A Generalization on Some New Types of Hardy-Hilbert’s Integral Inequalities

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