A Hardy–Littlewood integral inequality on finite intervals with a concave weight

摘要

We prove: For all concave functions ( w: [a,b] rightarrow [0,infty )) and for all functions (f in C^2[a,b]) with (f(a)=f(b)=0) we have $$begin{aligned} left( int _{a}^{b} w(x) f^{prime }(x)^2 ,dx right) ^2 le left( int _{a}^{b} w(x) f(x)^2 ,dx right) left( int _{a}^{b} w(x)f^{prime prime }(x)^2 ,dxright) . end{aligned}$$Moreover, we determine all cases of equality. Keywords Integral inequality HELP-type inequality Concave weight function Mathematics Subject Classification 26D10 26D15 Page %P Close Plain text Look Inside Reference tools Export citation EndNote (.ENW) JabRef (.BIB) Mendeley (.BIB) Papers (.RIS) Zotero (.RIS) BibTeX (.BIB) Add to Papers Other actions Register for Journal Updates About This Journal Reprints and Permissions Share Share this content on Facebook Share this content on Twitter Share this content on LinkedIn Related Content Supplementary Material (0) References (17) References1.V.F. Babenko, N.P. Korneichuk, V.A. Kofanov, S.A. Pichugov, Inequalities for Derivatives and their Applications (Naukova Dumka, Kiev, 2003)2.V.I. Berdyshev, The best approximation in ( L(0,infty ) ) to the differentiation operator. Mat. Zametki 5, 477–481 (1971)3.H.J. Bremermann, Complex convexity. Trans. Am. Math. Soc. 82, 17–51 (1956)MATHMathSciNetCrossRef4.W.D. Evans, W.N. Everitt, HELP inequalities for limit-circle and regular problems. Proc. R. Soc. London Ser. A 432, 367–390 (1991)MATHMathSciNetCrossRef5.W.N. Everitt, On an extension to an integro-differential inequality of Hardy, Littlewood and Pólya. Proc. R. Soc. Edinb., 69, 295–333 (1971/1972)6.G.H. Hardy, J.E. Littlewood, Some integral inequalities connected with the calculus of variations. Q. J. Math, Oxford Ser. 2 3, 241–252 (1932)CrossRef7.G.H. Hardy, J.E. Littlewood, G. Pólya, Inequalities (Cambridge University Press, Cambridge, 1934)8.T. Kato, On an inequality of Hardy, Littlewood and Pólya. Adv. Math. 7, 217–218 (1971)MATHCrossRef9.M.K. Kwong, A. Zettl, An extension of the Hardy–Littlewood inequality. Proc. Am. Math. Soc. 77, 117–118 (1979)MATHMathSciNetCrossRef10.M.K. Kwong, A. Zettl, Remarks on best constants for norm inequalities among powers of an operator. J. Approx. Theory 26(3), 249–258 (1979)MATHMathSciNetCrossRef11.M.K. Kwong, A. Zettl, Norm inequalities of product form in weighted (L^p) spaces. Proc. R. Soc. Edinb. 89A, 293–307 (1981)MathSciNetCrossRef12.M.K. Kwong, A. Zettl, Norm Inequalities for Derivatives and Differences, Lecture Notes in Mathematics 1536. (Springer, New York, 1992)13.M.K. Kwong, A. Zettl, An alternate proof of Kato’s inequality. Evolution Equations. Lecture Notes in Pure and Applied Mathematics, vol. 234 (Dekker, New York, 2003), pp. 275–27914.C. Niculescu, L.-E. Persson, Convex Functions and Their Applications, CMS Books in Mathematics, vol 23 (Springer, New York, 2006)15.A.W. Roberts, D.E. Varberg, Convex Functions (Academic Press, New York, 1973)MATH16.H.L. Royden, P.M. Fitzpatrick, Real Analysis, 4th edn. (Prentice Hall, Boston, 2010)MATH17.E.C. Titchmarsh, The Theory of Functions (Oxford University Press, London, 1939)MATH About this Article Title A Hardy–Littlewood integral inequality on finite intervals with a concave weight Journal Periodica Mathematica Hungarica Volume 71, Issue 2 , pp 184-192 Cover Date2015-12 DOI 10.1007/s10998-015-0096-x Print ISSN 0031-5303 Online ISSN 1588-2829 Publisher Springer Netherlands Additional Links Register for Journal Updates Editorial Board About This Journal Manuscript Submission Topics Mathematics, general Keywords Integral inequality HELP-type inequality Concave weight function 26D10 26D15 Authors Horst Alzer (1) Man Kam Kwong (2) Author Affiliations 1. Morsbacher Str. 10, 51545, Waldbröl, Germany 2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hunghom, Hong Kong Continue reading... To view the rest of this content please follow the download PDF link above.