Opuscula Math. 31, no. 1 (2011), 15-26
On strongly midconvex functions
José Luis Sánchez
Abstract. In this paper we collect some properties of strongly midconvex functions. First, counterparts of the classical theorems of Bernstein-Doetsch, Ostrowski and Sierpinski are presented. A version of Rodé support theorem for strongly midconvex functions and a Kuhn-type result on the relation between strongly midconvex functions and strongly \(t\)-convex functions are obtained. Finally, a connection between strong midconvexity and generalized convexity in the sense of Beckenbach is established.
Keywords: strongly convex functions, strongly midconvex functions, Bernstein-Doetsch-type theorem, Kuhn theorem, Rodé support theorem, Beckenbach convexity.
Mathematics Subject Classification: 26B25, 39B62.