On a generalization of the Opial inequality
\textit{Z. Opial} [Ann. Pol. Math. 9, 145--155 (1960; Zbl 0098.29102)] stated and proved the following classical inequality\N\[\N\int^h_0\mid f(x)f'(x)\mid dx\le\frac1{4}\int^h_0\mid f'(x)\mid^2dx,\N\]\Nwhere \(f\in C^1[0,h]\) and satisfies \(f(0)=f(h)=0\) and \(f(x) >0\) for all \(x\in (0,h)\). The above inequality is referred to as Opial's inequality. In this paper, the authors derive and prove some new Opial-type inequalities and give applications to the generalized Riemann-Liouville-type integral operators. The key results of the paper are given in Theorems 1 and 3 and several consequences of their results are pointed out and well discussed.
scientific article; zbMATH DE number 7863324
scientific article