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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
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