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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
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