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The Schur geometrical convexity of integral arithmetic mean. (English) Zbl 1134.26306
Summary: Suppose that $$f:[a,b]\subseteq (0,\infty)\to (0,\infty)$$ is a second order differentiable function, and
$G(x,y)=\begin{cases} \frac{1}{y-x}\int^y_x f(t)\,dt,\quad & x,y\in [a,b],\;x\neq y,\\ f(x), & x=y\in [a,b].\end{cases}$
If $$3f'(x)+xf''(x)\geq 0$$ (or $$\leq 0$$, resp.) for all $$x\in [a,b]$$, then $$G(x,y)$$ is a Schur geometrical convex (or concave, resp.) function on $$[a,b]$$.

##### MSC:
 26A51 Convexity of real functions in one variable, generalizations