Two mappings in connection to Hadamard’s inequalities. (English) Zbl 0758.26014

The author studies certain convex mappings which are connected to a given convex function \(f:[a,b]\to\mathbb{R}\). For example, let \[ H(t)={1\over b- a}\int^ b_ af(tx+(1-t)(a+b)/2)dx. \] Then \(H\) is convex on \([0,1]\); \(\inf H(t)=H(0)\), \(\sup H(t)=H(1)\) and \(H\) increases monotonically on \([0,1]\). This contains a refinement of the Jensen-Hadamard inequality \(H(0)\leq H(1)\leq(f(a)+f(b))/2\).


26D15 Inequalities for sums, series and integrals
26A51 Convexity of real functions in one variable, generalizations
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