Some Extensions Of Hilbert Integral Inequalities

Mathematic inequality plays a very important role in theoretical research and practi-cal application. Although as early as in 1911, Mathematician Schur completed the proof of Hilbert integral inequality, at present, the inequality still receives the attention of many scholars because it is widely used in Analytic number theory, functional analysis, differ-ential equation, approximation theory and other branches of mathematics. Especially in recent 20 years, many nature extension of the Hilbert inequalities have been proved.This paper mainly studies below three questions. Firstly, we prove a new class of inequalities by introducing the ? function and some different parameters; Secondly, the extension of Hardy-Hilbert inequality was given according to some basic Hilbert inequal-ities that have been proved; Finally, we set up a extension of multiple linear Hilbert inte-gral inequality through applying the weighted Holder inequality. Especially, in the special case of n=2, we return the classic Hilbert integral inequalities .The content of this paper is organized as follows:In the first chapter, we introduce the development of the Hilbert integral inequality,the research background and some other related topics.In the second chapter, we obtain two theorems and give their corollaries respectively.Kernels of the inequalities proved by two theorems are multiplications or divisions of two basic Hilbert-type inequalities' kernels, and the kernels of two basic Hilbert-type inequal-ities are respectively 1/max{x?,y?} and 1/|x-y|. In the first theorem, we establish a new generalized Hilbert-type inequality, which has the best constant factor and linkes two basic Hilbert-type inequalities. We prove the theorem by introducing two independent parameters y?; two pairs of conjugate index (p, q) (r, s). The kernel of inequality is 1/min{x?,y?}·|x-y|?-?·Meanwhile, great progress has been made due to the improvement of weight coefficien-t and application of parametrization methods. So by introducing ?-functions and other independent parameter ?, and by estimating the weight function ??(p, x), we establish-es a new Hilbert-type inequality that has the best constant and A homogeneous kernel in the second theorem. As applications, we establish some equivalents of the Hilbert-type inequality and obtain some very meaningful results by taking special parameters.In the third chapter, we obtain three extended Hilbert-type inequalities by introduc-ing ?-function, weighted Holder inequality. In the first two inequalities, we extend the inequality by introducing multiple integration, also establish a link between Hilbert in-equality and Hardy inequality, and obtain the corresponding Hardy-Hilbert inequality that has n integrable functions, which based on the conclusion of L. E. AZAR[59]. In the third theorem, we extend the inequality from the aspect of kernel function's form,and obtain a new Hilbert-type inequality by extending the original kernel functiony lnx/y/x+y to|lnx/y|?/x?+y?.Obviously, we can get some classical inequalities by taking some special values of?. Meanwhile, we can easily prove that the constant of the inequality is the best.In the forth chapter, we summarize our whole work and some research methods we frequently used when proving our theorems, which we believe may contribute to the future study of this subjiet.