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An inequality improving the second Hermite-Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products. (English) Zbl 0995.26009

Let \(f:[a,b]\to \mathbb{R}\) be a continuous convex function. The main result is the following (partial) refinement of the Hermite-Hadamard inequality: \[ \begin{aligned} 0 &\leq {1\over 8}\Biggl[f_+'\Biggl({a+ b\over 2}\Biggr)- f_-'\Biggl({a+ b\over 2}\Biggr)\Biggr](b- a)\\ &\leq {f(a)+ f(b)\over 2}- {1\over b-a} \int^b_a f(t) dt\\ &\leq{1\over 8} [f_-'(b)- f_+'(a)](b- a).\end{aligned} \] The constant \(1/8\) is sharp. Several applications are included.

MSC:

26D15 Inequalities for sums, series and integrals
26D10 Inequalities involving derivatives and differential and integral operators
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