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SEVERAL EXTENDED ANALOGUES OF HILBERT'S INEQUALITIES

SEVERAL EXTENDED ANALOGUES OF HILBERT'S INEQUALITIES DEMONSTRATIO MATHEMATICAVol. XLIINo 22009Hongxia Du, Yu MiaoS E V E R A L E X T E N D E D A N A L O G U E S OFHILBERT'S INEQUALITIESIqct £ — lOE 11By introducing the function — ! —J- with real numbersax + py + mm{a;, y}a, ¡3,7, we get several extended analogues of Hilbert's inequalities.Abstract.1. IntroductionIf / , g are real functions such thatoo0 < 5 f2(x)dx0(1.1)oo< oo a n d 0 < \ g2(x)dxo< oo,then we have the following well known Hilbert's integral inequality [1],oo oo f(\ \56\ ( \S[X)9{y)x +dxdyy00f°°1/2< 7T< \ f \ x ) d x \ g\x)dxU6\iwhere the constant factor 7r is the best possible. Furthermore, we have alsothe following Hardy-Hilbert's type inequality [1, Th 341, Th 342],00 00ooooi 0S 566g X'x~yf(r)n(iA,0gyf(x)g(y)dxdy00r°°11/2soooo< t t 2 \ \ f ( x ) d x \ g\x)dxU6>| 1/2J,where the constant factors 4 and 7r2 are both the best possible.There are numerous literatures to study the Hilbert's and Hardy-Hilbert'stype inequalities from different directions [4, 5, 6, 7]. Recently, Li-Wu-He [3]2000 Mathematics Subject Classification: 26D15.Key words and phrases: Hilbert's inequalities, best constant.298H. X. Du, Y. Miaoobtained the following inequality: if (1.1) is satisfied, then http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Demonstratio Mathematica de Gruyter

SEVERAL EXTENDED ANALOGUES OF HILBERT'S INEQUALITIES

Du, Hongxia; Miao, Yu
Demonstratio Mathematica , Volume 42 (2): 6 – Apr 1, 2009
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References (8)

Publisher
de Gruyter
Copyright
© by Hongxia Du
ISSN
0420-1213
eISSN
2391-4661
DOI
10.1515/dema-2009-0209
Publisher site
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Abstract

DEMONSTRATIO MATHEMATICAVol. XLIINo 22009Hongxia Du, Yu MiaoS E V E R A L E X T E N D E D A N A L O G U E S OFHILBERT'S INEQUALITIESIqct £ — lOE 11By introducing the function — ! —J- with real numbersax + py + mm{a;, y}a, ¡3,7, we get several extended analogues of Hilbert's inequalities.Abstract.1. IntroductionIf / , g are real functions such thatoo0 < 5 f2(x)dx0(1.1)oo< oo a n d 0 < \ g2(x)dxo< oo,then we have the following well known Hilbert's integral inequality [1],oo oo f(\ \56\ ( \S[X)9{y)x +dxdyy00f°°1/2< 7T< \ f \ x ) d x \ g\x)dxU6\iwhere the constant factor 7r is the best possible. Furthermore, we have alsothe following Hardy-Hilbert's type inequality [1, Th 341, Th 342],00 00ooooi 0S 566g X'x~yf(r)n(iA,0gyf(x)g(y)dxdy00r°°11/2soooo< t t 2 \ \ f ( x ) d x \ g\x)dxU6>| 1/2J,where the constant factors 4 and 7r2 are both the best possible.There are numerous literatures to study the Hilbert's and Hardy-Hilbert'stype inequalities from different directions [4, 5, 6, 7]. Recently, Li-Wu-He [3]2000 Mathematics Subject Classification: 26D15.Key words and phrases: Hilbert's inequalities, best constant.298H. X. Du, Y. Miaoobtained the following inequality: if (1.1) is satisfied, then

Journal

Demonstratio Mathematica – de Gruyter

Published: Apr 1, 2009

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