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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 \(H(0)\leq H(1)\leq(f(a)+f(b))/2\).

MSC:

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

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[2] {\scS. S. Dragomir, J. E. Pečarić, and J. Sándor}, A note on the Jensen-Hadamard inequality, Anal. Numér. Théor. Approx., in press.
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[5] Sándor, J, Some integral inequalities, Elem. math., 43, 177-180, (1988) · Zbl 0702.26016
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