On some functional equations connected to Hadamard inequalities

Summary.

We consider some functional equations connected with the Hadamard inequalities

$$f\left(\frac{x+y}{2}\right) \leq \frac{1}{y-x}\int^y_x f(t)dt \leq \frac{f(x) + f(y)}{2}$$

. Namely, we observe that for the function f(x) = x ² the middle expression from these inequalities satisfies the additional condition

$$\frac{f(x) + f(y)}{2} - \frac{1}{y-x} \int^y_x f(t)dt = 2 \left(\frac{1}{y-x}\int^y_x f(t)dt - f\left(\frac{x+y}{2}\right)\right)$$

. We ask for all functions having properties of this kind. Moreover, we consider some generalizations of the problem for functions acting on more general structures than ${\mathbb{R}}$ (integral domains).