Hermite-Hadamard type inequalities for increasing radiant functions. (English) Zbl 1055.26021

Sharikov, E. V.

JIPAM, J. Inequal. Pure Appl. Math. 4, No. 2, Paper No. 47, 13 p. (2003).

The author calls “radiant function” a subhomogeneous one, i.e., satisfying \(f(ax)\leq af(x)\) for any \(a\in (0,1)\), and \(x\in\mathbb{R}^n_+\). Then, assuming also that if \(f> 0\) is increasing (i.e., \(x\geq y\) implies \(f(x)\geq f(y)\), where \(x\geq y\) means the coordinate-wise order relation), he proves Hadamard-type integral inequalities on \(\mathbb{R}^n_+\). We quote the following result in \(\mathbb{R}_+\): If \(f> 0\) is increasing and subhomogeneous on \([a,b]\), then \[ f(\sqrt{(b- a)^2+ b^2}- (b- a))\leq {1\over b-a} \int^b_a f(x)\,dx. \]

Reviewer: József Sándor (Cluj-Napoca)

MSC:

26D15	Inequalities for sums, series and integrals
26A51	Convexity of real functions in one variable, generalizations
26B25	Convexity of real functions of several variables, generalizations

Keywords:

integral inequalities; convex functions; subhomogeneous functions; Hadamard type inequality