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DEMONR1AHO MATHEMAUCAMmMItmB.O. PacfaptUcON INTEGRAL INEQUALITIES SIMILAR TO OPIAL'S INEQUALITY1« IntrodactionIn 1960 Z. Opial [6] proved an inequality which has r e ceived considerable attention over the laat twenty five years.In [5] C. Oleoh obtained the following version of the Opialinequality established in [ 6 ] .I f s is absolutely continuous on [ a ( b ] and a(a) •* » ( b ) • 0,then(1)bfab[|a'(x)|2dx,awhere the oonstant t i ^ l ± g the best possible*After the appeeotanoe of the papers [5]> [6] in 1960,a number of papers had as their aim to give simpler proofs,various extensions and generalisations of the Opial inequality (1), (see [1], [ 3 ] - [ l 2 ] and the referenoea given therein).She integral inequalities of the form (1) are of considerableinterest and also have important applications in the theory ofordinary differential equations and boundary value problems(ase [4], [10], [11])• She main aim of this paper ia to establish some new Integral inequalities involving functions andtheir derivatives which olaim their origin to the Opial i n equality given in ( 1 ) . In fact, our reaults are motivated bythe interesting variants of inequality (1) given by Godunovaand Levin in [ l
Demonstratio Mathematica – de Gruyter
Published: Jan 1, 1989
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