Vol. 26, No. 2, 1968

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Extensions of Opial’s inequality

Paul Richard Beesack and Krishna M. Das

Vol. 26 (1968), No. 2, 215–232
Abstract

In this paper certain inequalities involving integrals of powers of a function and of its derivative are proved. The prototype of such inequalities is Opial’s Inequality which states that 2 0X|yy′|dx X 0Xy2 dx whenever y is absolutely continuous on [0,X] with y(0) = 0. The extensions dealt with here are all integral inequalities of the form

∫ b   p ′q           ∫ b  ′p+q
a s|y| |y |dx ≦ K (p,q) a r|y |   dx,

(or with replaced by ), where r, s are nonnegative, measurable functions on I = [a,b], and y is absolutely continuous on I with either y(a) = 0, or y(b) = 0, or both. In some cases y may be complex-valued, while in other cases ymust not change sign on I. The inequality (as stated) is obtained in case pq > 0 and either p + q 1 or p + q < 0, while the opposite inequality is obtained in case p < 0, q 1, p + q < 0, or p > 0, p + q < 0. In all cases, necessary and sufficient conditions are obtained for equality to hold.

Mathematical Subject Classification
Primary: 26.70
Milestones
Received: 17 November 1967
Published: 1 August 1968
Authors
Paul Richard Beesack
Krishna M. Das