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A note on Hadamard’s inequality. (English) Zbl 0880.26019

Let \(f\) be a real-valued convex function on \([a,b]\) and let \[ F(t)= {1\over 2(b-a)} \int^b_a \Biggl(f\Biggl({1+ t\over 2} a+ {1-t\over 2} x\Biggr)+ f\Biggl({1+ t\over 2} b+{1- t\over 2} x\Biggr)\Biggr)dx,\quad t\in [0,1]. \] The authors prove that \(F\) is convex and increasing on \([0,1]\), and \[ {1\over b-a} \int^b_a f(x)dx\leq F(t)\leq {1\over 2} (f(a)+ f(b)). \]

MSC:

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