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On multivariate Greene type inequalities

On multivariate Greene type inequalities DEMONSTRATIO MATHEMATICAVol. XXVIINo 21994B. P. PachpatteON MULTIVARIATE GREENE TYPE INEQUALITIESIn the present paper we establish a new integral inequality of the Greenetype involving functions of several variables. A corresponding inequality onthe discrete analogue of the main result is also given.1. IntroductionIn 1977, D. E. Greene [3] proved the following lemma.L E M M A 1.be continuousLet k\, k2 and // be nonnegative constants and let f , g and hinonnegative functions for all t > 0 with hi bounded such thaif(t)<k!+tJ h1(s)f(s)ds0g(t)<k2+t+ fe^h2(s)g(s)ds,0tf e-f°h3(s)f(s)ds+f00th4(s)g(s)ds,for all t > 0. Then there exist constants Ci and Mi such that/(<)<^ieClt,g(t)<M2ec^,for all t > 0.The proof of these inequalities given in [3] was elementary but long, andmuch shorter proofs and further generalizations were found in [2], [8], [12].In 1957, while investigating the boundedness of solutions of certain secondorder differential equations, Liang Ou-Iang [7] established the following inequality.A MS (MOS) Subject Classification (1980): Primary 26D15, Secondary 26D20.Keywords and phrases: Greene type inequalities, several variables, discrete analogue,Sum-difference equations.302B. P. P a c h p a t t eLet u and p be real-valued nonnegative continuous functionsdefined for all t>0. Iftu2(t) < c2 + 2 fp(s)u(s)ds,0for all t > 0, where c > 0 is a http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Demonstratio Mathematica de Gruyter

On multivariate Greene type inequalities

Pachpatte, B. P.
Demonstratio Mathematica , Volume 27 (2): 10 – Apr 1, 1994
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References (10)

Publisher
de Gruyter
Copyright
© by B. P. Pachpatte
ISSN
0420-1213
eISSN
2391-4661
DOI
10.1515/dema-1994-0208
Publisher site
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Abstract

DEMONSTRATIO MATHEMATICAVol. XXVIINo 21994B. P. PachpatteON MULTIVARIATE GREENE TYPE INEQUALITIESIn the present paper we establish a new integral inequality of the Greenetype involving functions of several variables. A corresponding inequality onthe discrete analogue of the main result is also given.1. IntroductionIn 1977, D. E. Greene [3] proved the following lemma.L E M M A 1.be continuousLet k\, k2 and // be nonnegative constants and let f , g and hinonnegative functions for all t > 0 with hi bounded such thaif(t)<k!+tJ h1(s)f(s)ds0g(t)<k2+t+ fe^h2(s)g(s)ds,0tf e-f°h3(s)f(s)ds+f00th4(s)g(s)ds,for all t > 0. Then there exist constants Ci and Mi such that/(<)<^ieClt,g(t)<M2ec^,for all t > 0.The proof of these inequalities given in [3] was elementary but long, andmuch shorter proofs and further generalizations were found in [2], [8], [12].In 1957, while investigating the boundedness of solutions of certain secondorder differential equations, Liang Ou-Iang [7] established the following inequality.A MS (MOS) Subject Classification (1980): Primary 26D15, Secondary 26D20.Keywords and phrases: Greene type inequalities, several variables, discrete analogue,Sum-difference equations.302B. P. P a c h p a t t eLet u and p be real-valued nonnegative continuous functionsdefined for all t>0. Iftu2(t) < c2 + 2 fp(s)u(s)ds,0for all t > 0, where c > 0 is a

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Demonstratio Mathematica – de Gruyter

Published: Apr 1, 1994

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