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 \[ f(\sqrt{(b- a)^2+ b^2}- (b- a))\leq {1\over b-a} \int^b_a f(x)\,dx. \]

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
Full Text: EuDML