On a generalization of the Opial inequality. (English) Zbl 07863324
Z. Opial [Ann. Pol. Math. 9, 145–155 (1960; Zbl 0098.29102)] stated and proved the following classical inequality
\[
\int^h_0\mid f(x)f'(x)\mid dx\le\frac1{4}\int^h_0\mid f'(x)\mid^2dx,
\]
where \(f\in C^1[0,h]\) and satisfies \(f(0)=f(h)=0\) and \(f(x) >0\) for all\(x\in (0,h).\) The above inequality is referred to as the Opial’s inequality. In this paper the authors derived and proves some new Opial type-inequalities and gave their 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 and pointed out and well discussed.
Reviewer: James Adedayo Oguntuase (Abeokuta)
Keywords:
fractional derivatives and integrals; fractional integral inequalities; Opial-type inequalitiesCitations:
Zbl 0098.29102References:
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