Multiplicative concavity of the integral of multiplicatively concave functions (Q608009)

A real-valued function \(f:I\subseteq {\mathbb R} \rightarrow (0,\infty)\) is said to be multiplicatively convex if \[ f(x^{1/2}y^{1/2}) \leq f^{1/2}(x)f^{1/2}(y) \] for all \(x,y \in I\). And \(f\) is called multiplicatively concave if \(1/f\) is multiplicatively convex. Multiplicatively convexity on \(E\subseteq {\mathbb R}_+^2\) is defined similarly. The authors prove that \(G(x,y)=|\int_x^y f(t)dt|\) is multiplicatively concave on \([a,b]\times [a,b]\) if \(f:[a,b]\subset (0,\infty) \rightarrow (0,\infty)\) is continuous and multiplicatively concave.