A more accurate multidimensional Hardy-Hilbert-type inequality

Bicheng Yang

doi:10.1016/j.jksus.2018.01.004

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Original article

31 (

); 164-170

doi:

10.1016/j.jksus.2018.01.004

A more accurate multidimensional Hardy-Hilbert-type inequality

Bicheng Yang

Department of Mathematics, Guangdong University of Education, Guangzhou, Guangdong 510303, PR China

Received: 2017-8-30, Accepted: 2018-1-15, Published: 2018-04-01

Disclaimer:
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.

Peer review under responsibility of King Saud University.

Abstract

Keywords

26D15

47A05

Hardy-Hilbert’s inequality

Weight coefficient

Hermite-Hadamard’s inequality

Equivalent form

Operator

1 Introduction

If $p > 1, \frac{1}{p} + \frac{1}{q} = 1, a_{m}, b_{n} ⩾ 0, a = {a_{m}}_{m = 1}^{\infty} \in l^{p}, b = {b_{n}}_{n = 1}^{\infty} \in l^{q}$ , $‖ a ‖_{p} = {(\sum_{m = 1}^{\infty} a_{m}^{p})}^{\frac{1}{p}} > 0, ‖ b ‖_{q} > 0$ , then we have the following well-known Hardy-Hilbert’s inequality

(1)

\sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} \frac{a_{m} b_{n}}{m + n} < \frac{π}{\sin (π / p)} ‖ a ‖_{p} ‖ b ‖_{q},

and the following more accurate Hardy-Hilbert’s inequality with the same best possible constant factor

\frac{π}{\sin (π / p)}

(cf. Hardy et al., 1934, Theorem 315, 323):

(2)

\sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} \frac{a_{m} b_{n}}{m + n - 1} < \frac{π}{\sin (π / p)} ‖ a ‖_{p} ‖ b ‖_{q} .

We still have the following Hilbert-type inequality with the best possible constant factor

{[\frac{π}{\sin (π / p)}]}^{2}

(cf. Hardy et al., 1934, Theorem 342):

(3)

\sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} \frac{\ln (m / n) a_{m} b_{n}}{m - n} < {[\frac{π}{\sin (π / p)}]}^{2} ‖ a ‖_{p} ‖ b ‖_{q} .

Inequalities (1), (2) and (3) are important in Analysis and its applications (cf. Hardy et al., 1934; Mitrinović et al., 1991; Yang, 2011 ).

Assuming that ${μ_{m}}_{m = 1}^{\infty}$ and ${υ_{n}}_{n = 1}^{\infty}$ are positive sequences with $U_{m} = \sum_{i = 1}^{m} μ_{i}, V_{n} = \sum_{j = 1}^{n} υ_{j} (m, n \in N = {1, 2, \dots}),$ We have the following Hardy-Hilbert-type inequality (cf. Hardy et al., 1934, Theorem 321):

(4)

\sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} \frac{a_{m} b_{n}}{U_{m} + V_{n}} < \frac{π}{\sin (π / p)} {(\sum_{m = 1}^{\infty} \frac{a_{m}^{p}}{μ_{m}^{p - 1}})}^{\frac{1}{p}} {(\sum_{n = 1}^{\infty} \frac{b_{n}^{q}}{ν_{n}^{q - 1}})}^{\frac{1}{q}} .

For

μ_{i} = υ_{j} = 1 (i, j \in N)

, inequality (4) reduces to (1).

In 2015, by using the transfer formula, Yang (2015) gave the following multidimensional Hilbert’s inequality: For $i_{0}, j_{0} \in N, α, β > 0$ , $\begin{matrix} ‖ x ‖_{α} : & = {(\sum_{k = 1}^{i_{0}} | x^{(k)} |^{α})}^{\frac{1}{α}} (x = (x^{(1)}, \dots, x^{(i_{0})}) \in R^{i_{0}}), \\ ‖ y ‖_{β} : & = {(\sum_{k = 1}^{j_{0}} | y^{(k)} |^{β})}^{\frac{1}{β}} (y = (y^{(1)}, \dots, y^{(j_{0})}) \in R^{j_{0}}), \end{matrix}$ $0 < λ_{1} ⩽ i_{0}, 0 < λ_{2} ⩽ j_{0}, λ_{1} + λ_{2} = λ, a_{m}, b_{n} ⩾ 0$ , we have

(5)

\sum_{n} \sum_{m} \frac{1}{‖ m ‖_{α}^{λ} + ‖ n ‖_{β}^{λ}} a_{m} b_{n} < K_{1}^{\frac{1}{p}} K_{2}^{\frac{1}{q}} {[\sum_{m} ‖ m ‖_{α}^{p (i_{0} - λ_{1}) - i_{0}} a_{m}^{p}]}^{\frac{1}{p}} {[\sum_{n} ‖ n ‖_{β}^{q (j_{0} - λ_{2}) - j_{0}} b_{n}^{q}]}^{\frac{1}{q}},

where

\sum_{m} = \sum_{m_{i_{0}} = 1}^{\infty} \dots \sum_{m_{1} = 1}^{\infty}

\sum_{n} = \sum_{n_{j_{0}} = 1}^{\infty} \dots \sum_{n_{1} = 1}^{\infty}

, the series in the right-hand side of (5) are positive values, and the best possible constant factor

K_{1}^{\frac{1}{p}} K_{2}^{\frac{1}{q}}

is indicated by

K_{1}^{\frac{1}{p}} K_{2}^{\frac{1}{q}} = {[\frac{Γ^{j_{0}} (\frac{1}{β})}{β^{j_{0} - 1} Γ (\frac{j_{0}}{β})}]}^{\frac{1}{p}} {[\frac{Γ^{i_{0}} (\frac{1}{α})}{α^{i_{0} - 1} Γ (\frac{i_{0}}{α})}]}^{\frac{1}{q}} \frac{π}{λ \sin (\frac{π λ_{1}}{λ})} .

With regards to the above assumptions, we still have the following multidimensional Hilbert-type inequality (cf. Yang, 2014):

(6)

\sum_{n} \sum_{m} \frac{\ln (‖ m ‖_{α} / ‖ n ‖_{β})}{‖ m ‖_{α}^{λ} - ‖ n ‖_{β}^{λ}} a_{m} b_{n} < K {[\sum_{m} ‖ m ‖_{α}^{p (i_{0} - λ_{1}) - i_{0}} a_{m}^{p}]}^{\frac{1}{p}} {[\sum_{n} ‖ n ‖_{β}^{q (j_{0} - λ_{2}) - j_{0}} b_{n}^{q}]}^{\frac{1}{q}},

where,

K = {[\frac{Γ^{j_{0}} (\frac{1}{β})}{β^{j_{0} - 1} Γ (\frac{j_{0}}{β})}]}^{\frac{1}{p}} {[\frac{Γ^{i_{0}} (\frac{1}{α})}{α^{i_{0} - 1} Γ (\frac{i_{0}}{α})}]}^{\frac{1}{q}} {[\frac{π}{λ \sin (\frac{π λ_{1}}{λ})}]}^{2}

is the best possible. For

i_{0} = j_{0} = λ = 1, λ_{1} = \frac{1}{q}, λ_{2} = \frac{1}{p}

, inequality (5) ((6)) reduces to (1) ((3)). Some other results on this type of inequalities and multiple inequalities were provided by Hong (2005), Krnić et al. (2008), Krnić and Vuković (2012), Rassias and Yang (2014), Shi and Yang (2015), Hong (2006, 2010), Perić and Vuković (2011), He (2015), Adiyasuren et al. (2016).

Recently, by using the weight coefficients, Huang (2015) gave an extension of (3) as follows: For $0 < λ_{1}, λ_{2} ⩽ 1, λ_{1} + λ_{2} = λ, a_{m}, b_{n} ⩾ 0$ ,

(7)

\sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} \frac{\ln (U_{m} / V_{n})}{U_{m}^{λ} - V_{n}^{λ}} a_{m} b_{n} < {[\frac{π}{λ \sin (\frac{π λ_{1}}{λ})}]}^{2} {[\sum_{m = 1}^{\infty} \frac{U_{m}^{p (1 - λ_{1}) - 1} a_{m}^{p}}{μ_{m}^{p - 1}}]}^{\frac{1}{p}} {[\sum_{n = 1}^{\infty} \frac{V_{n}^{q (1 - λ_{2}) - 1} b_{n}^{q}}{ν_{n}^{q - 1}}]}^{\frac{1}{q}},

where, the constant factor

{[\frac{π}{λ \sin (\frac{π λ_{1}}{λ})}]}^{2}

is the best possible (the series in the right-hand side of (7) are positive values). Another results on Hardy-Hilbert-type inequalities and Hilbert-type inequalities were given by Yang (2015), Shi and Yang (2015), Huang (2015), Wang et al. (2015), Yang and Chen (2016), Brnetić and Pečarić (2004), Krnić and Pečarić (2005), Krnic et al. (2005), Li et al. (2007), Laith (2008), Agarwal et al. (2015).

In this paper, by means of the weight coefficients, the transfer formula, Hermite-Hadamard’s inequality and the technique of analysis, a more accurate multidimensional Hardy-Hilbert-type inequality with a best possible constant factor is given, which is an extension of (6) and (7). Meanwhile, the equivalent forms and the operator expressions are considered.

2 Some lemmas

If $μ_{i}^{(k)} > 0, 0 ⩽ {\tilde{μ}}_{i}^{(k)} ⩽ \frac{1}{2} μ_{i}^{(k)} (k = 1, \dots, i_{0}; i = 1, \dots, m), υ_{j}^{(l)} > 0, 0 ⩽ {\tilde{υ}}_{j}^{(l)} ⩽ \frac{1}{2} υ_{j}^{(l)}$ $(l = 1, \dots, j_{0}; j = 1, \dots, n)$ , then we set $\begin{matrix} U_{m}^{(k)} : & = \sum_{i = 1}^{m} μ_{i}^{(k)}, {\tilde{U}}_{m}^{(k)} ≔ U_{m}^{(k)} - {\tilde{μ}}_{m}^{(k)} (k = 1, \dots, i_{0}), \\ V_{n}^{(l)} : & = \sum_{j = 1}^{n} υ_{j}^{(l)}, {\tilde{V}}_{n}^{(l)} ≔ V_{n}^{(l)} - {\tilde{υ}}_{n}^{(l)} (l = 1, \dots, j_{0}), \end{matrix}$

(8)

\begin{matrix} U_{m} ≔ (U_{m}^{(1)}, \dots, U_{m}^{(i_{0})}), {\tilde{μ}}_{m} : = ({\tilde{μ}}_{m}^{(1)}, \dots, {\tilde{μ}}_{m}^{(i_{0})}), \\ {\tilde{U}}_{m} ≔ ({\tilde{U}}_{m}^{(1)}, \dots, {\tilde{U}}_{m}^{(i_{0})}) = U_{m} - {\tilde{μ}}_{m}, \\ V_{n} ≔ (V_{n}^{(1)}, \dots, V_{n}^{(j_{0})}), {\tilde{υ}}_{n} : = ({\tilde{υ}}_{n}^{(1)}, \dots, {\tilde{υ}}_{n}^{(j_{0})}), \\ {\tilde{V}}_{n} ≔ ({\tilde{V}}_{n}^{(1)}, \dots, {\tilde{V}}_{n}^{(j_{0})}) = V_{n} - {\tilde{υ}}_{n} (m, n \in N) . \end{matrix}

We also set functions

μ_{k} (t) ≔ μ_{m}^{(k)}, t \in (m - \frac{1}{2}, m + \frac{1}{2}) (m \in N); υ_{l} (t) ≔ υ_{n}^{(l)}, t \in (n - \frac{1}{2}, n + \frac{1}{2}) (n \in N)

, and

(9)

\begin{matrix} U_{k} (x) : & = \int_{\frac{1}{2}}^{x} μ_{k} (t) dt (k = 1, \dots, i_{0}), \\ V_{l} (y) : & = \int_{\frac{1}{2}}^{y} υ_{l} (t) dt (l = 1, \dots, j_{0}), \end{matrix}

(10)

\begin{matrix} U (x) : & = (U_{1} (x), \dots, U_{i_{0}} (x)), \\ V (y) : & = (V_{1} (y), \dots, V_{j_{0}} (y)) (x, y ⩾ \frac{1}{2}) . \end{matrix}

It follows that

U_{k} (m) = \int_{\frac{1}{2}}^{m} μ_{k} (t) dt = \int_{\frac{1}{2}}^{m + \frac{1}{2}} μ_{k} (t) dt - \frac{1}{2} μ_{m}^{(k)}

\begin{matrix} ⩽ & {\tilde{U}}_{m}^{(k)} ⩽ U_{k} (m + \frac{1}{2}) (k = 1, \dots, i_{0}; m \in N), \\ V_{l} (n) ⩽ & {\tilde{V}}_{n}^{(l)} ⩽ V_{l} (n + \frac{1}{2}) (l = 1, \dots, j_{0}; n \in N), \end{matrix}

and for

x \in (m - \frac{1}{2}, m + \frac{1}{2}), U_{k}^{'} (x) = μ_{k} (x) = μ_{m}^{(k)}

(k = 1, \dots, i_{0}; m \in N)

; for

y \in (n - \frac{1}{2}, n + \frac{1}{2}), V_{l}^{'} (y) = υ_{l} (y) = υ_{n}^{(l)}

(l = 1, \dots, j_{0}; n \in N)

Lemma 1

(cf. Yang and Chen, 2016) Suppose that $g (t) (> 0)$ is strictly decreasing and strictly convex in $(\frac{1}{2}, \infty)$ , satisfying $\int_{\frac{1}{2}}^{\infty} g (t) dt \in R_{+}$ . We have the following Hermite-Hadamard’s inequality

(11)

\int_{n}^{n + 1} g (t) dt < g (n) < \int_{n - \frac{1}{2}}^{n + \frac{1}{2}} g (t) dt (n \in N),

and then

(12)

\int_{1}^{\infty} g (t) dt < \sum_{n = 1}^{\infty} g (n) < \int_{\frac{1}{2}}^{\infty} g (t) dt .

Lemma 2

If $i_{0} \in N, α, M > 0, Ψ (u)$ is a non-negative measurable function in $(0, 1]$ , and

(13)

D_{M} ≔ \{x = (x_{1}, \dots, x_{i_{0}}) \in R_{+}^{i_{0}}; u = \sum_{i = 1}^{i_{0}} {(\frac{x_{i}}{M})}^{α} ⩽ 1\},

then we have the following transfer formula (cf. Hong, 2005):

(14)

\int \dots \int_{D_{M}} Ψ (\sum_{i = 1}^{i_{0}} {(\frac{x_{i}}{M})}^{α}) {dx}_{1} \dots {dx}_{s} = \frac{M^{i_{0}} Γ^{i_{0}} (\frac{1}{α})}{α^{i_{0}} Γ (\frac{i_{0}}{α})} \int_{0}^{1} Ψ (u) u^{\frac{i_{0}}{α} - 1} du .

Lemma 3

If $i_{0}, j_{0} \in N, α, β, ε > 0, μ_{m}^{(k)} ⩾ μ_{m + 1}^{(k)} (m \in N; k = 1, \dots, i_{0}), υ_{n}^{(l)} ⩾ υ_{n + 1}^{(l)}$ $(n \in N; l = 1, \dots, i_{0}), b ≔ \min_{1 ⩽ i ⩽ i_{0}, 1 ⩽ j ⩽ j_{0}} {μ_{1}^{(i)}, υ_{1}^{(j)}} (> 0)$ , then we have

(15)

\sum_{m} | {\tilde{U}}_{m} ‖_{α}^{- i_{0} - ε} \prod_{k = 1}^{i_{0}} μ_{m}^{(k)} ⩽ \frac{Γ^{i_{0}} (\frac{1}{α})}{ε b^{ε} i_{0}^{ε / α} α^{i_{0} - 1} Γ (\frac{i_{0}}{α})} + O (1),

(16)

\sum_{n} ‖ {\tilde{V}}_{n} ‖_{β}^{- j_{0} - ε} \prod_{k = 1}^{j_{0}} υ_{n}^{(k)} ⩽ \frac{Γ^{j_{0}} (\frac{1}{β})}{ε b^{ε} j_{0}^{ε / β} β^{j_{0} - 1} Γ (\frac{j_{0}}{β})} + \tilde{O} (1) (ε \to 0^{+}) .

Proof

For $M > {bi}_{0}^{1 / α}$ , we set $Ψ (u) = \{\begin{matrix} 0, & 0 < u < \frac{b^{α} i_{0}}{M^{α}}, \\ \frac{1}{{({Mu}^{1 / α})}^{i_{0} + ε}}, & \frac{b^{α} i_{0}}{M^{α}} ⩽ u ⩽ 1 . \end{matrix}$ By (14), it follows that $\begin{matrix} \int_{{x \in R_{+}^{i_{0}}; x_{i} ⩾ b}} \frac{dx}{‖ x ‖_{α}^{i_{0} + ε}} = \lim_{M \to \infty} \int \dots \int_{D_{M}} Ψ (\sum_{i = 1}^{i_{0}} {(\frac{x_{i}}{M})}^{α}) {dx}_{1} \dots {dx}_{i_{0}} \\ = & \lim_{M \to \infty} \frac{M^{i_{0}} Γ^{i_{0}} (\frac{1}{α})}{α^{i_{0}} Γ (\frac{i_{0}}{α})} \int_{b^{α} i_{0} / M^{α}}^{1} \frac{u^{\frac{i_{0}}{α} - 1}}{{({Mu}^{1 / α})}^{i_{0} + ε}} du = \frac{Γ^{i_{0}} (\frac{1}{α})}{ε b^{ε} i_{0}^{ε / α} α^{i_{0} - 1} Γ (\frac{i_{0}}{α})} . \end{matrix}$ In view of (12) and the above result, since $U_{k} (m) ⩽ {\tilde{U}}_{m}^{(k)}$ , we find $\begin{matrix} 0 & < \sum_{{m \in N^{i_{0}}; m_{i} ⩾ 2}} ‖ {\tilde{U}}_{m} ‖_{α}^{- i_{0} - ε} \prod_{k = 1}^{i_{0}} μ_{m}^{(k)} \\ ⩽ \sum_{{m \in N^{i_{0}}; m_{i} ⩾ 2}} \int_{{x \in R_{+}^{i_{0}}; m_{i} - \frac{1}{2} ⩽ x_{i} < m_{i} + \frac{1}{2}}} ‖ U (m) ‖_{α}^{- i_{0} - ε} \prod_{k = 1}^{i_{0}} μ_{m}^{(k)} dx \\ < \sum_{{m \in N^{i_{0}}; m_{i} ⩾ 2}} \int_{{x \in R_{+}^{i_{0}}; m_{i} - \frac{1}{2} ⩽ x_{i} < m_{i} + \frac{1}{2}}} ‖ U (x) ‖_{α}^{- i_{0} - ε} \prod_{k = 1}^{i_{0}} μ_{k} (x) dx \end{matrix}$ $\begin{matrix} = \int_{{x \in R_{+}^{i_{0}}; x_{i} ⩾ \frac{3}{2}}} ‖ U (x) ‖_{α}^{- i_{0} - ε} \prod_{k = 1}^{i_{0}} μ_{k} (x) dx \\ \overset{v = U (x)}{=} \int_{{v \in R_{+}^{i_{0}}; v_{i} ⩾ μ_{1}^{(i)}}} ‖ v ‖_{α}^{- i_{0} - ε} dv ⩽ \int_{{v \in R_{+}^{i_{0}}; v_{i} ⩾ b}} ‖ v ‖_{α}^{- i_{0} - ε} dv \\ = \frac{Γ^{i_{0}} (\frac{1}{α})}{ε b^{ε} i_{0}^{ε / α} α^{i_{0} - 1} Γ (\frac{i_{0}}{α})} . \end{matrix}$ For $i_{0} = 1, 0 < \sum_{{m \in N^{i_{0}}; m_{i} = 1}} ‖ {\tilde{U}}_{m} ‖_{α}^{- i_{0} - ε} \prod_{k = 1}^{i_{0}} μ_{m}^{(k)} ⩽ {(μ_{1}^{(1)})}^{- ε} < \infty$ ; for $i_{0} ⩾ 2$ , we set $H_{i} ≔ \sum_{{m \in N^{i_{0}}; m_{i} = 1}} ‖ {\tilde{U}}_{m} ‖_{α}^{- i_{0} - ε} \prod_{k = 1}^{i_{0}} μ_{m}^{(k)} (i = 1, \dots, i_{0}) .$ Without lose of generality, we estimate $H_{i_{0}}$ as follows: $\begin{matrix} H_{i_{0}} & ⩽ \sum_{{m \in N^{i_{0}}; m_{i_{0}} = 1}} ‖ U (m) ‖_{α}^{- i_{0} - ε} \prod_{k = 1}^{i_{0}} μ_{m}^{(k)} \\ = μ_{1}^{(i_{0})} \sum_{m \in N^{i_{0} - 1}} \frac{\prod_{k = 1}^{i_{0} - 1} μ_{m}^{(k)}}{{[\sum_{i = 1}^{i_{0} - 1} U_{i}^{α} (m) + {(\frac{1}{2} μ_{1}^{(i_{0})})}^{α}]}^{\frac{1}{α} (i_{0} + ε)}} \\ < \sum_{m \in N^{i_{0} - 1}} \int_{{x \in R_{+}^{i_{0} - 1}; m_{i} - \frac{1}{2} ⩽ x_{i} < m_{i} + \frac{1}{2}}} \frac{μ_{1}^{(i_{0})} \prod_{k = 1}^{i_{0} - 1} μ_{k} (x) dx}{{[\sum_{i = 1}^{i_{0} - 1} U_{i}^{α} (x) + {(\frac{1}{2} μ_{1}^{(i_{0})})}^{α}]}^{\frac{1}{α} (i_{0} + ε)}} \\ = μ_{1}^{(i_{0})} \int_{{x \in R^{i_{0} - 1}; x_{i} ⩾ \frac{1}{2}}} \frac{\prod_{k = 1}^{i_{0} - 1} μ_{k} (x)}{{[\sum_{i = 1}^{i_{0} - 1} U_{i}^{α} (x) + {(\frac{1}{2} μ_{1}^{(i_{0})})}^{α}]}^{\frac{1}{α} (i_{0} + ε)}} dx \\ \overset{v = U (x)}{⩽} μ_{1}^{(i_{0})} \int_{R_{+}^{i_{0} - 1}} \frac{1}{{[M^{α} \sum_{i = 1}^{i_{0} - 1} {(\frac{v_{i}}{M})}^{α} + {(\frac{1}{2} μ_{1}^{(i_{0})})}^{α}]}^{\frac{1}{α} (i_{0} + ε)}} dv . \end{matrix}$ By (14), we find $0 < H_{i_{0}} ⩽ μ_{1}^{(i_{0})} \lim_{M \to \infty} \frac{M^{i_{0} - 1} Γ^{i_{0} - 1} (\frac{1}{α})}{α^{i_{0} - 1} Γ (\frac{i_{0} - 1}{α})} \int_{0}^{1} \frac{u^{\frac{i_{0} - 1}{α} - 1} du}{{[M^{α} u + {(\frac{1}{2} μ_{1}^{(i_{0})})}^{α}]}^{\frac{i_{0} + ε}{α}}}$ $\begin{matrix} \overset{t = \frac{M^{α} u}{{(\frac{1}{2} μ_{1}^{(i_{0})})}^{α}}}{=} \frac{2^{1 + ε}}{{(μ_{1}^{(i_{0})})}^{ε}} \frac{Γ^{i_{0} - 1} (\frac{1}{α})}{α^{i_{0} - 1} Γ (\frac{i_{0} - 1}{α})} \int_{0}^{\infty} \frac{t^{\frac{i_{0} - 1}{α} - 1} dt}{{(t + 1)}^{\frac{i_{0} + ε}{α}}} \\ = & \frac{2^{1 + ε}}{{(μ_{1}^{(i_{0})})}^{ε}} \frac{Γ^{i_{0} - 1} (\frac{1}{α})}{α^{i_{0} - 1} Γ (\frac{i_{0} - 1}{α})} B (\frac{i_{0} - 1}{α}, \frac{1 + ε}{α}) < \infty, \end{matrix}$ namely, $H_{i_{0}} = O_{i_{0}} (1)$ . Hence, we have $\sum_{m} ‖ {\tilde{U}}_{m} ‖_{α}^{- i_{0} - ε} \prod_{k = 1}^{i_{0}} μ_{m}^{(k)} ⩽ \sum_{{m \in N^{i_{0}}; m_{i} ⩾ 2}} ‖ {\tilde{U}}_{m} ‖_{α}^{- i_{0} - ε} \prod_{k = 1}^{i_{0}} μ_{m}^{(k)} + \sum_{i = 1}^{i_{0}} H_{i} ⩽ \frac{Γ^{i_{0}} (\frac{1}{α})}{ε b^{ε} i_{0}^{ε / α} α^{i_{0} - 1} Γ (\frac{i_{0}}{α})} + \sum_{i = 1}^{i_{0}} O_{i} (1) (ε \to 0^{+}),$ and then (15) follows. In the same way, we have (16). □

Definition 1

For $0 < α, β, λ ⩽ 1, λ_{1}, λ_{2} > 0, λ_{1} + λ_{2} = λ$ , we define weight coefficients $w (λ_{1}, n)$ and $W (λ_{2}, m)$ as follows:

(17)

w (λ_{1}, n) : = \sum_{m} \frac{\ln (‖ {\tilde{U}}_{m} ‖_{α} / ‖ {\tilde{V}}_{n} ‖_{β})}{‖ {\tilde{U}}_{m} ‖_{α}^{λ} - ‖ {\tilde{V}}_{n} ‖_{β}^{λ}} \frac{‖ {\tilde{V}}_{n} ‖_{β}^{λ_{2}}}{‖ {\tilde{U}}_{m} ‖_{α}^{i_{0} - λ_{1}}} \prod_{k = 1}^{i_{0}} μ_{m}^{(k)},

(18)

W (λ_{2}, m) : = \sum_{n} \frac{\ln (‖ {\tilde{U}}_{m} ‖_{α} / ‖ {\tilde{V}}_{n} ‖_{β})}{‖ {\tilde{U}}_{m} ‖_{α}^{λ} - ‖ {\tilde{V}}_{n} ‖_{β}^{λ}} \frac{‖ {\tilde{U}}_{m} ‖_{α}^{λ_{1}}}{‖ {\tilde{V}}_{n} ‖_{β}^{j_{0} - λ_{2}}} \prod_{l = 1}^{j_{0}} υ_{n}^{(l)} .

Example 1

Setting $g (t) ≔ \frac{\ln t}{t - 1} (t > 0), g (1) = \lim_{t \to 1} \frac{\ln t}{t - 1} = 1$ , we find $\begin{matrix} g (t) = & \frac{\ln [1 + (t - 1)]}{t - 1} = \sum_{k = 0}^{\infty} {(- 1)}^{k} \frac{{(t - 1)}^{k}}{k + 1} \\ = & \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k} k!}{k + 1} \frac{{(t - 1)}^{k}}{k!} (- 1 < t - 1 ⩽ 1), \end{matrix}$ and then, $g^{(k)} (1) = \frac{{(- 1)}^{k} k!}{k + 1} (k = 0, 1, \dots), g^{'} (1) = - \frac{1}{2}, g^{″} (1) = \frac{2}{3}$ .

We put $h (t) ≔ t - 1 - t \ln t$ , and find $g^{'} (t) = \frac{h (t)}{t {(t - 1)}^{2}} (t \in R_{+} ⧹ {1})$ . Since $h^{'} (t) = - \ln t > 0 (0 < t < 1); h^{'} (t) = - \ln t < 0 (t > 1)$ , we have $\max h (t) = h (1) = 0$ and $g^{'} (t) < 0 (t > 0)$ , with $g^{'} (1) = - \frac{1}{2} < 0$ .

We put $J (t) ≔ - {(t - 1)}^{2} - 2 t (t - 1) + 2 t^{2} \ln t$ , and find $g^{″} (t) = \frac{J (t)}{t^{2} {(t - 1)}^{3}} (t \in R_{+} ⧹ {1})$ . Since $J^{'} (t) = - 4 (t - 1) + 4 t \ln t, J^{″} (t) = 4 \ln t < 0 (0 < t < 1); J^{″} (t) = 4 \ln t > 0 (t > 1)$ , we have $\min J^{'} (t) = J^{'} (1) = 0, J^{'} (t) > 0 (t \in R_{+} ⧹ {1})$ and $J (t)$ is strict decreasing in $R_{+}$ . Since $J (1) = 0$ , we have $J (t) < 0 (0 < t < 1); J (t) > 0 (t > 1)$ and $g^{″} (t) = \frac{J (t)}{t^{2} {(t - 1)}^{3}} > 0 (t > 0)$ , with $g^{″} (1) = \frac{2}{3} > 0$ .

For $0 < λ ⩽ 1$ , we set $G (u) = \frac{1}{λ} g (u^{λ}) = \frac{\ln u}{u^{λ} - 1} (u > 0)$ . Then we obtain $G^{'} (u) = g^{'} (u^{λ}) u^{λ - 1} < 0; G^{″} (u) = g^{″} (u^{λ}) u^{2 λ - 2} + (λ - 1) g^{'} (u^{λ}) u^{λ - 2} > 0$ .

With regards to the assumptions of Definition 1, we set $k_{λ} (x, y) = \frac{\ln (x / y)}{x^{λ} - y^{λ}} (x, y > 0)$ , and find $\begin{matrix} \frac{\partial}{\partial x} k_{λ} (x, y) = & \frac{\partial}{\partial x} y^{λ} G (\frac{x}{y}) < 0, \\ \frac{\partial^{2}}{\partial x^{2}} k_{λ} (x, y) = & \frac{\partial^{2}}{\partial x^{2}} y^{λ} G (\frac{x}{y}) > 0; \end{matrix}$ $\begin{matrix} \frac{\partial}{\partial y} k_{λ} (x, y) = & \frac{\partial}{\partial y} x^{λ} G (\frac{y}{x}) < 0, \\ \frac{\partial^{2}}{\partial y^{2}} k_{λ} (x, y) = & \frac{\partial^{2}}{\partial y^{2}} x^{λ} G (\frac{y}{x}) > 0 . \end{matrix}$

In the same way, since $0 < λ_{1} < 1 ⩽ i_{0}, 0 < λ_{2} < 1 ⩽ j_{0}$ , we still can find that $k_{λ} (x, y) \frac{1}{x^{i_{0} - λ_{1}}}$ $(k_{λ} (x, y) \frac{1}{y^{j_{0} - λ_{2}}})$ is strictly decreasing and strictly convex in $x \in (0, \infty) (y \in (0, \infty))$ , satisfying $\begin{matrix} \frac{\partial}{\partial x} (k_{λ} (x, y) \frac{1}{x^{i_{0} - λ_{1}}}) < & 0, \frac{\partial^{2}}{\partial x^{2}} (k_{λ} (x, y) \frac{1}{x^{i_{0} - λ_{1}}}) > 0; \\ \frac{\partial}{\partial y} (k_{λ} (x, y) \frac{1}{y^{j_{0} - λ_{2}}}) < & 0, \frac{\partial^{2}}{\partial y^{2}} (k_{λ} (x, y) \frac{1}{y^{j_{0} - λ_{2}}}) > 0 . \end{matrix}$

We obtain

(19)

\begin{matrix} k (λ_{1}) : & = \int_{0}^{\infty} k_{λ} (u, 1) \frac{du}{u^{1 - λ_{1}}} = \int_{0}^{\infty} \frac{u^{λ_{1} - 1} \ln u}{u^{λ} - 1} du \\ \overset{v = u^{λ}}{=} & \frac{1}{λ^{2}} \int_{0}^{\infty} \frac{v^{(λ_{1} / λ) - 1} dv}{v - 1} = {[\frac{π}{λ \sin (\frac{π λ_{1}}{λ})}]}^{2} \in R_{+} . \end{matrix}

(ii) If ${(- 1)}^{i} h^{(i)} (t) > 0 (t > 0; i = 0, 1, 2), A > 0, 0 < α ⩽ 1$ , then we have $\frac{d}{dx} h ({(A + x^{α})}^{\frac{1}{α}}) = h^{'} ({(A + x^{α})}^{\frac{1}{α}}) {(A + x^{α})}^{\frac{1}{α} - 1} x^{α - 1} < 0,$ $\begin{matrix} \frac{d^{2}}{{dx}^{2}} h ({(A + x^{α})}^{\frac{1}{α}}) = h^{″} ({(A + x^{α})}^{\frac{1}{α}}) {(A + x^{α})}^{\frac{2}{α} - 2} x^{2 α - 2} \\ + (1 - α) h^{'} ({(A + x^{α})}^{\frac{1}{α}}) {(A + x^{α})}^{\frac{1}{α} - 2} x^{2 α - 2} \\ + (α - 1) h^{'} ({(A + x^{α})}^{\frac{1}{α}}) {(A + x^{α})}^{\frac{1}{α} - 1} x^{α - 2} \\ = & h^{″} ({(A + x^{α})}^{\frac{1}{α}}) {(A + x^{α})}^{\frac{2}{α} - 2} x^{2 α - 2} \\ + A (α - 1) h^{'} ({(A + x^{α})}^{\frac{1}{α}}) {(A + x^{α})}^{\frac{1}{α} - 2} x^{α - 2} > 0 (x > 0) . \end{matrix}$

Hence, by (11), for $m_{i} - \frac{1}{2} < x_{i} < m_{i} + \frac{1}{2} (i = 1, \dots, i_{0}; m \in N)$ , we have $\prod_{k = 1}^{i_{0}} μ_{m}^{(k)} = \prod_{k = 1}^{i_{0}} μ_{k} (x)$ and $\frac{\ln (‖ U (m) ‖_{α} / ‖ {\tilde{V}}_{n} ‖_{β})}{‖ U (m) ‖_{α}^{λ} - ‖ {\tilde{V}}_{n} ‖_{β}^{λ}} ‖ U (m) ‖_{α}^{λ_{1} - i_{0}} \prod_{k = 1}^{i_{0}} μ_{m}^{(k)} < \int_{{x \in R_{+}^{i_{0}}; m_{i} - \frac{1}{2} < x_{i} < m_{i} - \frac{1}{2}}} \frac{\ln (‖ U (x) ‖_{α} / ‖ {\tilde{V}}_{n} ‖_{β})}{‖ U (x) ‖_{α}^{λ} - ‖ {\tilde{V}}_{n} ‖_{β}^{λ}} ‖ U (x) ‖_{α}^{λ_{1} - i_{0}} \prod_{k = 1}^{i_{0}} μ_{k} (x) dx .$

Lemma 4

With regards to the assumptions of Definition 1, (i) we have

(20)

w (λ_{1}, n) < K_{β} (λ_{1}) (n \in N^{j_{0}}),

(21)

W (λ_{2}, m) < K_{α} (λ_{1}) (m \in N^{i_{0}}),

where,

(22)

K_{β} (λ_{1}) = \frac{Γ^{j_{0}} (\frac{1}{β})}{β^{j_{0} - 1} Γ (\frac{j_{0}}{β})} k (λ_{1}), K_{α} (λ_{1}) = \frac{Γ^{i_{0}} (\frac{1}{α})}{α^{i_{0} - 1} Γ (\frac{i_{0}}{α})} k (λ_{1});

(ii) for $μ_{m}^{(k)} ⩾ μ_{m + 1}^{(k)}$ $(m \in N), υ_{n}^{(l)} ⩾ υ_{n + 1}^{(l)} (n \in N), U_{\infty}^{(k)} = V_{\infty}^{(l)} = \infty (k = 1, \dots, i_{0}, l = 1, \dots, j_{0})$ , we have

(23)

0 < K_{α} (λ_{1}) (1 - θ_{λ} (n)) < w (λ_{1}, n) (n \in N^{j_{0}}),

where, for

c ≔ \max_{1 ⩽ k ⩽ i_{0}} {μ_{1}^{(k)}} (> 0)

(24)

0 < θ_{λ} (n) ≔ \frac{1}{λ^{2} k (λ_{1})} \int_{0}^{c^{λ} i_{0}^{λ / α} / ‖ {\tilde{V}}_{n} ‖_{β}^{λ}} \frac{\ln t}{t - 1} t^{\frac{λ_{1}}{λ} - 1} dt = O (\frac{1}{‖ {\tilde{V}}_{n} ‖_{β}^{λ_{1} / 2}}) .

Proof

(i) Since $‖ {\tilde{U}}_{m} ‖_{α} ⩾ ‖ U (m) ‖_{α}$ , by (12), (14) and Example 1 (ii), for $0 < λ_{1} < 1 ⩽ i_{0}, λ > 0$ , it follows that $\begin{matrix} w (λ_{1}, n) & ⩽ \sum_{m} \frac{\ln (‖ U (m) ‖_{α} / ‖ {\tilde{V}}_{n} ‖_{β})}{‖ U (m) ‖_{α}^{λ} - ‖ {\tilde{V}}_{n} ‖_{β}^{λ}} \frac{‖ {\tilde{V}}_{n} ‖_{β}^{λ_{2}}}{‖ U (m) ‖_{α}^{i_{0} - λ_{1}}} \prod_{k = 1}^{i_{0}} μ_{m}^{(k)} \\ < \sum_{m} \int_{{x \in R_{+}^{i_{0}}; m_{i} - \frac{1}{2} < x_{i} ⩽ m_{i} + \frac{1}{2}}} \frac{\ln (‖ U (x) ‖_{α} / ‖ {\tilde{V}}_{n} ‖_{β})}{‖ U (x) ‖_{α}^{λ} - ‖ {\tilde{V}}_{n} ‖_{β}^{λ}} \\ \times \frac{‖ {\tilde{V}}_{n} ‖_{β}^{λ_{2}}}{‖ U (x) ‖_{α}^{i_{0} - λ_{1}}} \prod_{k = 1}^{i_{0}} μ_{k} (x) dx \end{matrix}$ $\begin{matrix} = & \int_{{x \in R_{+}^{i_{0}}; x_{i} > \frac{1}{2}}} \frac{\ln (‖ U (x) ‖_{α} / ‖ {\tilde{V}}_{n} ‖_{β})}{‖ U (x) ‖_{α}^{λ} - ‖ {\tilde{V}}_{n} ‖_{β}^{λ}} \frac{‖ {\tilde{V}}_{n} ‖_{β}^{λ_{2}}}{‖ U (x) ‖_{α}^{i_{0} - λ_{1}}} \prod_{k = 1}^{i_{0}} μ_{k} (x) dx \\ \overset{v = U (x)}{=} & \int_{R_{+}^{i_{0}}} \frac{\ln (‖ v ‖_{α} / ‖ {\tilde{V}}_{n} ‖_{β})}{‖ v ‖_{α}^{λ} - ‖ {\tilde{V}}_{n} ‖_{β}^{λ}} ‖ v ‖_{α}^{λ_{1} - i_{0}} ‖ {\tilde{V}}_{n} ‖_{β}^{λ_{2}} dv \end{matrix}$ $\begin{matrix} = & \lim_{M \to \infty} \int_{D_{M}} \frac{\ln (M {[\sum_{i = 1}^{i_{0}} {(\frac{v_{i}}{M})}^{α}]}^{1 / α} / ‖ {\tilde{V}}_{n} ‖_{β}) ‖ {\tilde{V}}_{n} ‖_{β}^{λ_{2}}}{M^{λ} {[\sum_{i = 1}^{i_{0}} {(\frac{v_{i}}{M})}^{α}]}^{λ / α} - ‖ {\tilde{V}}_{n} ‖_{β}^{λ}} {\{M {[\sum_{i = 1}^{j_{0}} {(\frac{v_{i}}{M})}^{α}]}^{\frac{1}{α}}\}}^{λ_{1} - i_{0}} dv \\ = & \lim_{M \to \infty} \frac{M^{i_{0}} Γ^{i_{0}} (\frac{1}{α})}{α^{i_{0}} Γ (\frac{i_{0}}{α})} \int_{0}^{1} \frac{\ln ({Mu}^{1 / α} / ‖ {\tilde{V}}_{n} ‖_{β}) ‖ {\tilde{V}}_{n} ‖_{β}^{λ_{2}}}{M^{λ} u^{λ / α} - ‖ {\tilde{V}}_{n} ‖_{β}^{λ}} {({Mu}^{1 / α})}^{λ_{1} - i_{0}} u^{\frac{i_{0}}{α} - 1} du \end{matrix}$ $\overset{t = \frac{M^{λ} u^{λ / α}}{‖ {\tilde{V}}_{n} ‖_{β}^{λ}}}{=} \frac{Γ^{i_{0}} (\frac{1}{α})}{λ^{2} α^{i_{0} - 1} Γ (\frac{i_{0}}{α})} \int_{0}^{\infty} \frac{t^{\frac{λ_{1}}{λ} - 1} \ln t}{t - 1} dt = \frac{Γ^{i_{0}} (\frac{1}{α}) k (λ_{1})}{α^{i_{0} - 1} Γ (\frac{i_{0}}{α})} = K_{α} (λ_{1}) .$ Hence, we have (20). In the same way, we have (21).

(ii) Since for $m_{i} ⩽ x_{i} < m_{i} + \frac{1}{2}, μ_{m_{i}}^{(k)} ⩾ μ_{m_{i} + 1}^{(k)} = μ_{k} (x + \frac{1}{2})$ ; for $m_{i} + \frac{1}{2} ⩽ x_{i} < m_{i} + 1, μ_{m}^{(k)} = μ_{k} (x + \frac{1}{2})$ , by (12) and in the same way, for $c = \max_{1 ⩽ k ⩽ i_{0}} {μ_{1}^{(k)}} (> 0)$ , we have $\begin{matrix} w (λ_{1}, n) & ⩾ \sum_{m} \frac{\ln (‖ U (m + \frac{1}{2}) ‖_{α} / ‖ {\tilde{V}}_{n} ‖_{β}) ‖ {\tilde{V}}_{n} ‖_{β}^{λ_{2}}}{‖ U (m + \frac{1}{2}) ‖_{α}^{λ} - ‖ {\tilde{V}}_{n} ‖_{β}^{λ}} ‖ U (m + \frac{1}{2}) ‖_{α}^{λ_{1} - i_{0}} \prod_{k = 1}^{i_{0}} μ_{m}^{(k)} \\ > \sum_{m} \int_{{x \in R_{+}^{i_{0}}; m_{i} ⩽ x_{i} < m_{i} + 1}} \frac{\ln (‖ U (x + \frac{1}{2}) ‖_{α} / ‖ {\tilde{V}}_{n} ‖_{β}) ‖ {\tilde{V}}_{n} ‖_{β}^{λ_{2}}}{‖ U (x + \frac{1}{2}) ‖_{α}^{λ} - ‖ {\tilde{V}}_{n} ‖_{β}^{λ}} \times ‖ U (x + \frac{1}{2}) ‖_{α}^{λ_{1} - i_{0}} \prod_{k = 1}^{i_{0}} μ_{k} (x + \frac{1}{2}) dx \\ = \int_{{[1, \infty)}^{i_{0}}} \frac{\ln (‖ U (x + \frac{1}{2}) ‖_{α} / ‖ {\tilde{V}}_{n} ‖_{β}) ‖ {\tilde{V}}_{n} ‖_{β}^{λ_{2}}}{‖ U (x + \frac{1}{2}) ‖_{α}^{λ} - ‖ {\tilde{V}}_{n} ‖_{β}^{λ}} \times ‖ U (x + \frac{1}{2}) ‖_{α}^{λ_{1} - i_{0}} \prod_{k = 1}^{i_{0}} μ_{k} (x + \frac{1}{2}) dx \\ \overset{v = U (x + \frac{1}{2})}{⩾} \int_{{[c, \infty)}^{i_{0}}} \frac{\ln (‖ v ‖_{α} / ‖ {\tilde{V}}_{n} ‖_{β}) ‖ {\tilde{V}}_{n} ‖_{β}^{λ_{2}}}{‖ v ‖_{α}^{λ} - ‖ {\tilde{V}}_{n} ‖_{β}^{λ}} ‖ v ‖_{α}^{λ_{1} - i_{0}} dv . \end{matrix}$ For $M > {ci}_{0}^{1 / α}$ , we set $Ψ (u) = \{\begin{matrix} 0, & 0 < u ⩽ \frac{c^{α} i_{0}}{M^{α}}, \\ \frac{\ln ({Mu}^{1 / α} / ‖ {\tilde{V}}_{n} ‖_{β})}{M^{λ} u^{λ / α} - ‖ {\tilde{V}}_{n} ‖_{β}^{λ}} {({Mu}^{1 / α})}^{λ_{1} - i_{0}}, & \frac{c^{α} i_{0}}{M^{α}} < u ⩽ 1 . \end{matrix}$ By (14), it follows that $\begin{matrix} \int_{{x \in R_{+}^{i_{0}}; x_{i} ⩾ c}} \frac{\ln (‖ x ‖_{α} / ‖ {\tilde{V}}_{n} ‖_{β}) ‖ {\tilde{V}}_{n} ‖_{β}^{λ_{2}}}{‖ x ‖_{α}^{λ} - ‖ {\tilde{V}}_{n} ‖_{β}^{λ}} ‖ x ‖_{α}^{λ_{1} - i_{0}} dx & = \lim_{M \to \infty} \int \dots \int_{D_{M}} Ψ (\sum_{i = 1}^{i_{0}} {(\frac{x_{i}}{M})}^{α}) {dx}_{1} \dots {dx}_{i_{0}} \\ = \lim_{M \to \infty} \frac{M^{i_{0}} Γ^{i_{0}} (\frac{1}{α})}{α^{i_{0}} Γ (\frac{i_{0}}{α})} \times \int_{c^{α} i_{0} / M^{α}}^{1} \frac{\ln ({Mu}^{1 / α} / ‖ {\tilde{V}}_{n} ‖_{β}) ‖ {\tilde{V}}_{n} ‖_{β}^{λ_{2}}}{M^{λ} u^{λ / α} - ‖ {\tilde{V}}_{n} ‖_{β}^{λ}} {({Mu}^{1 / α})}^{λ_{1} - i_{0}} u^{\frac{i_{0}}{α} - 1} du \\ \overset{t = \frac{M^{λ} u^{λ / α}}{‖ {\tilde{V}}_{n} ‖_{β}^{λ}}}{=} \frac{Γ^{i_{0}} (\frac{1}{α})}{α^{i_{0} - 1} Γ (\frac{i_{0}}{α}) λ^{2}} \int_{c^{λ} i_{0}^{λ / α} / ‖ {\tilde{V}}_{n} ‖_{β}^{λ}}^{\infty} \frac{\ln t}{t - 1} t^{\frac{λ_{1}}{λ} - 1} dt . \end{matrix}$ Hence, we have $\begin{matrix} w (λ_{1}, n) > & \frac{Γ^{i_{0}} (\frac{1}{α})}{α^{i_{0} - 1} Γ (\frac{i_{0}}{α}) λ^{2}} \int_{c^{λ} i_{0}^{λ / α} / ‖ {\tilde{V}}_{n} ‖_{β}^{λ}}^{\infty} \frac{\ln t}{t - 1} t^{\frac{λ_{1}}{λ} - 1} dt \\ = & K_{α} (λ_{1}) (1 - θ_{λ} (n)) > 0 . \end{matrix}$

Since $\frac{t^{λ_{1} / (2 λ)} \ln t}{t - 1} \to 0 (t \to 0^{+})$ , there exists a constant $M > 0$ , such that $\frac{t^{λ_{1} / (2 λ)} \ln t}{t - 1} ⩽ M (t \in (0, c^{λ} i_{0}^{λ / α} / ‖ {\tilde{V}}_{1} ‖_{β}^{λ})$ . We obtain $\begin{matrix} 0 & < θ_{λ} (n) = \frac{1}{λ^{2} k (λ_{1})} \int_{0}^{c^{λ} i_{0}^{λ / α} / ‖ {\tilde{V}}_{n} ‖_{β}^{λ}} \frac{\ln t}{t - 1} t^{\frac{λ_{1}}{λ} - 1} dt \\ ⩽ \frac{M}{λ^{2} k (λ_{1})} \int_{0}^{c^{λ} i_{0}^{λ / α} / ‖ {\tilde{V}}_{n} ‖_{β}^{λ}} t^{\frac{λ_{1}}{2 λ} - 1} dt = \frac{2 M}{λ λ_{1} k (λ_{1})} {(\frac{{ci}_{0}^{1 / α}}{‖ {\tilde{V}}_{n} ‖_{β}})}^{\frac{λ_{1}}{2}}, \end{matrix}$ and then (23) and (24) follow. □

3 Main results

Setting functions $\tilde{Φ} (m) ≔ \frac{‖ {\tilde{U}}_{m} ‖_{α}^{p (i_{0} - λ_{1}) - i_{0}}}{{(\prod_{k = 1}^{i_{0}} μ_{m}^{(k)})}^{p - 1}}, \tilde{Ψ} (n) ≔ \frac{‖ {\tilde{V}}_{n} ‖_{β}^{q (j_{0} - λ_{2}) - j_{0}}}{{(\prod_{l = 1}^{j_{0}} υ_{n}^{(l)})}^{q - 1}} (m \in N^{i_{0}}, n \in N^{j_{0}}),$ and the following normed spaces $\begin{matrix} l_{p, \tilde{Φ}} : & = \{a = {a_{m}}; ‖ a ‖_{p, \tilde{Φ}} ≔ {(\sum_{m} \tilde{Φ} (m) | a_{m} |^{p})}^{\frac{1}{p}} < \infty\}, \\ l_{q, \tilde{Ψ}} : & = \{b = {b_{n}}; ‖ b ‖_{q, \tilde{Ψ}} ≔ {(\sum_{n} \tilde{Ψ} (n) | b_{n} |^{q})}^{\frac{1}{q}} < \infty\}, \\ l_{p, {\tilde{Ψ}}^{1 - p}} : & \{c = {c_{n}}; ‖ c ‖_{p, {\tilde{Ψ}}^{1 - p}} ≔ {(\sum_{n} {\tilde{Ψ}}^{1 - p} (n) | c_{n} |^{p})}^{\frac{1}{p}} < \infty\}, \end{matrix}$ we have

Theorem 1

If $p > 1, \frac{1}{p} + \frac{1}{q} = 1, 0 < α, β, λ ⩽ 1, λ_{1}, λ_{2} > 0, λ_{1} + λ_{2} = λ$ , then for $a_{m}, b_{n} ⩾ 0, a = {a_{m}} \in l_{p, \tilde{Φ}}, b = {b_{n}} \in l_{q, \tilde{Ψ}}, ‖ a ‖_{p, \tilde{Φ}}, ‖ b ‖_{q, \tilde{Ψ}} > 0$ , we have the following equivalent inequalities

(25)

I ≔ \sum_{n} \sum_{m} \frac{\ln (‖ {\tilde{U}}_{m} ‖_{α} / ‖ {\tilde{V}}_{n} ‖_{β}) a_{m} b_{n}}{‖ {\tilde{U}}_{m} ‖_{α}^{λ} - ‖ {\tilde{V}}_{n} ‖_{β}^{λ}} < K_{β}^{\frac{1}{p}} (λ_{1}) K_{α}^{\frac{1}{q}} (λ_{1}) ‖ a ‖_{p, \tilde{Φ}} ‖ b ‖_{q, \tilde{Ψ}},

(26)

J : = {\{\sum_{n} \frac{\prod_{k = 1}^{j_{0}} υ_{n}^{(k)}}{‖ {\tilde{V}}_{n} ‖_{β}^{j_{0} - p λ_{2}}} {[\sum_{m} \frac{\ln (‖ {\tilde{U}}_{m} ‖_{α} / ‖ {\tilde{V}}_{n} ‖_{β}) a_{m}}{‖ {\tilde{U}}_{m} ‖_{α}^{λ} - ‖ {\tilde{V}}_{n} ‖_{β}^{λ}}]}^{p}\}}^{\frac{1}{p}} < K_{β}^{\frac{1}{p}} (λ_{1}) K_{α}^{\frac{1}{q}} (λ_{1}) ‖ a ‖_{p, \tilde{Φ}} .

where,

(27)

K_{β}^{\frac{1}{p}} (λ_{1}) K_{α}^{\frac{1}{q}} (λ_{1}) = {[\frac{Γ^{j_{0}} (\frac{1}{β})}{β^{j_{0} - 1} Γ (\frac{j_{0}}{β})}]}^{\frac{1}{p}} {[\frac{Γ^{i_{0}} (\frac{1}{α})}{α^{i_{0} - 1} Γ (\frac{i_{0}}{α})}]}^{\frac{1}{q}} {[\frac{π}{λ \sin (\frac{π λ_{1}}{λ})}]}^{2} .

Proof

By Hölder’s inequality with weight (cf. Kuang, 2004), we have $\begin{matrix} I & = \sum_{n} \sum_{m} \frac{\ln (‖ {\tilde{U}}_{m} ‖_{α} / ‖ {\tilde{V}}_{n} ‖_{β})}{‖ {\tilde{U}}_{m} ‖_{α}^{λ} - ‖ {\tilde{V}}_{n} ‖_{β}^{λ}} [\frac{‖ {\tilde{U}}_{m} ‖_{α}^{\frac{i_{0} - λ_{1}}{q}}}{‖ {\tilde{V}}_{n} ‖_{β}^{\frac{j_{0} - λ_{2}}{p}}} \frac{{(\prod_{l = 1}^{j_{0}} υ_{n}^{(l)})}^{\frac{1}{p}} a_{m}}{{(\prod_{k = 1}^{i_{0}} μ_{m}^{(k)})}^{\frac{1}{q}}}] \\ \times [\frac{‖ {\tilde{V}}_{n} ‖_{β}^{\frac{j_{0} - λ_{2}}{p}}}{‖ {\tilde{U}}_{m} ‖_{α}^{\frac{i_{0} - λ_{1}}{q}}} \frac{{(\prod_{k = 1}^{i_{0}} μ_{m}^{(k)})}^{\frac{1}{q}}}{{(\prod_{l = 1}^{j_{0}} υ_{n}^{(l)})}^{\frac{1}{p}}} b_{n}] \\ ⩽ {[\sum_{m} W (λ_{2}, m) \frac{‖ {\tilde{U}}_{m} ‖_{α}^{p (i_{0} - λ_{1}) - i_{0}} a_{m}^{p}}{{(\prod_{k = 1}^{i_{0}} μ_{m}^{(k)})}^{p - 1}}]}^{\frac{1}{p}} {[\sum_{n} w (λ_{1}, n) \frac{‖ {\tilde{V}}_{n} ‖_{β}^{q (j_{0} - λ_{2}) - j_{0}} b_{n}^{q}}{{(\prod_{l = 1}^{j_{0}} υ_{n}^{(l)})}^{q - 1}}]}^{\frac{1}{q}} . \end{matrix}$ Then by (20) and (21), we have (25). We set $b_{n} ≔ \frac{\prod_{l = 1}^{j_{0}} υ_{n}^{(l)}}{‖ {\tilde{V}}_{n} ‖_{β}^{j_{0} - p λ_{2}}} {[\sum_{m} \frac{\ln (‖ {\tilde{U}}_{m} ‖_{α} / ‖ {\tilde{V}}_{n} ‖_{β}) a_{m}}{‖ {\tilde{U}}_{m} ‖_{α}^{λ} - ‖ {\tilde{V}}_{n} ‖_{β}^{λ}}]}^{p - 1}, n \in N^{j_{0}} .$ Then we have $J = ‖ b ‖_{q, \tilde{Ψ}}^{q - 1}$ . Since the right-hand side of (26) is finite, it follows that $J < \infty$ . If $J = 0$ , then ( 26) is trivially valid; if $J > 0$ , then by (25), we have $\begin{matrix} ‖ b ‖_{q, \tilde{Ψ}}^{q} = & J^{p} = I < K_{β}^{\frac{1}{p}} (λ_{1}) K_{α}^{\frac{1}{q}} (λ_{1}) ‖ a ‖_{p, \tilde{Φ}} ‖ b ‖_{q, \tilde{Ψ}}, \\ ‖ b ‖_{q, \tilde{Ψ}}^{q - 1} = & J < K_{β}^{\frac{1}{p}} (λ_{1}) K_{α}^{\frac{1}{q}} (λ_{1}) ‖ a ‖_{p, \tilde{Φ}}, \end{matrix}$ namely, (26) follows. On the other hand, assuming that (26) is valid, by Hölder’s inequality (cf. Kuang, 2004), we have

(28)

I = \sum_{n} \frac{{(\prod_{l = 1}^{j_{0}} υ_{n}^{(l)})}^{1 / p}}{‖ {\tilde{V}}_{n} ‖_{β}^{(j_{0} / p) - λ_{2}}} \sum_{m} \frac{\ln (‖ {\tilde{U}}_{m} ‖_{α} / ‖ {\tilde{V}}_{n} ‖_{β}) a_{m}}{‖ {\tilde{U}}_{m} ‖_{α}^{λ} - ‖ {\tilde{V}}_{n} ‖_{β}^{λ}} \times \frac{‖ {\tilde{V}}_{n} ‖_{β}^{(j_{0} / p) - λ_{2}}}{{(\prod_{l = 1}^{j_{0}} υ_{n}^{(l)})}^{1 / p}} b_{n} ⩽ J ‖ b ‖_{q, \tilde{Ψ}} .

Then by (26), we have (25), which is equivalent to (26 ). □

Theorem 2

With regards to the assumptions of Theorem 1, if $μ_{m}^{(k)} ⩾ μ_{m + 1}^{(k)}$ $(m \in N), υ_{n}^{(l)} ⩾ υ_{n + 1}^{(l)}$ $(n \in N), U_{\infty}^{(k)} = V_{\infty}^{(l)} = \infty (k = 1, \dots, i_{0}, l = 1, \dots, j_{0})$ , then the constant factor $K_{β}^{\frac{1}{p}} (λ_{1}) K_{α}^{\frac{1}{q}} (λ_{1})$ in (25) and (26) is the best possible.

Proof

For $0 < ε < \min {p λ_{1}, p (1 - λ_{2})}, {\tilde{λ}}_{1} = λ_{1} - \frac{ε}{p} (\in (0, 1)), {\tilde{λ}}_{2} = λ_{2} + \frac{ε}{p} (< 1)$ , we set $\begin{matrix} \tilde{a} = & {{\tilde{a}}_{m}}, {\tilde{a}}_{m} ≔ ‖ {\tilde{U}}_{m} ‖_{α}^{- i_{0} + {\tilde{λ}}_{1}} \prod_{k = 1}^{i_{0}} μ_{m}^{(k)} (m \in N^{i_{0}}), \\ \tilde{b} = & {{\tilde{b}}_{n}}, {\tilde{b}}_{n} ≔ ‖ {\tilde{V}}_{n} ‖_{β}^{- j_{0} + {\tilde{λ}}_{2} - ε} \prod_{l = 1}^{j_{0}} υ_{n}^{(l)} (n \in N^{j_{0}}) . \end{matrix}$ Then by (15) and (16), we obtain $\begin{matrix} ‖ \tilde{a} ‖_{p, \tilde{Φ}} ‖ \tilde{b} ‖_{q, \tilde{Φ}} & = {[\sum_{m} \frac{‖ {\tilde{U}}_{m} ‖_{α}^{p (i_{0} - λ_{1}) - i_{0}} {\tilde{a}}_{m}^{p}}{{(\prod_{k = 1}^{i_{0}} μ_{m}^{(k)})}^{p - 1}}]}^{\frac{1}{p}} {[\sum_{n} \frac{‖ {\tilde{V}}_{n} ‖_{β}^{q (j_{0} - λ_{2}) - j_{0}} {\tilde{b}}_{n}^{q}}{{(\prod_{l = 1}^{j_{0}} υ_{n}^{(l)})}^{q - 1}}]}^{\frac{1}{q}} \\ = {(\sum_{m} ‖ {\tilde{U}}_{m} ‖_{α}^{- i_{0} - ε} \prod_{k = 1}^{i_{0}} μ_{m}^{(k)})}^{\frac{1}{p}} {(\sum_{n} ‖ {\tilde{V}}_{n} ‖_{β}^{- j_{0} - ε} \prod_{l = 1}^{j_{0}} υ_{n}^{(j)})}^{\frac{1}{q}} \\ ⩽ \frac{1}{ε} {(\frac{Γ^{i_{0}} (\frac{1}{α})}{b^{ε} i_{0}^{ε / α} α^{i_{0} - 1} Γ (\frac{i_{0}}{α})} + ε O (1))}^{\frac{1}{p}} {(\frac{Γ^{j_{0}} (\frac{1}{β})}{b^{ε} j_{0}^{ε / β} β^{j_{0} - 1} Γ (\frac{j_{0}}{β})} + ε \tilde{O} (1))}^{\frac{1}{q}} . \end{matrix}$ By (23) and (24), we find $\begin{matrix} \tilde{I} : & = \sum_{n} [\sum_{m} \frac{\ln (‖ {\tilde{U}}_{m} ‖_{α} / ‖ {\tilde{V}}_{n} ‖_{β}) {\tilde{a}}_{m}}{‖ {\tilde{U}}_{m} ‖_{α}^{λ} - ‖ {\tilde{V}}_{n} ‖_{β}^{λ}}] {\tilde{b}}_{n} = \sum_{n} \frac{w ({\tilde{λ}}_{1}, n)}{‖ {\tilde{V}}_{n} ‖_{β}^{j_{0} + ε}} \prod_{l = 1}^{j_{0}} υ_{n}^{(l)} \\ > & K_{α} ({\tilde{λ}}_{1}) \sum_{n} (1 - O (\frac{1}{‖ {\tilde{V}}_{n} ‖_{β}^{{\tilde{λ}}_{1} / 2}})) \frac{1}{‖ {\tilde{V}}_{n} ‖_{β}^{j_{0} + ε}} \prod_{l = 1}^{j_{0}} υ_{n}^{(l)} \\ = & K_{α} ({\tilde{λ}}_{1}) (\frac{Γ^{j_{0}} (\frac{1}{β})}{ε b^{ε} j_{0}^{ε / β} β^{j_{0} - 1} Γ (\frac{j_{0}}{β})} + \tilde{O} (1) - O_{1} (1)) . \end{matrix}$

If there exists a constant $K ⩽ K_{β}^{\frac{1}{p}} (λ_{1}) K_{α}^{\frac{1}{q}} (λ_{1})$ , such that (25) is valid when replacing $K_{β}^{\frac{1}{p}} (λ_{1}) K_{α}^{\frac{1}{q}} (λ_{1})$ by K, then we have $ε \tilde{I} < ε K ‖ \tilde{a} ‖_{p, \tilde{Φ}} ‖ \tilde{b} ‖_{q, \tilde{Φ}}$ , namely, $K_{α} (λ_{1} - \frac{ε}{p}) (\frac{Γ^{j_{0}} (\frac{1}{β})}{b^{ε} j_{0}^{ε / β} β^{j_{0} - 1} Γ (\frac{j_{0}}{β})} + ε \tilde{O} (1) - ε O_{1} (1)) < K {(\frac{Γ^{i_{0}} (\frac{1}{α})}{b^{ε} i_{0}^{ε / α} α^{i_{0} - 1} Γ (\frac{i_{0}}{α})} + ε O (1))}^{\frac{1}{p}} {(\frac{Γ^{j_{0}} (\frac{1}{β})}{b^{ε} j_{0}^{ε / β} β^{j_{0} - 1} Γ (\frac{j_{0}}{β})} + ε \tilde{O} (1))}^{\frac{1}{q}} .$ For $ε \to 0^{+}$ , it follows that $\frac{Γ^{j_{0}} (\frac{1}{β})}{β^{j_{0} - 1} Γ (\frac{j_{0}}{β})} \frac{Γ^{i_{0}} (\frac{1}{α}) k (λ_{1})}{α^{i_{0} - 1} Γ (\frac{i_{0}}{α})} ⩽ K {[\frac{Γ^{i_{0}} (\frac{1}{α})}{α^{i_{0} - 1} Γ (\frac{i_{0}}{α})}]}^{\frac{1}{p}} {[\frac{Γ^{j_{0}} (\frac{1}{β})}{β^{j_{0} - 1} Γ (\frac{j_{0}}{β})}]}^{\frac{1}{q}},$ and then $K_{β}^{\frac{1}{p}} (λ_{1}) K_{α}^{\frac{1}{q}} (λ_{1}) ⩽ K$ . Hence, $K = K_{β}^{\frac{1}{p}} (λ_{1}) K_{α}^{\frac{1}{q}} (λ_{1})$ is the best possible constant factor of (25). The constant factor in (26) is still the best possible. Otherwise, we would reach a contradiction by (28) that the constant factor in (25) is not the best possible. □

4 Operator expressions

With regards to the assumptions of Theorem 1, in view of $\begin{matrix} c_{n} : & = \frac{\prod_{k = 1}^{j_{0}} υ_{n}^{(k)}}{‖ {\tilde{V}}_{n} ‖_{β}^{j_{0} - p λ_{2}}} {[\sum_{m} \frac{\ln (‖ {\tilde{U}}_{m} ‖_{α} / ‖ {\tilde{V}}_{n} ‖_{β})}{‖ {\tilde{U}}_{m} ‖_{α}^{λ} - ‖ {\tilde{V}}_{n} ‖_{β}^{λ}} a_{m}]}^{p - 1}, n \in N^{j_{0}} \\ c = & {c_{n}}, ‖ c ‖_{p, {\tilde{Ψ}}^{1 - p}} = J < K_{β}^{\frac{1}{p}} (λ_{1}) K_{α}^{\frac{1}{q}} (λ_{1}) ‖ a ‖_{p, \tilde{Φ}} < \infty, \end{matrix}$ we can set the following definition:

Definition 2

Define a multidimensional Hilbert’s operator $T : l_{p, \tilde{Φ}} \to l_{p, {\tilde{Ψ}}^{1 - p}}$ as follows: For any $a \in l_{p, \tilde{Φ}}$ , there exists a unique representation $Ta = c \in l_{p, {\tilde{Ψ}}^{1 - p}}$ , satisfying

(29)

Ta (n) ≔ \sum_{m} \frac{\ln (‖ {\tilde{U}}_{m} ‖_{α} / ‖ {\tilde{V}}_{n} ‖_{β})}{‖ {\tilde{U}}_{m} ‖_{α}^{λ} - ‖ {\tilde{V}}_{n} ‖_{β}^{λ}} a_{m} (n \in N^{j_{0}}) .

For

b \in l_{q, \tilde{Ψ}}

, we define the following formal inner product of Ta and b as follows:

(30)

(Ta, b) ≔ \sum_{n} [\sum_{m} \frac{\ln (‖ {\tilde{U}}_{m} ‖_{α} / ‖ {\tilde{V}}_{n} ‖_{β})}{‖ {\tilde{U}}_{m} ‖_{α}^{λ} - ‖ {\tilde{V}}_{n} ‖_{β}^{λ}} a_{m}] b_{n} .

Then by Theorem 1, we have the following equivalent inequalities:

(31)

(Ta, b) < K_{β}^{\frac{1}{p}} (λ_{1}) K_{α}^{\frac{1}{q}} (λ_{1}) ‖ a ‖_{p, \tilde{Φ}} ‖ b ‖_{q, \tilde{Ψ}},

(32)

‖ Ta ‖_{p, {\tilde{Ψ}}^{1 - p}} < K_{β}^{\frac{1}{p}} (λ_{1}) K_{α}^{\frac{1}{q}} (λ_{1}) ‖ a ‖_{p, \tilde{Φ}} .

It follows that T is bounded with

(33)

‖ T ‖ ≔ \sup_{a (\neq θ) \in l_{p, \tilde{Φ}}} \frac{‖ Ta ‖_{p, {\tilde{Ψ}}^{1 - p}}}{‖ a ‖_{p, \tilde{Φ}}} ⩽ K_{β}^{\frac{1}{p}} (λ_{1}) K_{α}^{\frac{1}{q}} (λ_{1}) .

Since by Theorem 2, the constant factor

K_{β}^{\frac{1}{p}} (λ_{1}) K_{α}^{\frac{1}{q}} (λ_{1})

in (32) is the best possible, we have

(34)

\begin{matrix} ‖ T ‖ = & K_{β}^{\frac{1}{p}} (λ_{1}) K_{α}^{\frac{1}{q}} (λ_{1}) \\ = & {[\frac{Γ^{j_{0}} (\frac{1}{β})}{β^{j_{0} - 1} Γ (\frac{j_{0}}{β})}]}^{\frac{1}{p}} {[\frac{Γ^{i_{0}} (\frac{1}{α})}{α^{i_{0} - 1} Γ (\frac{i_{0}}{α})}]}^{\frac{1}{q}} {[\frac{π}{λ \sin (\frac{π λ_{1}}{λ})}]}^{2} . \end{matrix}

Remark 1

(i) For ${\tilde{μ}}_{i}^{(k)} = 0 (k = 1, \dots, i_{0}; i = 1, \dots, m), {\tilde{υ}}_{j}^{(l)} = 0$ $(l = 1, \dots, j_{0}; j = 1, \dots, n)$ , setting $\begin{matrix} Φ (m) : & = \frac{‖ U_{m} ‖_{α}^{p (i_{0} - λ_{1}) - i_{0}}}{{(\prod_{k = 1}^{i_{0}} μ_{m}^{(k)})}^{p - 1}} (m \in N^{i_{0}}), \\ Ψ (n) : & = \frac{‖ V_{n} ‖_{β}^{q (j_{0} - λ_{2}) - j_{0}}}{{(\prod_{l = 1}^{j_{0}} υ_{n}^{(l)})}^{q - 1}} (n \in N^{j_{0}}), \end{matrix}$ then (25) and (26) reduce the following equivalent inequalities with the same best possible constant factor $K_{β}^{\frac{1}{p}} (λ_{1}) K_{α}^{\frac{1}{q}} (λ_{1})$ :

(35)

\sum_{n} \sum_{m} \frac{\ln (‖ U_{m} ‖_{α} / ‖ V_{n} ‖_{β}) a_{m} b_{n}}{‖ U_{m} ‖_{α}^{λ} - ‖ V_{n} ‖_{β}^{λ}} < K_{β}^{\frac{1}{p}} (λ_{1}) K_{α}^{\frac{1}{q}} (λ_{1}) ‖ a ‖_{p, Φ} ‖ b ‖_{q, Ψ},

(36)

{\{\sum_{n} \frac{\prod_{k = 1}^{j_{0}} υ_{n}^{(k)}}{‖ V_{n} ‖_{β}^{j_{0} - p λ_{2}}} {[\sum_{m} \frac{\ln (‖ U_{m} ‖_{α} / ‖ V_{n} ‖_{β}) a_{m}}{‖ U_{m} ‖_{α}^{λ} - ‖ V_{n} ‖_{β}^{λ}}]}^{p}\}}^{\frac{1}{p}} < K_{β}^{\frac{1}{p}} (λ_{1}) K_{α}^{\frac{1}{q}} (λ_{1}) ‖ a ‖_{p, Φ} .

Hence, (25) and (26) are more accurate extensions of (35) and (36).

(ii) For $μ_{i}^{(k)} = 1 (k = 1, \dots, i_{0}; i = 1, \dots, m), υ_{j}^{(l)} = 1 (l = 1, \dots, j_{0}; j = 1, \dots, n),$ (35) reduces to (5); for $i_{0} = j_{0} = 1$ , (35) reduces to (6). Hence, (35) is an extension of (5) and (6); so is (25).

5 Conclusions

In this paper, by means of the weight coefficients, the transfer formula, Hermite-Hadamard’s inequality and the technique of real analysis, a more accurate multidimensional Hardy-Hilbert-type inequality with a best possible constant factor is given by Theorems 1 and 2, which is an extension of some published results. Moreover, the equivalent forms with the best possible constant factor are obtained by Theorems 1 and 2, and the operator expressions are also considered. The method of weight coefficients is very important, which helps us to prove the main inequalities with the best possible constant factor. The lemmas and theorems of this paper provide an extensive account of this type of inequalities.

Acknowledgments

This work is supported by the National Natural Science Foundation (No. 61370186, No. 61640222), and Appropriative Researching Fund for Professors and Doctors, Guangdong University of Education (No. 2015ARF25). We are grateful for their help.

References

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Mitrinović D.S., Pečarić J.E., Fink A.M., . Inequalities Involving Functions and their Integrals and Derivatives. Boston: Kluwer Acaremic Publishers; 1991.
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Introduction

Some lemmas

Main results

Operator expressions

Conclusions

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[1] Hardy G.H., Littlewood J.E., Pólya G., . Inequalities. Cambridge: Cambridge University Press; 1934.

[2] Mitrinović D.S., Pečarić J.E., Fink A.M., . Inequalities Involving Functions and their Integrals and Derivatives. Boston: Kluwer Acaremic Publishers; 1991.

[3] Yang B.C., . Discrete Hilbert-type Inequalities. The United Arab Emirates: Bentham Science Publishers Ltd.; 2011.

[4] Yang B.C., . On a more accurate multidimensional Hilbert-type inequality with parameters. Math. Inequalities Appl.. 2015;18(2):429-441.
[Google Scholar]

[5] Yang B.C., . A multidimensional discrete Hilbert-type inequality. Int. J. Nonlinear Anal. Appl.. 2014;5(1):80-88.
[Google Scholar]

[6] Hong Y., . On Hardy-Hilbert integral inequalities with some parameters. J. Ineq. Pure Appl. Math.. 2005;6(4):1-10. Art. 92
[Google Scholar]

[7] Krnić M., Pečarić J.E., Vuković P., . On some higher-dimensional Hilbert’s and Hardy-Hilbert’s type integral inequalities with parameters. Math. Inequal. Appl.. 2008;11:701-716.
[Google Scholar]

[8] Krnić M., Vuković P., . On a multidimensional version of the Hilbert-type inequality. Anal. Math.. 2012;38:291-303.
[Google Scholar]

[9] Rassias M.Th., Yang B.C., . On a multidimensional half - discrete Hilbert -type inequality related to the hyperbolic cotangent function. Appl. Math. Comput.. 2014;242:800-813.
[Google Scholar]

[10] Shi Y.P., Yang B.C., . On a multidimensional Hilbert-type inequality with parameters. J. Inequalities Appl.. 2015;2015:371.
[Google Scholar]

[11] Hong Y., . On multiple Hardy-Hilbert integral inequalities with some parameters. J. Inequalities Appl.. 2006;2006 Article ID 94960, 11 pages
[Google Scholar]

[12] Huang Q.L., . On a multiple Hilbert’s inequality with parameters. J. Inequalities Appl.. 2010;2010 Article ID 309319, 12 pages
[Google Scholar]

[13] Perić I., Vuković P., . Multiple Hilbert’s type inequalities with a homogeneous kernel. Banach J. Math. Anal.. 2011;5(2):33-43.
[Google Scholar]

[14] He B., . A multiple Hilbert-type discrete inequality with a new kernel and best possible constant factor. J. Math. Anal. Appl.. 2015;431 990–902
[Google Scholar]

[15] Adiyasuren V., Batbold T., Krnić M., . Multiple Hilbert-type inequalities involving some differential operators. Banach J. Math. Anal.. 2016;10(2):320-337.
[Google Scholar]

[16] Huang Q.L., . A new extension of a Hardy-Hilbert-type inequality. J. Inequalities Appl.. 2015;2015:397.
[Google Scholar]

[17] Yang B.C., . An extension of a Hardy-Hilbert-type inequality. J. Guangdong Univ. Educ.. 2015;35(3):1-7.
[Google Scholar]

[18] Shi Y.P., Yang B.C., . A new Hardy-Hilbert-type inequality with multi-parameters and a best possible constant factor. J. Inequalities Appl.. 2015;2015:380.
[Google Scholar]

[19] Huang Q.L., . A new extension of Hardy-Hilbert-type inequality. J. Inequalities Appl.. 2015;2015:397.
[Google Scholar]

[20] Wang A.Z., Huang Q.L., Yang B.C., . A strengthened Mulholland-type inequality with parameters. J. Inequalities Appl.. 2015;2015:329.
[Google Scholar]

[21] Yang B.C., Chen Q., . On a more accurate Hardy-Mulholland-type inequality. J. Inequalities Appl.. 2016;2016:82.
[Google Scholar]

[22] Brnetić I., Pečarić J., . Generalization of Hilbert’s integral inequality. Math. Inequalities Appl.. 2004;7(2):199-205.
[Google Scholar]

[23] Krnić M., Pečarić J., . General Hilbert’s and Hardy’s inequalities. Math. Inequalities Appl.. 2005;8(1):29-51.
[Google Scholar]

[24] Krnic M., Gao M.Z., Pecaric J., . On the best constant in Hilbert’s inequality. Math. Inequalities Appl.. 2005;8(2):317-329.
[Google Scholar]

[25] Li Y.J., Wang J.P., He B., . On further analogs of Hilbert’s inequality. J. Inequalities Appl.. 2007;2007 Article ID 76329, 6 pages
[Google Scholar]

[26] Laith E.A., . On some extensions of Hardy-Hilbert’s inequality and applications. J. Inequalities Appl.. 2008;2008 Article ID 546828, 14 pages
[Google Scholar]

[27] Agarwal R.P., O’Regan D., Saker S.H., . Some Hardy-type inequalities with weighted functions via Opial type inequalities. Adv. Dyn. Syst. Appl.. 2015;10:1-9.
[Google Scholar]

[28] Kuang J.C., . Applied Inequalities. Jinan, China: Shangdong Science Technic Press; 2004.