Parameterized Multidimensional Hilbert Type Inequalities
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Parameterized Multidimensional Hilbert-Type Inequalities
Author: Bicheng Yang
language: en
Publisher: Scientific Research Publishing, Inc. USA
Release Date: 2020-04-27
In 1934, G. H. Hardy et al. published a famous book entitled “Inequalities”, in which a theory about Hardy-Hilbert-type inequalities with the general homogeneous kernels of degree-1 and the best possible constant factors was built by introducing one pair of conjugate exponents. In January 2009, for generalized theory of Hardy-Hilbert-type inequalities, a book entitled “The Norm of Operator and Hilbert-Type Inequalities” (by Bicheng Yang) was published by Science Press of China, which considered the theory of Hilbert-type inequalities and operators with the homogeneous kernels of degree negative numbers and the best possible constant factors, by introducing two pairs of conjugate exponents and a few independent parameters. In October 2009 and January 2011, two books entitled “Hilbert-Type Integral Inequalities” and “Discrete Hilbert-Type Inequalities” (by Bicheng Yang) were published by Bentham Science Publishers Ltd., which considered mainly Hilbert-type integral and discrete inequalities with the homogeneous kernels of degree real numbers and applications. In 2012, a book entitled “Nonlinear Analysis: Stability, Approximation, and Inequality” was published by Springer, which contained Chapter 42 entitled “Hilbert-Type Operator: Norms and Inequalities” (by Bicheng Yang). In this chapter, the author defined a general Yang-Hilbert-type integral operator and studied six particular kinds of this operator with different measurable kernels in several normed spaces. In 2014, a book entitled “Half-Discrete Hilbert-Type Inequalities” was published in World Scientific Publishing Co. Pte. Ltd. (in Singapore), in which, the authors Bicheng Yang and L. Debnath considered some kinds of half-discrete Yang-Hilbert-type inequalities and their applications. In a word, the theory of Hilbert-type integral, discrete and half- discrete inequalities is almost built by Bicheng Yang et al. in the above stated books.
On Extended Hardy-hilbert Integral Inequalities And Applications
Hilbert-type inequalities, including Hilbert's inequalities proved in 1908, Hardy-Hilbert-type inequalities proved in 1934, and Yang-Hilbert-type inequalities first proved around 1998, play an important role in analysis and its applications. These inequalities are mainly divided in three classes: integral, discrete and half-discrete. During the last twenty years, there have been many research advances on Hilbert-type inequalities, and especially on Yang-Hilbert-type inequalities.In the present monograph, applying weight functions, the idea of parametrization as well as techniques of real analysis and functional analysis, we prove some new Hilbert-type integral inequalities as well as their reverses with parameters. These inequalities constitute extensions of the well-known Hardy-Hilbert integral inequality. The equivalent forms and some equivalent statements of the best possible constant factors associated with several parameters are considered. Furthermore, we also obtain the operator expressions with the norm and some particular inequalities involving the Riemann-zeta function and the Hurwitz-zeta function. In the form of applications, by means of the beta function and the gamma function, we use the extended Hardy-Hilbert integral inequalities to consider several Hilbert-type integral inequalities involving derivative functions and upper limit functions. In the last chapter, we consider the case of Hardy-type integral inequalities. The lemmas and theorems within provide an extensive account of these kinds of integral inequalities and operators.Efforts have been made for this monograph hopefully to be useful, especially to graduate students of mathematics, physics and engineering, as well as researchers in these domains.
A Kind of Half-Discrete Hardy-Hilbert-Type Inequalities Involving Several Applications
Author: CV-Bicheng Yang
language: en
Publisher: Scientific Research Publishing, Inc. USA
Release Date: 2023-12-22
In this book, applying the weight functions, the idea of introduced parameters and the techniques of real analysis and functional analysis, we provide a new kind of half-discrete Hilbert-type inequalities named in Mulholland-type inequality. Then, we consider its several applications involving the derivative function of higher-order or the multiple upper limit function. Some new reverses with the partial sums are obtained. We also consider some half-discrete Hardy-Hilbert’s inequalities with two internal variables involving one derivative function or one upper limit function in the last chapter. The lemmas and theorems provide an extensive account of these kinds of half-discrete inequalities and operators.