TAILIEUCHUNG - Báo cáo hóa học: " Research Article A Multiple Hilbert-Type Integral Inequality with the Best Constant Factor"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article A Multiple Hilbert-Type Integral Inequality with the Best Constant Factor | Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2007 Article ID 71049 14 pages doi 2007 71049 Research Article A Multiple Hilbert-Type Integral Inequality with the Best Constant Factor Baoju Sun Received 9 February 2007 Accepted 29 April 2007 Recommended by Eugene H. Dshalalow By introducing the norm x a x e R and two parameters a A we give a multiple Hilbert-type integral inequality with a best possible constant factor. Also its equivalent form is considered. Copyright 2007 Baoju Sun. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction If p 1 1 p 1 q 1 f g 0 satisfy 0 v fp t dt TO and 0 J0 gq t dt TO then the well-known Hardy-Hilbert s integral inequality is given by see 1 2 f x g y x y dxdy r TO 1 p f TO 1 q n fp t dt gq t dt sin n p 0 0 where the constant factor n sin n p is the best possible. Its equivalent form is nr dxỴdy 0 0 x y pTO sn p Jc fp x dx where the constant factor n sin n p p is still the best possible. Hardy-Hilbert integral inequality is important in analysis and applications. During the past few years many researchers obtained various generalizations variants and extensions of inequality see 3-9 and the references cited therein . 2 Journal of Inequalities and Applications Hardy et al. 1 gave a Hilbert-type integral inequality similar to as T f x g y dxdy 2 fp x dx yp f gỉ x dx yq 0 x - y sin n p 0 0 where the constant factor n sin n p 2 is the best possible. Recently Yang gave a generalization of as see 9 T intfyif x g y dxdy 0 xA - yA L . n 1 íí x p-1 1-A fp x dx pff x q-1 1-A gq x dx q A sin n p 0 0 where the constant factor n Asin n p 2 is the best possible. Its equivalent form is -u w djdy r j x dx where the constant factor n Asin n p 2p is the best possible. At present because of the requirement of .

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