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

Dragomir, S. S.

JIPAM, J. Inequal. Pure Appl. Math. 3, No. 3, Paper No. 35, 8 p. (2002).

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.

Reviewer: Constantin Niculescu (Craiova)

MSC:

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

Keywords:

Hermite-Hadamard integral inequality; convex functions; semi-inner products; Hermite-Hadamard inequality