## A refinement of Hadamard’s inequality for isotonic linear functionals.(English)Zbl 0799.26016

Let $$E$$ be a nonempty set and let $$L$$ be a linear class of real valued functions. Further, let $$A: L\to\mathbb{R}$$ be an isotonic linear functional with $$A(\text{\textbf{1}})= 1$$. Let $$f: C (\subseteq X)\to \mathbb{R}$$ be a convex function on a convex subset $$C$$ of a real linear space $$X$$. If $$h: E\to\mathbb{R}$$, $$0\leq h(t)\leq 1$$ $$(t\in E)$$, $$h\in L$$ is so that $$f(hx+ (1- h)y)$$, $$f((1- h)x+ hy)$$ belong to $$L$$ for $$x$$, $$y$$ fixed in $$C$$, then the following inequalities are proved: \begin{aligned} f\left({x+ y\over 2}\right) & \leq{1\over 2} \bigl[f(A(h)x+ (1- A(h))y)+ f((1- A(h))x+ A(h)y)\bigr]\\ & \leq {1\over 2} \bigl(A[f(hx+ (1- h)y)]+ A[f((1- h)x+ hy)]\bigr)\\ & \leq {f(x)+ f(y)\over 2}.\end{aligned} {}.

### MSC:

 26D15 Inequalities for sums, series and integrals 26A51 Convexity of real functions in one variable, generalizations 39B72 Systems of functional equations and inequalities