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

Abstract

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 Hardy–Hilbert-Type Inequality Involving Parameters Composed of a Pair of Weight Coefficients with Their Sums

“…and the following inequality. (14) reduces to inequality (2); Furthermore, for η = η 1 = η 2 = 0, inequality (14) reduces to inequality (7). Hence, inequalities (11) and ( 14) are the generalizations of inequalities ( 2) and (7), respectively.…”

“…is the classical Beta function. Obviously, for λ = 1, λ 1 = 1 q , λ 2 = 1 p , inequality (7) reduces to the Hardy-Hilbert's inequality (1); for p = q = 2, λ 1 = λ 2 = λ 2 ,(7) reduces to (6). Recently, Huang, Wu and Yang [7] provided a half-discrete Hardy-Hilbert-type inequality, as follows:…”

“…Motivated by the above-mentioned inequalities (2), ( 7) and (8), in this paper, we establish a new inequality that contains parameters composed of a pair of weight coefficients η 1 and η 2 with their sum η (η ∈ [0, 1 2 ]). The obtained inequality is a unified generalization of inequalities ( 2) and (7), as well as a more accurate version of inequalities (2) and (7). The main technical approaches include the construction of weight coefficients and the use of Hermite-Hadamard's inequality and the Euler-Maclaurin summation formula for estimation.…”

Special Issue Editorial “Symmetry in the Mathematical Inequalities”

“…X. Huang et al, in the paper "A More Accurate Half-Discrete Hilbert-Type Inequality Involving One upper Limit Function and One Partial Sum" [6], constructed proper weight coefficients by virtue of the symmetry principle and used them to establish a more accurate half-discrete, Hilbert-type inequality involving one upper limit function and one partial sum. The authors proved the new inequality with the help of the Euler-Maclaurin summation formula and Abel's partial summation formula.…”