Generalization of an inequality for integral transforms with kernel and related results
The authors establish a generalization of the inequality introduced by \textit{D. S. Mitrinović} and \textit{J. E. Pečarić} [Bull. Pol. Acad. Sci., Math. 36, No.~9--10, 645--648 (1988; Zbl 0758.26011)]. Specifically, they prove that \[ \int^ b _ a r(x)\phi\Big(\frac{u_ 1 (x)}{u_ 2 (x)}\Big)dx \leq \int^ b _ a q(x)\phi\Big(\frac{v_ 1 (x)}{v_ 2 (x)}\Big)dx, \] where \[ q(x)=v_ 2 (x)\int^ b _ a \frac{r(x)K(t,x)}{u_ 2 (t)}dt \quad u_ 2(t)\neq 0, \] \(\phi\) is a convex function and \(u_ i \in {\mathcal U}(\nu,K)\), \(i=1,2\). Mean-value theorems of Cauchy type for the above inequality are established and proved by taking its difference. Furthermore, the positive semidefiniteness of the matrices generated by the difference of the inequality which implies the exponential convexity and logarithmic convexity is also proved. Also, new means of Cauchy type are defined and the monotonicity of these means is proved. Several consequences of the results obtained are presented and discussed.
scientific article; zbMATH DE number 5847785