On Hardy-type integral inequalities.pdfVIP

Appl. Math. Mech. -Engl. Ed., 34(10), 1297–1304 (2013) Applied Mathematics DOI 10.1007/s10483-013-1746-x and Mechanics c Shanghai University and Springer-Verlag Berlin Heidelberg 2013 (English Edition) On Hardy-type integral inequalities∗ Tuo LENG ( ), Yong FENG ( ) (Chengdu Institute of Computer Applications, Chinese Academy of Sciences, Chengdu 610042, P. R. China) Abstract The Hardy integral inequality is one of the most important inequalities in analysis. The present paper establishes some new Copson-Pachpatte (C-P) type inequal- ities, which are the generalizations of the Hardy integral inequalities on binary functions. Key words Hardy inequality, H¨older inequality, Copson inequality, Izumi inequality, Pachpatte inequality Chinese Library Classification O178 2010 Mathematics Subject Classification 26D15 1 Introduction The well-known Hardy’s integral inequality is as follows: Theorem 1.1 If p 1, f (x) 0 for 0 x ∞, and F (x) = 1 x f (t)dt, then x 0 ∞ p p p ∞ p F (x) dx f (x) dx (1) 0 p − 1 0 unless f ≡ 0. The constant is the best possible. Theorem 1.1 was first proved by Hardy[1] in an attempt to give a simple proof of Hilbert’s double series theorem (see [2]). One of the best known and interesting generalization of the inequality (1) was g