A new reverse half-discrete Mulholland-type inequality with a nonhomogeneous kernel

In this paper, a new reverse half-discrete Mulholland-type inequality with the nonhomogeneous kernel of the form h(v(x)lnn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$h(v(x)\ln n)$\end{document} and the best possible constant factor is obtained by using the weight functions and the technique of real analysis. The equivalent reverses are considered. As corollaries, we deduce some new equivalent reverse inequalities with the homogeneous kernel of the form kλ(v(x),lnn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$k_{\lambda }(v(x),\ln n)$\end{document}. A few particular cases are provided. Our new reverse half-discrete Mulholland-type inequality which has a nonhomogeneous kernel is more general than in the previous homogeneous kernel work. The harmonized integration will have more applications.


Introduction
Hilbert-type inequalities are a class of mathematical inequalities that generalize the classical analytic inequality.They have applications in various areas of mathematics such as functional analysis, operator theory, and time scales [1][2][3][4][5].
In 2016, Hong [21] discussed an equivalent description of (1) with a general homogeneous kernel related to some parameters and the optimal constant factors.Similar works were considered in the papers [22]- [23].Recently, Mulholland-type inequalities with homogeneous kernel were obtained in [24][25][26].However, only a few reverse Mulholland-type inequalities with parameters were given in [27].
In this paper, by means of the weight functions and the techniques of real analysis, a new reverse half-discrete Mulholland-type inequality with a general nonhomogeneous kernel of the form h(v(x) ln n) is given.The best possible constant factor and some equivalent reverses are considered.As a corollary, we deduce some new equivalent reverse inequalities with a general homogeneous kernel of the form k λ (v(x), ln n).A few particular cases are provided.Our new reverse Mulholland-type inequality with a nonhomogeneous kernel is more inclusive and encompasses previous studies that focused on homogeneous kernels.This advancement in harmonized integration is expected to have broader applications and implications.

Some lemmas
For N = {1, 2, . ..}, we define the following weight functions: Setting u = v(x) ln n, we find Lemma 1 Under the assumptions of Definition 1, we have the following inequalities: where θ (x) := 1 Proof From the assumption that h(u)u σ -1 (u > 0) is decreasing, we find that 1 y h(v(x) ln y) × ln σ -1 y is strictly decreasing with respect to y ∈ (1, ∞).In view of the decreasingness property of the series, for x ∈ R + , we have . Hence, we have (8).The lemma is proved.
By the reverse Hölder's inequality and (6), we have .
where F(σ ) (σ ∈ I) is called the dominating function.By Lebesgue dominated convergence theorem (cf.[29]), we have The lemma is proved.
Proof For any ε > 0 such that σ -ε q ∈ I, we set the following functions: (i) If p < 0 (0 < q < 1), then in view of the decreasingness property of the series, we have If there exists a positive constant k ≥ k(σ ) such that ( 13) is valid when we replace k(σ ) by k, then, in particular, we have εI 1 > εkL 1 , namely v( 1) In view of Lemma 3, k(σ ) is continuous.When ε → 0 + in the above inequality, we find k(σ ) ≥ k.Hence, k = k(σ ) is the best possible constant factor in (13).
(ii) If 0 < p < 1 (q < 0), then in view of the assumption and the decreasingness property of the series, we obtain If there exists a positive constant k ≥ k(σ ) such that ( 14) is valid when we replace k(σ ) by k, then, in particular, we have εI 1 > εkL 2 and v( 1) As ε → 0 + , in view of the continuity of k(σ ), we find k(σ ) ≥ k.Hence, k = k(σ ) is the best possible constant factor of (14).
The theorem is proved.

Some corollaries and particular cases
Replacing v(x) by v -1 (x) in Theorems 1 and 2, by Remark 1, setting we have the following corollary: (i) If p < 0 (0 < q < 1), then we have the following equivalent inequalities with the best possible constant factor k(σ ): .
(ii) If 0 < p < 1 (q < 0), and there exists a constant a > 0 such that θ λ (x) = O( 1 v a (x) ) (0 < x < 1), then we have the following equivalent reverse inequalities with the best possible constant factor k λ (σ ): Then we find that h(u) = 1 (1+u) λ is decreasing as a function of u > 0, σ , a = σ > 0, and I = (0, λ).In view of Theorems 1 and 2, as well as Corollary 2, we have two classes of equivalent reverse inequalities with the particular kernels and the best possible constant factor B(μ, σ ).
In particular, for v(x) = ln(1 + x), the related inequalities are called half-discrete Mulholland-type inequalities.
Then we find that h(u) = ln u u λ -1 is decreasing with respect to u > 0 (cf.[7]), We obtain In the same way, we can find that So we set a = σ 2 > 0 and I = (0, λ).In view of Theorems 1 and 2, as well as Corollary 2, we have two classes of equivalent reverse inequalities with the particular kernels and the best possible constant factor [ In view of the assumptions, we still find that is decreasing with respect to u > 0, and We consider For x ∈ (0, 1), there exist constants b, M > 0 such that bv(1) ln 2 ≤ 1 and In the same way, we can find that

Conclusions
In this paper, using the weight functions and the techniques of real analysis, a new reverse half-discrete Mulholland-type inequality with the nonhomogeneous kernel of the form h(v(x) ln n) is obtained in Theorem 1.The best possible constant factor and some equivalent reverses are considered in Theorem 2. In Corollary 2, we deduce some new equivalent reverse inequalities with the homogeneous kernel of the form k λ (v(x), ln n).Some particular cases are provided in Example 1.These lemmas and theorems provide an extensive account of such inequalities.
In contrast to the extensive research conducted on the homogeneous kernel, the investigation of nonhomogeneous kernel is complex and applied to obtain a reverse half-discrete Hilbert-type inequality.The new reverse half-discrete Mulholland-type inequality with the nonhomogeneous kernel transforms the field of study into a multidimensional space.It becomes even more comprehensive in applications and consequences.The potential impact of our research is to inequalities involving higher-order derivative functions and multiple upper limit functions.It would be a remarkable achievement in the framework of a half-discrete Hilbert-type inequality with a nonhomogeneous kernel.