scholarly journals On a class of Hilbert-type inequalities in the whole plane related to exponent function

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Minghui You
Keyword(s):  
Euler Number ◽  
Bernoulli Number ◽  
Type Inequality ◽  
Partial Fraction ◽  
Multiple Parameters ◽  
Partial Fraction Expansion ◽  
Hilbert Type

AbstractBy introducing a kernel involving an exponent function with multiple parameters, we establish a new Hilbert-type inequality and its equivalent Hardy form. We also prove that the constant factors of the obtained inequalities are the best possible. Furthermore, by introducing the Bernoulli number, Euler number, and the partial fraction expansion of cotangent function and cosecant function, we get some special and interesting cases of the newly obtained inequality.

scholarly journals On a new generalization of some Hilbert-type inequalities

2021 ◽  
Vol 19 (1) ◽  
pp. 569-582
Author(s):  
Minghui You ◽  
Wei Song ◽  
Xiaoyu Wang
Keyword(s):  
Kernel Function ◽  
Euler Number ◽  
Bernoulli Number ◽  
Type Inequality ◽  
Trigonometric Functions ◽  
Hardy Type Inequality ◽  
Hardy Type ◽  
Higher Derivatives ◽  
Derivatives Of ◽  
Hilbert Type

Abstract In this work, by introducing several parameters, a new kernel function including both the homogeneous and non-homogeneous cases is constructed, and a Hilbert-type inequality related to the newly constructed kernel function is established. By convention, the equivalent Hardy-type inequality is also considered. Furthermore, by introducing the partial fraction expansions of trigonometric functions, some special and interesting Hilbert-type inequalities with the constant factors represented by the higher derivatives of trigonometric functions, the Euler number and the Bernoulli number are presented at the end of the paper.

scholarly journals On a half-discrete Hilbert-type inequality related to hyperbolic functions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Minghui You
Keyword(s):  
Type Inequality ◽  
Weight Coefficient ◽  
Bernoulli Numbers ◽  
Rational Fraction ◽  
Hyperbolic Functions ◽  
Multiple Parameters ◽  
Tangent Function ◽  
Hilbert Type

AbstractBy the introduction of a new half-discrete kernel which is composed of several exponent functions, and using the method of weight coefficient, a Hilbert-type inequality and its equivalent forms involving multiple parameters are established. In addition, it is proved that the constant factors of the newly obtained inequalities are the best possible. Furthermore, by the use of the rational fraction expansion of the tangent function and introducing the Bernoulli numbers, some interesting and special half-discrete Hilbert-type inequalities are presented at the end of the paper.

scholarly journals On a more accurate Hilbert-type inequality in the whole plane with the general homogeneous kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Xingshou Huang ◽  
Bicheng Yang
Keyword(s):  
Equivalent Form ◽  
Type Inequality ◽  
Constant Factor ◽  
Real Analysis ◽  
Homogeneous Kernel ◽  
Hilbert’S Inequality ◽  
The Best Possible Constant ◽  
Hilbert Type ◽  
Weight Coefficients

AbstractBy the use of the weight coefficients, the idea of introduced parameters and the technique of real analysis, a more accurate Hilbert-type inequality in the whole plane with the general homogeneous kernel is given, which is an extension of the more accurate Hardy–Hilbert’s inequality. An equivalent form is obtained. The equivalent statements of the best possible constant factor related to several parameters, the operator expressions and a few particular cases are considered.

scholarly journals A NEW HALF-DISCRETE HILBERT-TYPE INEQUALITY IN THE WHOLE PLANE

2017 ◽  
Vol 7 (3) ◽  
pp. 977-991
Author(s):  
Bicheng Yang ◽  
◽  
Bing He
Keyword(s):  
Type Inequality ◽  
Hilbert Type

scholarly journals Chern Class Inequalities on Polarized Manifolds and Nef Vector Bundles

2020 ◽  
Author(s):  
Ping Li ◽  
Fangyang Zheng
Keyword(s):  
Chern Class ◽  
Vector Bundles ◽  
Euler Number ◽  
Type Inequality ◽  
Chern Number ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Bisectional Curvature ◽  
Chern Numbers ◽  
Nef Vector Bundles

Abstract This article is concerned with Chern class and Chern number inequalities on polarized manifolds and nef vector bundles. For a polarized pair $(M,L)$ with $L$ very ample, our 1st main result is a family of sharp Chern class inequalities. Among them the 1st one is a variant of a classical result and the equality case of the 2nd one is a characterization of hypersurfaces. The 2nd main result is a Chern number inequality on it, which includes a reverse Miyaoka–Yau-type inequality. The 3rd main result is that the Chern numbers of a nef vector bundle over a compact Kähler manifold are bounded below by the Euler number. As an application, we classify compact Kähler manifolds with nonnegative bisectional curvature whose Chern numbers are all positive. A conjecture related to the Euler number of compact Kähler manifolds with nonpositive bisectional curvature is proposed, which can be regarded as a complex analogue to the Hopf conjecture.

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