Half-discrete Hilbert-type Inequality Research Articles

Construction of a class of half-discrete Hilbert-type inequalities in the whole plane

Abstract In this work, we first define two special sets of real numbers, and then, we construct a half-discrete kernel function where the variables are defined in the whole plane, and the parameters in the kernel function are limited to the newly constructed special sets. Estimate the kernel function in the whole plane by converting it to the first quadrant, and then, a class of new Hilbert-type inequality is established. Additionally, it is proved that the constant factor of the newly established inequality is the best possible. Furthermore, assigning special values to the parameters and using rational fraction expansion of cosecant function, some special results are presented at the end of this article.

A new reverse half-discrete Hilbert-type inequality with one partial sum involving one derivative function of higher order

Abstract In this article, a new reverse half-discrete Hilbert-type inequality with one partial sum involving one derivative function of higher order is obtained, by using the weight functions, the mid-value theorem, and the techniques of real analysis. A few equivalent statements of the best possible constant factor related to several parameters are considered. As applications, the equivalent forms and some particular inequalities are provided.

A half-discrete Hilbert-type inequality in the whole plane with the constant factor related to a cotangent function

In this work, by the introduction of some parameters, a new half-discrete kernel function in the whole plane is defined, which involves both the homogeneous and the nonhomogeneous cases. By employing some techniques of real analysis, especially the method of a weight function, a new half-discrete Hilbert-type inequality with the new kernel function, as well as its equivalent Hardy-type inequalities are established. Moreover, it is proved that the constant factors of the newly obtained inequalities are the best possible. Finally, assigning special values to the parameters, some new half-discrete Hilbert-type inequalities with special kernels are presented at the end of the paper.

More accurate form of half-discrete Hilbert-type inequality with a general kernel

More accurate form of half-discrete Hilbert-type inequality with a general kernel

A class of half-discrete Hilbert-type inequalities in the whole plane involving some classical kernels

A class of half-discrete Hilbert-type inequalities in the whole plane involving some classical kernels

On a more accurate half-discrete Hilbert-type inequality involving hyperbolic functions

Abstract In this work, by the introduction of a new kernel function composed of exponent functions with several parameters, and using the method of weight coefficient, Hermite-Hadamard’s inequality, and some other techniques of real analysis, a more accurate half-discrete Hilbert-type inequality including both the homogeneous and non-homogeneous cases is established. Furthermore, by introducing the Bernoulli number and the rational fraction expansion of tangent function, some special and interesting Hilbert-type inequalities and their equivalent hardy-type inequalities are presented at the end of the paper.

More accurate and strengthened forms of half-discrete Hilbert inequality

In this work, by introducing a new half-discrete kernel function with parameters, and using the method of weight coefficient, Euler-Maclaurin summation formula and some other techniques of real analysis, a more accurate half-discrete Hilbert-type inequality as well as its strengthened form are established. Additionally, some equivalent inequalities in Hardy's form are also considered. Furthermore, it is proved that the constant factors in the newly obtained inequalities are the best possible. At last, by taking some special values of parameters and using the numerical calculation function of Matlab, some special cases are presented at the end of the paper.

ON A NEW HALF-DISCRETE HILBERT-TYPE INEQUALITY WITH THE MULTIPLE UPPER LIMIT FUNCTION AND THE PARTIAL SUMS

<abstract> By means of the weight coefficients, the Euler-Maclaurin summation formula and Abel's summation by parts formula, a new half-discrete Hilbert-type inequality with the power function as the interval variables as well as one multiple upper limit function and one partial sums is given. As applications, the equivalent conditions of the best possible constant factor in a particular inequality related to a few parameters and some particular cases are considered. We also obtain the equivalent forms and the operator expression in the case of <italic>m</italic>=0. </abstract>

A MORE ACCURATE HALF-DISCRETE HILBERT-TYPE INEQUALITY INVOLVING ONE HIGHER-ORDER DERIVATIVE FUNCTION

<abstract> By means of the weight functions, Hermite-Hadamards inequality and the techniques of real analysis, a new more accurate half-discrete Hilberttype inequality involving one higher-order derivative function is given. The equivalent conditions of the best possible constant factor related to a few parameters, the equivalent forms, several particular inequalities and the operator expressions are considered. </abstract>

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

By 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.

A More Accurate Half-Discrete Hilbert-Type Inequality Involving One upper Limit Function and One Partial Sum

In this paper, by virtue of the symmetry principle, we construct proper weight coefficients and use them to establish a more accurate half-discrete Hilbert-type inequality involving one upper limit function and one partial sum. Then, we prove the new inequality with the help of the Euler–Maclaurin summation formula and Abel’s partial summation formula. Finally, we illustrate how the obtained results can generate some new half-discrete Hilbert-type inequalities.

A more accurate half-discrete Hilbert-type inequality in the whole plane and the reverses

A more accurate half-discrete Hilbert-type inequality in the whole plane with multi-parameters is established by the use of Hermite–Hadamard’s inequality and weight functions. Furthermore, some equivalent forms and some special types of inequalities and operator representations as well as reverses are considered.

The equivalent parameter conditions for constructing multiple integral half-discrete Hilbert-type inequalities with a class of nonhomogeneous kernels and their applications

Abstract In this paper, we establish equivalent parameter conditions for the validity of multiple integral half-discrete Hilbert-type inequalities with the nonhomogeneous kernel G ( n λ 1 ∥ x ∥ m , ρ λ 2 ) G\left({n}^{{\lambda }_{1}}\parallel x{\parallel }_{m,\rho }^{{\lambda }_{2}}\hspace{-0.16em}) ( λ 1 λ 2 &gt; 0 {\lambda }_{1}{\lambda }_{2}\gt 0 ) and obtain best constant factors of the inequalities in specific cases. In addition, we also discuss their applications in operator theory.

On a New Half-Discrete Hilbert-Type Inequality Involving the Variable Upper Limit Integral and Partial Sums

In this paper we establish a new half-discrete Hilbert-type inequality involving the variable upper limit integral and partial sums. As applications, an inequality obtained from the special case of the half-discrete Hilbert-type inequality is further investigated; moreover, the equivalent conditions of the best possible constant factor related to several parameters are proved.

A half-discrete Hilbert-type inequality in the whole plane related to the Riemann zeta function

By the use of Hermite–Hadamard’s inequality and weight functions, a half-discrete Hilbert-type inequality in the whole plane with the kernel of hyperbolic cotangent function and multi-parameters is given. The constant factor related to the Riemann zeta function is proved to be the best possible. The equivalent forms, two kinds of particular inequalities, the operator expressions and some equivalent reverses are considered.

A Half-Discrete Hilbert-Type Inequality in the Whole Plane with Multiparameters

By the use of weight functions and technique of real analysis, a new half-discrete Hilbert-type inequality in the whole plane with multiparameters and the best possible constant factor is given. Furthermore, the equivalent forms, two kinds of particular inequalities, and the operator expressions with the norm are considered.

A generalization of a half-discrete Hilbert’s inequality

Abstract Considering different parameters and by means of Hadamard’s inequality, we obtain new and more general half-discrete Hilbert-type inequalities. Then we extract from our results some special cases that have been proved previously by other authors.

A more accurate reverse half-discrete Hilbert-type inequality

Using the way of weight functions and the idea of introducing parameters, and by means of Hermite-Hadamard’s inequality, a more accurate reverse half-discrete Hilbert-type inequality with the non-homogeneous kernel and a best constant factor is established. In addition, its best extension with parameters, the equivalent forms, as well as some particular cases are given.

On more accurate reverse multidimensional half-discrete Hilbert-type inequalities

On more accurate reverse multidimensional half-discrete Hilbert-type inequalities

A New Multiple Half-Discrete Hilbert-Type Inequality with Parameters and a Best Possible Constant Factor

Using the proper parameters and the way of weight functions, a new multiple half-discrete Hilbert-type inequality with a best constant factor is given. The operator expressions, equivalent forms, reverse inequalities and particular cases are considered.