several sufficient conditions which guarantee the continuity of such functions. The main result of the paper is a counterpart of a celebrated Hahn-Banach theorem for $$\omega $$ -convex functions. Several consequences are also contained in the paper.