On certain integral inequalities related to Opial's inequality

Abstract

The aim of the present paper is to establish some new integral inequalities involving three functions and their derivatives which in the special cases yield the well known Opial inequality and some of its generalizations.

Sharp Opial‐Type Inequalities Involving Higher Order Derivatives of Two Functions

“…Extensions of the above inequalities involving three or more functions are immediate. For example, an addition of (82) and the inequalities similar to (82) for the pair For v = 1 inequality (84) reduces to a result proved in [15].…”

Some new generalizations of Maroni inequality on time scales

“…For some classical results on Opial inequality, the readers are referred to the articles (see [2]- [9], [15]). Now, we give some of earliest version of the Opial inequlality as follows: The following inequality holds true:…”

Still more generalizations of Hardy's inequality

“…In [5] B. G. Pachpatte added one more to the multitude of papers generalizing Hardy's Inequality; and the present paper, inspired by [5], is yet another. The proof of our leading theorem is modelled on that of Pachpatte, and our theorem generalizes his.…”

“…REMARK. Pachpatte's Theorem 1 in[5] is the case of Theorem 3 (above) in which p > l,q = l,c = a,a is replaced by 1 and fix) by fix) log(x).…”