An integral inequality of convolution type
The authors prove that if \(f\) is nondecreasing and nonnegative, \(g\) is convex and the function \(y\mapsto h_ c(y)= g(cy)-\int_ 0^ c k(s) ds g(y)\) is nonnegative and nondecreasing for each \(c\in (0,b]\), or, if \(f\) is nonincreasing and nonnegative, \(g\) is concave and \(y\mapsto h_ c(y)\) is nonnegative and nonincreasing for each \(c\in(0,b]\), then \[ \int^ x_ 0 k(x-s)g(f(s)) ds\leq g\left(\int^ x_ 0 f(s) ds\right),\tag{*} \] when \(0<x\leq b\). Some special cases are considered in a number of corollaries. Furthermore, it is shown that if \(f\) is nonnegative and nondecreasing, \(g\) is nondecreasing and concave with \(\int^ x_ 0 k(s) ds g(y)\leq g(yx)\) and \(\int^ x_ 0 k(s) ds\leq x\) for \(0\leq x\leq 1\) and \(y\geq 0\), then (*) holds when \(0< x\leq 1\).
scientific article; zbMATH DE number 493223