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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
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