An inequality similar to Opial’s inequality

Abstract

rb 1 b (1) fJ |yy(n) I dx (b ? a)n1 I y(n) 12dx. 2 (He employs (1) to prove uniqueness of the initial value problem for linear differential equations of order n.) Also there are generalizations of Opial's original inequality in other directions. (See, for instance, Calvert [2] and Yang [6].) The purpose of this note is to obtain a sharper version of (1) and other generalizations. THEOREM 1. Let yE C(n-1) [a, b ] be such that y() (a) = O for i=O, 1, n -1 where n ? 1. Let y(n-1) be absolutely continuous and y (n) 1 2 dx < oo . Then,