Extensions and refinements of the Hermite-Hadamard inequality for the triangle

Abstract

The paper deals with the Hermite-Hadamard inequality for convex functions on the triangle. A key role has the mid-triangle whose vertices are midpoints of the given triangle sides. The Hermite-Hadamard inequality is extended to convex combinations of all mid-triangle points except vertices. Discrete and integral refinements of this extension are also considered.