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In this paper, by using hardy inequality, we establish some new integral inequalities of Hardy-Hilbert type with general kernel. As applications, equivalent forms and some particular results are built; the corresponding to the double series inequalities are given.reverse forms are considered also.

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We give an improvement of some inequalities similar to Hilbert's inequality involving series of nonnegative terms. The integral analogies of the main results are also given.

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The aim of this paper is to establish a new extension of Hilbert's inequality and Hardy-Hilbert's inequality for multifunctions with best constants factors. Also, we present some applications for Hilbert's inequality which give new integral inequalities.

Abstract:
The aim of this paper is to establish a new extension of Hilbert's inequality and Hardy-Hilbert's inequality for multifunctions with best constants factors. Also, we present some applications for Hilbert's inequality which give new integral inequalities.

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By applying the way of weight functions and a Hardy's integral inequality, a Hilbert-Hardy-type integral operator is defined, and the norm of operator is obtained. As applications, a new Hilbert-Hardy-type inequality similar to Hilbert-type integral inequality is given, and two equivalent inequalities with the best constant factors as well as some particular examples are considered.

Abstract:
Sulaiman presented, in 2008, new kinds of Hardy-Hilbert’s integral inequality in which the weight function is homogeneous. In this paper, we present a generalization on the kinds of Hardy-Hilbert’s integral inequality. 1. Introduction and Preliminaries For any two nonnegative measurable functions and such that we have the Hilbert’s integral inequality [1] that The constant is the best possible. In 1925, Hardy [2] extended the Hilbert’s integral inequality into the integral inequality as follows. If , , and such that then we have the Hardy-Hilbert’s integral inequality that The constant is the best possible. Both the two inequalities are important in mathematical analysis and its applications [3]. In 1938, Widder [4] studied on the Stieltjes Transform . Now, we recall the beta function as follows: In 2001, Yang [5] extended the Hardy-Hilbert’s integral inequality into the following integral inequality. If , , , and such that then we have where . The constant is the best possible. We also recall that a nonnegative function which is said to be homogeneous function of degree if for all . And we say that is increasing if and are increasing functions. In 2008, Sulaiman [6] gave new integral inequality similar to the Hardy-Hilbert’s integral inequality. If , , , , is a positive increasing homogeneous function of degree , and and then, for all , we have where In this paper, we present a generalization of the integral inequality (9) and its applications. Next proposition will be used in the next section. Proposition 1 (see [6]). Let be a positive increasing function, and . Then, for all , one has 2. Main Results Theorem 2. Let , , , , and let be positive increasing homogeneous function of degree , and and and let be a function such that for all . Then, for all , one has where Proof. Let and . By the H？lder inequality, the assumption of , and the Tonelli theorem, we have Now, we put and for the first integral, and then we put and for the second integral. And, by Proposition 1, one has Then, by the assumption, one has This proof is completed. 3. Applications Corollary 3. Let , , and , and let be a positive increasing homogeneous function of degree , and and Then, for all , one has where Proof. (a) This follows from Theorem 2 where for all . (b) This follows from Theorem 2 where for all . (c) This follows from Theorem 2 where for all . (d) This follows from Theorem 2 where for all . 4. Open Problem In this section, we pose a question that is how to generalize the integral inequality (13) if may not satisfy the property for all . Acknowledgments The author would

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In this paper, Hardy-Hilbert's integral inequalities are generalized. Hardy-Hilbert's integral inequality with optimal constant factor on unsymmetrical core function and weighted Hardy-Hilbert's integral inequality are obtained, and some special cases are also considered.

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In this paper, by estimating the weight coefficient effectively, we establish an improvement of a Hardy-Hilbert type inequality proved by B.C. Yang, our main tool is Euler-Maclaurin expansion for the zeta function. As applications, some particular results are considered

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By estimating the weight coefficient, a reverse Hardy-Hilbert-type inequality is proved. As applications, some equivalent forms and a number of particular cases are obtained.