×

zbMATH — the first resource for mathematics

Hermite-Hadamard type inequalities for increasing radiant functions. (English) Zbl 1055.26021
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 \[ f(\sqrt{(b- a)^2+ b^2}- (b- a))\leq {1\over b-a} \int^b_a f(x)\,dx. \]

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
PDF BibTeX XML Cite
Full Text: EuDML