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A mapping in connection to Hadamard’s inequalities. (English) Zbl 0747.26015

The following generalization of the well-known Hadamard inequalities for convex functions is given: Let \(f:[a,b]\to R\) be a convex function on the interval \([a,b]\) from \(R\). Let the function \(H:[0,1]\to R\) be defined by \[ H(t):={1\over b-a}\int^ b_ af\left(tx+(1-t){a+b\over 2}\right)dx. \] Then (i) \(H\) is convex on \([0,1]\), (ii) we have \(\inf_{t\in[0,1]}H(t)=H(0)=f\left({a+b\over 2}\right)\), \(\sup_{t\in [0,1]}H(t)=H(1)={1\over b-a}\int^ b_ af(x)dx\), (iii) \(H\) is monotonously increasing on \([0,1]\).

MSC:

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