On Vector Hermite-Hadamard Differences Controlled by Their Scalar Counterparts

Abstract

We present a new, in a sense direct, proof that the system of two functional inequalities

$$\biggl \Vert F \biggl(\frac{x+y}{2} \biggr) - \frac{1}{y-x} \int_{x}^{y} F(t)\,dt \biggr \Vert \leq\frac{1}{y-x} \int_{x}^{y} f(t)\,dt - f \biggl(\frac{x+y}{2} \biggr) $$

and

$$\biggl \Vert \frac{F(x) + F(y)}{2} - \frac{1}{y-x} \int_{x}^{y} F(t)\,dt \biggr \Vert \leq\frac{f(x) + f(y)}{2} -\frac{1}{y-x} \int_{x}^{y} f(t)\,dt $$

is satisfied for functions F and f mapping an open interval I of the real line ℝ into a Banach space and into ℝ, respectively, if and only if F yields a delta-convex mapping with a control function f.

A similar result is obtained for delta-convexity of higher orders with detailed proofs given in the case of delta-convexity of the second order, i.e. when the functional inequality

holds true provided that x,y∈I, x≤y.