A Mulholland-type inequality in the whole plane with multi parameters

Bing He; Bicheng Yang

doi:10.1016/j.jksus.2018.04.029

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Original article

32 (

); 245-250

doi:

10.1016/j.jksus.2018.04.029

A Mulholland-type inequality in the whole plane with multi parameters

Bing He, Bicheng Yang^⁎

Department of Mathematics, Guangdong University of Education, Guangzhou, Guangdong 51003, PR China

⁎Corresponding author. bcyang@gdei.edu.cn (Bicheng Yang)

Received: 2018-3-31, Accepted: 2018-4-23, Published: 2018-01-01

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

By introducing independent parameters, and applying the weight coefficients, we give a new Mulholland-type inequality in the whole plane with a best possible constant factor. Moreover, the equivalent forms, a few particular cases and the operator expressions are considered.

Keywords

26D15

47A07

Mulholland-type inequality

Parameter

Weight coefficient

Equivalent form

Operator expression

1 Introduction

If $p > 1, \frac{1}{p} + \frac{1}{q} = 1, a_{m}, b_{n} ⩾ 0, 0 < \sum_{m = 1}^{\infty} a_{m}^{p} < \infty$ and $0 < \sum_{n = 1}^{\infty} b_{n}^{q} < \infty$ , then we have the following Hardy-Hilbert’s inequality (cf. Hardy et al., 1934):

(1)

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

with the best possible constant factor

\frac{π}{\sin (π / p)}

. The Mulholland’s inequality with the same best possible constant factor

\frac{π}{\sin (π / p)}

was provided as follows (cf. Theorem 343 of Hardy et al. (1934), replacing

\frac{a_{m}}{m}, \frac{b_{n}}{n}

a_{m}, b_{n}

(2)

\sum_{n = 2}^{\infty} \sum_{m = 2}^{\infty} \frac{a_{m} b_{n}}{ln mn} < \frac{π}{\sin (π / p)} {(\sum_{m = 2}^{\infty} \frac{a_{m}^{p}}{m^{1 - p}})}^{1 / p} {(\sum_{n = 2}^{\infty} \frac{b_{n}^{q}}{n^{1 - q}})}^{1 / q} .

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

In 2007, a Hilbert-type integral inequality in the whole plane was given as follows (cf. Wang and Yang, 2011):

(3)

\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \frac{min {1, | xy |} f (x) g (y)}{\sqrt{1 + xy + {(xy)}^{2}}} dxdy < 2 ln (3 + 2 \sqrt{3}) {(\int_{- \infty}^{\infty} | x | f^{2} (x) dx \int_{- \infty}^{\infty} | y | f^{2} (y) dy)}^{\frac{1}{2}},

where the constant factor

2 ln (3 + 2 \sqrt{3})

is the best possible. Some new results on inequalities (1)–(3) were obtained by Gao and Yang (1998), Yang et al. (2011), Krnić and Pečarić (2005), Perić and Vuković (2011), He (2015), Adiyasuren et al. (2016), Yang (2007), Li and He (2007), Krnić and Vukovic (2012), Huang (2015), Huang and Yang (2013), Huang et al. (2014a,b), Huang (2010). In 2016, Yang and Chen gave a more accurate extension of (1) in the whole plane as follows (cf. Zhong et al., 2017):

(4)

\sum_{|n| = 1}^{\infty} \sum_{|m| = 1}^{\infty} \frac{a_{m} b_{n}}{{(|m - ξ| + |n - η|)}^{λ}} < 2 B (λ_{1}, λ_{2}) {[\sum_{|m| = 1}^{\infty} {|m - ξ|}^{p (1 - λ_{1}) - 1} a_{m}^{p}]}^{\frac{1}{p}} {[\sum_{|n| = 1}^{\infty} {|n - η|}^{q (1 - λ_{2}) - 1} b_{n}^{q}]}^{\frac{1}{q}},

where the constant factor

2 B (λ_{1}, λ_{2}) (0 < λ_{1}, λ_{2} ⩽ 1, λ_{1} + λ_{2} = λ, ξ, η \in [0, \frac{1}{2}])

is the best possible. Another result on this kind of inequalities was provided by Xin et al. (2016).

In this paper, by introducing independent parameters, applying the weight coefficients, we give a Mulholland-type inequality in the whole plane with a best possible constant factor similar to (4) and the main result of Xin et al. (2016). Moreover, the equivalent forms, a few particular cases and the operator expressions are considered.

2 Some lemmas

In the following, we agree that $p > 1, \frac{1}{p} + \frac{1}{q} = 1, σ \in R = (- \infty, \infty), - σ < λ_{1}, λ_{2} ⩽ 1 - σ, λ_{1} + λ_{2} = λ, ρ ⩾ 1$ , $\arccos \sqrt{1 - \frac{1}{ρ}} ⩽ γ ⩽ π - \arccos \sqrt{1 - \frac{1}{ρ}} (γ = α, β),$ $ζ \in (- 1, 1)$ , satisfying $\frac{1}{ρ (1 - \cos γ)} - 1 ⩽ ζ ⩽ 1 - \frac{1}{ρ (1 + \cos γ)}$ $((ζ, γ) = (ξ, α)$ or $(η, β))$ , and

(5)

h_{γ} (λ_{1}) : = \frac{2 (λ + 2 σ) {csc}^{2} γ}{(λ_{1} + σ) (λ_{2} + σ)} \in R_{+} = (0, \infty) (γ = α, β) .

Remark 1

With regards to the above assumptions, it follows that $ρ (1 \pm ζ) (1 \mp \cos γ) ⩾ 1$ $((ζ, γ) = (ξ, α)$ or $(η, β))$ . In particular, for $ρ = 1$ , we have $α = β = \frac{π}{2}$ and $ξ = η = 0$ .

For $|t| > 1, t = x, y, (ζ, γ, t) = (ξ, α, x)$ (or $(η, β, y))$ , we set $A_{ζ, γ} (t) : = |t - ζ| + (t - ζ) \cos γ,$ and the following function: $H (x, y) : = \frac{{[min {ln ρ A_{ξ, α} (x),ln ρ A_{η, β} (y)}]}^{σ}}{{[\max {ln ρ A_{ξ, α} (x),ln ρ A_{η, β} (y)}]}^{λ + σ}} .$

Definition 1

Define two weight coefficients as follows:

(6)

ω (λ_{2}, m) : = \sum_{|n| = 2}^{\infty} \frac{H (m, n)}{A_{η, β} (n)} \frac{\ln^{λ_{1}} ρ A_{ξ, α} (m)}{{ln}^{1 - λ_{2}} ρ A_{η, β} (n)}, |m| \in N ⧹ {1},

(7)

ϖ (λ_{1}, n) : = \sum_{|m| = 2}^{\infty} \frac{H (m, n)}{A_{ξ, α} (m)} \frac{\ln^{λ_{2}} ρ A_{η, β} (n)}{{ln}^{1 - λ_{1}} ρ A_{ξ, α} (m)}, |n| \in N ⧹ {1},

where

\sum_{|j|}^{= 2 \infty} \dots = \sum_{j = - 2}^{- \infty} \dots + \sum_{j = 2}^{\infty} \dots

(j = m, n)

Lemma 1

(cf. Xin et al., 2016) Suppose that $g (t) (> 0)$ is strictly decreasing in $(1, \infty)$ , satisfying $\int_{1}^{\infty} g (t) dt \in R_{+}$ . We have

(8)

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

If ${(- 1)}^{i} g^{(i)} (t) > 0$ $(i = 0, 1, 2; t \in (\frac{3}{2}, \infty)), \int_{\frac{3}{2}}^{\infty} g (t) dt \in R_{+}$ , then we have the following Hermite-Hadamard’s inequality (cf. Chen and Yang, 2016):

(9)

\sum_{n = 2}^{\infty} g (n) < \int_{\frac{3}{2}}^{\infty} g (t) dt .

Lemma 2

The following inequalities are valid for $λ_{1} > - σ, - σ < λ_{2} ⩽ 1 - σ$ :

(10)

h_{β} (λ_{1}) (1 - θ (λ_{2}, m)) < ω (λ_{2}, m) < h_{β} (λ_{1}), |m| \in N ⧹ {1},

where

(11)

\begin{matrix} θ (λ_{2}, m) : = & \frac{(λ_{1} + σ) (λ_{2} + σ)}{λ + 2 σ} \int_{0}^{\frac{\ln ρ (2 + η) (1 + \cos β)}{P \ln ρ A_{ξ, α} (m)}} \frac{{(min {1, u})}^{σ} u^{λ_{2} - 1}}{{(max {1, u})}^{λ + σ}} du \\ = & O (\frac{1}{\ln^{λ_{2} + σ} ρ A_{ξ, α} (m)}) \in (0, 1) . \end{matrix}

Proof

For $| m | \in N ⧹ {1}$ , we put $\begin{matrix} H^{(1)} (m, y) : = & \frac{{[min {\ln ρ A_{ξ, α} (m), \ln ρ (y - η) (\cos β - 1)}]}^{σ}}{{[\max {\ln ρ A_{ξ, α} (m), \ln ρ (y - η) (\cos β - 1)}]}^{λ + σ}}, y < - 1, \\ H^{(2)} (m, y) : = & \frac{{[min {\ln ρ A_{ξ, α} (m), \ln ρ (y - η) (\cos β + 1)}]}^{σ}}{{[\max {\ln ρ A_{ξ, α} (m), \ln ρ (y - η) (\cos β + 1)}]}^{λ + σ}}, y > 1, \end{matrix}$ wherefrom $H^{(1)} (m, - y) = \frac{{[min {\ln ρ A_{ξ, α} (m), \ln ρ (y + η) (1 - \cos β)}]}^{σ}}{{[\max {\ln ρ A_{ξ, α} (m), \ln ρ (y + η) (1 - \cos β)}]}^{λ + σ}}, y > 1 .$

We find

(12)

ω (λ_{2}, m) = \sum_{n = - 2}^{- \infty} \frac{H^{(1)} (m, n)}{(n - η) (\cos β - 1)} \frac{\ln^{λ_{1}} ρ A_{ξ, α} (m)}{\ln^{1 - λ_{2}} ρ (n - η) (\cos β - 1)} + \sum_{n = 2}^{\infty} \frac{H^{(2)} (m, n)}{(n - η) (\cos β + 1)} \frac{\ln^{λ_{1}} ρ A_{ξ, α} (m)}{\ln^{1 - λ_{2}} ρ (n - η) (\cos β + 1)} = \frac{\ln^{λ_{1}} ρ A_{ξ, α} (m)}{1 - \cos β} \sum_{n = 2}^{\infty} \frac{H^{(1)} (m, - n)}{(n + η) \ln^{1 - λ_{2}} ρ (n + η) (1 - \cos β)} + \frac{\ln^{λ_{1}} ρ A_{ξ, α} (m)}{1 + \cos β} \sum_{n = 2}^{\infty} \frac{H^{(2)} (m, n)}{(n - η) \ln^{1 - λ_{2}} ρ (n - η) (1 + \cos β)} .

In virtue of $λ_{1} > σ, - σ < λ_{2} ⩽ 1 - σ$ , and $λ_{1} + λ_{2} = λ$ , we find that for $y > 1, i = 1, 2$ , $\frac{H^{(i)} (m, {(- 1)}^{i} y)}{[y - {(- 1)}^{i} η] \ln^{1 - λ_{2}} ρ [y - {(- 1)}^{i} η] [1 + {(- 1)}^{i} \cos β]} = \{\begin{matrix} \frac{1}{[y - {(- 1)}^{i} η] {(\ln ρ A_{ξ, α} (m))}^{λ + σ} \ln^{1 - λ_{2} - σ} ρ [y - {(- 1)}^{i} η] [1 + {(- 1)}^{i} \cos β]}, \\ \frac{1}{ρ} < [y - {(- 1)}^{i} η] [1 + {(- 1)}^{i} \cos β] ⩽ A_{ξ, α} (m) \\ \frac{{(\ln ρ A_{ξ, α} (m))}^{σ}}{[y - {(- 1)}^{i} η] \ln^{1 + λ_{1} + σ} ρ [y - {(- 1)}^{i} η] [1 + {(- 1)}^{i} \cos β]}, \\ [y - {(- 1)}^{i} η] [1 + {(- 1)}^{i} \cos β] > A_{ξ, α} (m) \end{matrix}$ are strictly decreasing in $(1, \infty)$ . By (12) and (8), we find $ω (λ_{2}, m) < \frac{\ln^{λ_{1}} ρ A_{ξ, α} (m)}{1 - \cos β} \int_{1}^{\infty} \frac{H^{(1)} (m, - y) dy}{(y + η) \ln^{1 - λ_{2}} ρ (y + η) (1 - \cos β)} + \frac{\ln^{λ_{1}} ρ A_{ξ, α} (m)}{1 + \cos β} \int_{1}^{\infty} \frac{H^{(2)} (m, y) dy}{(y - η) \ln^{1 - λ_{2}} ρ (y - η) (1 + \cos β)} .$

Setting $u = \frac{\ln ρ (y + η) (1 - \cos β)}{\ln ρ A_{ξ, α} (m)} (u = \frac{\ln ρ (y - η) (1 + \cos β)}{\ln ρ A_{ξ, α} (m)})$ in the above first (second) integral, in view of Remark 1, by simplifications, we obtain $ω (λ_{2}, m) < (\frac{1}{1 - \cos β} + \frac{1}{1 + \cos β}) \int_{0}^{\infty} \frac{{(min {1, u})}^{σ} u^{λ_{2} - 1}}{{(\max {1, u})}^{λ + σ}} du = 2 {csc}^{2} β (\int_{0}^{1} u^{λ_{2} + σ - 1} du + \int_{1}^{\infty} \frac{u^{λ_{2} - 1}}{u^{λ + σ}} du) = h_{β} (λ_{1}) .$

By (12) and (8), in the same way, we still have $ω (λ_{2}, m) > \frac{\ln^{λ_{1}} ρ A_{ξ, α} (m)}{1 - \cos β} \int_{2}^{\infty} \frac{H^{(1)} (m, - y) dy}{(y + η) \ln^{1 - λ_{2}} ρ (y + η) (1 - \cos β)} + \frac{\ln^{λ_{1}} ρ A_{ξ, α} (m)}{1 + \cos β} \int_{2}^{\infty} \frac{H^{(2)} (m, y) dy}{(y - η) \ln^{1 - λ_{2}} ρ (y - η) (1 + \cos β)} ⩾ 2 {csc}^{2} β \int_{\frac{\ln ρ (2 + η) (1 + \cos β)}{\ln ρ A_{ξ, α} (m)}}^{\infty} \frac{{(min {1, u})}^{σ} u^{λ_{2} - 1}}{{(max {1, u})}^{λ + σ}} du = h_{β} (λ_{1}) - 2 {csc}^{2} β \int_{0}^{\frac{\ln ρ (2 + η) (1 + \cos β)}{\ln ρ A_{ξ, α} (m)}} \frac{{(min {1, u})}^{σ} u^{λ_{2} - 1}}{{(\max {1, u})}^{λ + σ}} du = h_{β} (λ_{1}) (1 - θ (λ_{2}, m)) > 0,$ where $θ (λ_{2}, m) (< 1)$ is indicated by (11). It follows that for $A_{ξ, α} (m) ⩾ (2 + η) (1 + \cos β)$ , $0 < θ (λ_{2}, m) = \frac{(λ_{1} + σ) (λ_{2} + σ)}{λ + 2 σ} \int_{0}^{\frac{\ln ρ (2 + η) (1 + \cos β)}{\ln ρ A_{ξ, α} (m)}} u^{λ_{2} + σ - 1} du = \frac{λ_{1} + σ}{λ + 2 σ} {[\frac{\ln ρ (2 + η) (1 + \cos β)}{\ln ρ A_{ξ, α} (m)}]}^{λ_{2} + σ} .$ Hence, (10) and (11) are valid. □

In the same way, we still have.

Lemma 3

The following inequalities are valid for $λ_{2} > - σ, - σ < λ_{1} ⩽ 1 - σ$ :

(13)

h_{α} (λ_{1}) (1 - \tilde{θ} (λ_{1}, n)) < ϖ (λ_{1}, n) < h_{α} (λ_{1}), |n| \in N ⧹ {1},

where

(14)

\begin{matrix} \tilde{θ} (λ_{1}, n) : = & \frac{(λ_{1} + σ) (λ_{2} + σ)}{λ + 2 σ} \int_{0}^{\frac{\ln ρ (2 + ξ) (1 + \cos α)}{\ln ρ A_{η, β} (n)}} \frac{{(min {1, u})}^{σ} u^{λ_{1} - 1}}{{(max {1, u})}^{λ + σ}} du \\ = & O (\frac{1}{\ln^{λ_{1} + σ} ρ A_{η, β} (n)}) \in (0, 1) . \end{matrix}

Lemma 4

If $(ζ, γ) = (ξ, α)$ (or $(η, β))$ , then for $ε > 0$ , we have

(15)

H_{ε} (ζ, γ) : = \sum_{|k| = 2}^{\infty} \frac{\ln^{- 1 - ε} ρ [|k - ζ| + (k - ζ) \cos γ]}{|k - ζ| (k - ζ) \cos γ} = \frac{1}{ε} (2 {csc}^{2} γ + o (1)) (ε \to 0^{+}) .

Proof

By (9), we find $H_{ε} (ζ, γ) = \sum_{k = - 2}^{- \infty} \frac{\ln^{- 1 - ε} ρ (k - ζ) (\cos γ - 1)}{(k - ζ) (\cos γ - 1)} + \sum_{k = 2}^{\infty} \frac{\ln^{- 1 - ε} ρ (k - ζ) (\cos γ + 1)}{(k - ζ) (\cos γ + 1)} = \sum_{k = 2}^{\infty} [\frac{\ln^{- 1 - ε} ρ (k + ζ) (1 - \cos γ)}{(k + ζ) (1 - \cos γ)} + \frac{\ln^{- 1 - ε} ρ (k - ζ) (\cos γ + 1)}{(k - ζ) (\cos γ + 1)}] < \int_{\frac{3}{2}}^{\infty} [\frac{\ln^{- 1 - ε} ρ (y + ζ) (1 - \cos γ)}{(y + ζ) (1 - \cos γ)} + \frac{\ln^{- 1 - ε} ρ (y - ζ) (\cos γ + 1)}{(y - ζ) (\cos γ + 1)}] dy = \frac{1}{ε} [\frac{\ln^{- ε} ρ (\frac{3}{2} + ζ) (1 - \cos γ)}{1 - \cos γ} + \frac{\ln^{- ε} ρ (\frac{3}{2} - ζ) (1 + \cos γ)}{1 + \cos γ}] = \frac{1}{ε} (\frac{1}{1 - \cos γ} + \frac{1}{1 + \cos γ} + o_{1} (1)) (ε \to 0^{+}) .$

By (8), we still can find that $H_{ε} (ζ, γ) = \sum_{k = 2}^{\infty} [\frac{\ln^{- 1 - ε} ρ (k + ζ) (1 - \cos γ)}{(k + ζ) (1 - \cos γ)} + \frac{\ln^{- 1 - ε} ρ (n - ζ) (\cos γ + 1)}{(n - ζ) (\cos γ + 1)}] > \int_{2}^{\infty} [\frac{\ln^{- 1 - ε} ρ (y + ζ) (1 - \cos γ)}{(y + ζ) (1 - \cos γ)} + \frac{\ln^{- 1 - ε} ρ (y - ζ) (\cos γ + 1)}{(y - ζ) (\cos γ + 1)}] dy = \frac{1}{ε} [\frac{\ln^{- ε} ρ (2 + ζ) (1 - \cos γ)}{1 - \cos γ} + \frac{\ln^{- ε} ρ (2 - ζ) (1 + \cos γ)}{1 + \cos γ}] = \frac{1}{ε} (\frac{1}{1 - \cos γ} + \frac{1}{1 + \cos γ} + o_{2} (1)) (ε \to 0^{+}) .$ Hence, we prove that (15) is valid. □

3 Main results and a few particular cases

We set

(16)

k_{α, β} (λ_{1}) : = h_{β}^{1 / p} (λ_{1}) h_{α}^{1 / q} (λ_{1}) = \frac{2 (λ + 2 σ)}{(λ_{1} + σ) (λ_{2} + σ)} {csc}^{2 / p} β {csc}^{2 / q} α .

Theorem 1

Suppose that $a_{m}, b_{n} ⩾ 0 (|m|, |n| \in N ⧹ {1})$ , satisfy $0 < \sum_{|m| = 2}^{\infty} \frac{\ln^{p (1 - λ_{1}) - 1} ρ A_{ξ, α} (m)}{{(A_{ξ, α} (m))}^{1 - p}} a_{m}^{p} < \infty, < \sum_{|n| = 2}^{\infty} \frac{\ln^{q (1 - λ_{2}) - 1} ρ A_{η, β} (n)}{{(A_{η, β} (n))}^{1 - q}} b_{n}^{q} < \infty .$

We have the following equivalent inequalities:

(17)

I : = \sum_{|n| = 2}^{\infty} \sum_{|m| = 2}^{\infty} H (m, n) a_{m} b_{n} < k_{α, β} (λ_{1}) {[\sum_{|m| = 2}^{\infty} \frac{\ln^{p (1 - λ_{1}) - 1} ρ A_{ξ, α} (m)}{{(A_{ξ, α} (m))}^{1 - p}} a_{m}^{p}]}^{\frac{1}{p}} \times {[\sum_{|n| = 2}^{\infty} \frac{\ln^{q (1 - λ_{2}) - 1} ρ A_{η, β} (n)}{{(A_{η, β} (n))}^{1 - q}} b_{n}^{q}]}^{\frac{1}{q}},

(18)

J : = {[\sum_{|n| = 2}^{\infty} \frac{\ln^{p λ_{2} - 1} ρ A_{η, β} (n)}{A_{η, β} (n)} {(\sum_{|m| = 2}^{\infty} H (m, n) a_{m})}^{p}]}^{\frac{1}{p}} < k_{α, β} (λ_{1}) {[\sum_{| m | = 2}^{\infty} \frac{\ln^{p (1 - λ_{1}) - 1} ρ A_{ξ, α} (m)}{{(A_{ξ, α} (m))}^{1 - p}} a_{m}^{p}]}^{1 / p} .

In particular, (i) for $α = β = \frac{π}{2}, \frac{1}{ρ} - 1 ⩽ ξ, η ⩽ 1 - \frac{1}{ρ}$ , we have the following equivalent inequalities similar to (4):

(19)

\sum_{|n| = 2}^{\infty} \sum_{|m| = 2}^{\infty} \frac{{(min {\ln ρ | m - ξ |, \ln ρ | n - η |})}^{σ} a_{m} b_{n}}{{(\max {\ln ρ | m - ξ |, \ln ρ | n - η |})}^{λ + σ}} < \frac{2 (λ + 2 σ)}{(λ_{1} + σ) (λ_{2} + σ)} \times {[\sum_{|m| = 2}^{\infty} \frac{\ln^{p (1 - λ_{1}) - 1} ρ | m - ξ |}{{|m - ξ|}^{1 - p}} a_{m}^{p}]}^{\frac{1}{p}} {[\sum_{|n| = 2}^{\infty} \frac{\ln^{q (1 - λ_{2}) - 1} ρ |n - η|}{{|n - η|}^{1 - q}} b_{n}^{q}]}^{\frac{1}{q}},

(20)

{\{\sum_{|n| = 2}^{\infty} \frac{\ln^{p λ_{2} - 1} ρ |n - η|}{|n - η|} {[\sum_{|m| = 2}^{\infty} \frac{{(min {\ln ρ | m - ξ |, \ln ρ | n - η |})}^{σ} a_{m}}{{(\max {\ln ρ | m - ξ |, \ln ρ | n - η |})}^{λ + σ}}]}^{p}\}}^{\frac{1}{p}} < \frac{2 (λ + 2 σ)}{(λ_{1} + σ) (λ_{2} + σ)} {[\sum_{|m| = 2}^{\infty} \frac{\ln^{p (1 - λ_{1}) - 1} ρ |m - ξ|}{{|m - ξ|}^{1 - p}} a_{m}^{p}]}^{1 / p} .

(ii) For $ξ = η = 0, \arccos \sqrt{1 - \frac{1}{ρ}}$ $⩽ α, β ⩽ π - \arccos \sqrt{1 - \frac{1}{ρ}}$ , we have the following equivalent inequalities:

(21)

\sum_{|n| = 2}^{\infty} \sum_{|m| = 2}^{\infty} \frac{{[min {\ln ρ (| m | + m \cos α), \ln ρ (| n | + n \cos β)}]}^{σ} a_{m} b_{n}}{{[\max {\ln ρ (| m | + m \cos α), \ln ρ (| n | + \cos β)}]}^{λ + σ}} < k_{α, β} (λ_{1}) {[\sum_{|m| = 2}^{\infty} \frac{\ln^{p (1 - λ_{1}) - 1} ρ (| m | + m \cos α)}{{(| m | + m \cos α)}^{1 - p}} a_{m}^{p}]}^{1 / p} \times {[\sum_{|n| = 2}^{\infty} \frac{\ln^{q (1 - λ_{2}) - 1} ρ (| n | + \cos β)}{{(| n | + \cos β)}^{1 - q}} b_{n}^{q}]}^{1 / q},

(22)

{\{\sum_{|n| = 2}^{\infty} \frac{\ln^{p λ_{2} - 1} ρ (| n | + \cos β)}{| n | + \cos β} \times {[\sum_{|m| = 2}^{\infty} \frac{{[min {\ln ρ (| m | + m \cos α), \ln ρ (| n | + n \cos β)}]}^{σ} a_{m}}{{[\max {\ln ρ (| m | + m \cos α), \ln ρ (| n | + \cos β)}]}^{λ + σ}}]}^{p}\}}^{\frac{1}{p}} < k_{α, β} (λ_{1}) {[\sum_{|m| = 2}^{\infty} \frac{\ln^{p (1 - λ_{1}) - 1} ρ (| m | + m \cos α)}{{(| m | + m \cos α)}^{1 - p}} a_{m}^{p}]}^{\frac{1}{p}} .

Proof

By Hölder’s inequality with weight (cf. Kuang, 2010) and (7), we find ${(\sum_{|m| = 2}^{\infty} H (m, n) a_{m})}^{p} = {\{\sum_{|m| = 2}^{\infty} H (m, n) [\frac{{(A_{ξ, α} (m))}^{1 / q} \ln^{(1 - λ_{1}) / q} ρ A_{ξ, α} (m)}{\ln^{(1 - λ_{2}) / p} ρ A_{η, β} (n)} a_{m}] \times [\frac{\ln^{(1 - λ_{2}) / p} ρ A_{η, β} (n)}{{(A_{ξ, α} (m))}^{1 / q} \ln^{(1 - λ_{1}) / q} ρ A_{ξ, α} (m)}]\}}^{p} ⩽ \sum_{|m| = 2}^{\infty} H (m, n) \frac{{(A_{ξ, α} (m))}^{p / q} \ln^{(1 - λ_{1}) p / q} ρ A_{ξ, α} (m)}{\ln^{1 - λ_{2}} ρ A_{η, β} (n)} a_{m}^{p} \times {[\sum_{|m| = 2}^{\infty} H (m, n) \frac{\ln^{(1 - λ_{2}) q / p} ρ A_{η, β} (n)}{A_{ξ, α} (m) \ln^{1 - λ_{1}} ρ A_{ξ, α} (m)}]}^{p - 1} = \frac{{(ϖ (λ_{1}, n))}^{p - 1} A_{η, β} (n)}{\ln^{p λ_{2} - 1} ρ A_{η, β} (n)} \sum_{|m| = 2}^{\infty} H (m, n) \frac{{(A_{ξ, α} (m))}^{p / q} \ln^{(1 - λ_{1}) p / q} ρ A_{ξ, α} (m)}{A_{η, β} (n) \ln^{1 - λ_{2}} ρ A_{η, β} (n)} a_{m}^{p} .$

By (13), it follows that

(23)

J < h_{α}^{1 / q} (λ_{1}) {[\sum_{|n| = 2}^{\infty} \sum_{|m| = 2}^{\infty} \frac{H (m, n) {(A_{ξ, α} (m))}^{p / q} \ln^{(1 - λ_{1}) p / q} ρ A_{ξ, α} (m)}{A_{η, β} (n) \ln^{1 - λ_{2}} ρ A_{η, β} (n)} a_{m}^{p}]}^{\frac{1}{p}} = h_{α}^{1 / q} (λ_{1}) {[\sum_{|m| = 2}^{\infty} \sum_{|n| = 2}^{\infty} \frac{H (m, n) {(A_{ξ, α} (m))}^{p / q} \ln^{(1 - λ_{1}) p / q} ρ A_{ξ, α} (m)}{A_{η, β} (n) \ln^{1 - λ_{2}} ρ A_{η, β} (n)} a_{m}^{p}]}^{\frac{1}{p}} = h_{α}^{1 / q} (λ_{1}) {[\sum_{|m| = 2}^{\infty} ω (λ_{2}, m) \frac{\ln^{p (1 - λ_{1}) - 1} ρ A_{ξ, α} (m)}{{(A_{ξ, α} (m))}^{1 - p}} a_{m}^{p}]}^{\frac{1}{p}} .

By (10) and (16), we have (18).

Using Hölder’s inequality again, we have

(24)

I = \sum_{|n| = 2}^{\infty} [\frac{\ln^{λ_{2} - (1 / p)} ρ A_{η, β} (n)}{{(A_{η, β} (n))}^{1 / p}} \sum_{|m| = 2}^{\infty} H (m, n) a_{m}] [\frac{\ln^{(1 / p) - λ_{2}} ρ A_{η, β} (n)}{{(A_{η, β} (n))}^{- 1 / p}} b_{n}] ⩽ J {[\sum_{|n| = 2}^{\infty} \frac{\ln^{q (1 - λ_{2}) - 1} ρ A_{η, β} (n)}{{(A_{η, β} (n))}^{1 - q}} b_{n}^{q}]}^{\frac{1}{q}},

and then by (18), we have (17).

On the other hand, assuming that (17) is valid, we set $b_{n} : = \frac{\ln^{p λ_{2} - 1} ρ A_{η, β} (n)}{A_{η, β} (n)} {(\sum_{|m| = 2}^{\infty} H (m, n) a_{m})}^{p - 1}, |n| \in N ⧹ {1},$ and find $J = {[\sum_{|n| = 2}^{\infty} \frac{\ln^{q (1 - λ_{2}) - 1} ρ A_{η, β} (n)}{{(A_{η, β} (n))}^{1 - q}} b_{n}^{q}]}^{1 / p} .$ By (23), it follows that $J < \infty$ . If $J = 0$ , then (18) is trivially valid; if $J > 0$ , then we have $0 < \sum_{[n] = 2}^{\infty} \frac{\ln^{q (1 - λ_{2} - 1)} ρ A_{η, β} (n)}{{(A_{η, β} (n))}^{1 - q}} b_{n}^{q} = J^{p} = I < k_{α, β} (λ_{1}) \times {[\sum_{| m | = 2}^{\infty} \frac{\ln^{p (1 - λ_{1} - 1)} ρ A_{ξ, β} (m)}{{(A_{ξ, α} (m))}^{1 - q}} α_{m}^{p}]}^{\frac{1}{p}} {[\sum_{| n | = 2}^{\infty} \frac{\ln^{q (1 - λ_{2} - 1)} ρ A_{η, β} (n)}{{(A_{η, β} (n))}^{1 - q}} b_{n}^{q}]}^{\frac{1}{q}}, J = {[\sum_{| n | = 2}^{\infty} \frac{\ln^{q (1 - λ_{2} - 1)} ρ A_{η, β} (n)}{{(A_{η, β} (n))}^{1 - q}} b_{n}^{q}]}^{\frac{1}{p}} < k_{α, β} (λ_{1}) {[\sum_{| m | = 2}^{\infty} \frac{\ln^{p (1 - λ_{1} - 1)} ρ A_{ξ, β} (m)}{{(A_{ξ, α} (m))}^{1 - q}} α_{m}^{p}]}^{\frac{1}{p}} .$ Hence, (18) is valid, which is equivalent to (17). □

Theorem 2

With regards to the assumptions of Theorem 1, the constant factor $k_{α, β} (λ_{1})$ in (17) and (18) is the best possible.

Proof

For $0 < ε < q (λ_{2} + σ)$ , we set ${\tilde{λ}}_{1} = λ_{1} + \frac{ε}{q} (> - σ), {\tilde{λ}}_{2} = λ_{2} - \frac{ε}{q}$ $(\in (- σ, 1 - σ))$ , and ${\tilde{a}}_{m} : = \frac{\ln^{λ_{1} - (ε / p) - 1} ρ A_{ξ, α} (m)}{A_{ξ, α} (m)} = \frac{\ln^{{\tilde{λ}}_{1} - ε - 1} ρ A_{ξ, α} (m)}{A_{ξ, α} (m)} (|m| \in N ⧹ {1}),$ ${\tilde{b}}_{n} : = \frac{\ln^{λ_{2} - (ε / q) - 1} ρ A_{η, β} (n)}{A_{η, β} (n)} = \frac{\ln^{{\tilde{λ}}_{2} - 1} ρ A_{η, β} (n)}{A_{η, β} (n)} (|n| \in N ⧹ {1}) .$ By (15) and (13), we find ${\tilde{I}}_{1} : = {[\sum_{|m| = 2}^{\infty} \frac{\ln^{p (1 - λ_{1}) - 1} ρ A_{ξ, α} (m)}{{(A_{ξ, α} (m))}^{1 - p}} {\tilde{a}}_{m}^{p}]}^{\frac{1}{p}} {[\sum_{|n| = 2}^{\infty} \frac{\ln^{q (1 - λ_{2}) - 1} ρ A_{η, β} (n)}{{(A_{η, β} (n))}^{1 - q}} {\tilde{b}}_{n}^{q}]}^{\frac{1}{q}} = {[\sum_{|m| = 2}^{\infty} \frac{\ln^{- 1 - ε} ρ A_{ξ, α} (m)}{A_{ξ, α} (m)}]}^{\frac{1}{p}} {[\sum_{|n| = 2}^{\infty} \frac{\ln^{- 1 - ε} ρ A_{η, β} (n)}{A_{η, β} (n)}]}^{\frac{1}{q}} = \frac{1}{ε} {(2 {csc}^{2} α + o (1))}^{1 / p} {(2 {csc}^{2} β + \tilde{o} (1))}^{1 / q} (ε \to 0^{+}),$ $\tilde{I} : = \sum_{|m| = 2}^{\infty} \sum_{|n| = 2}^{\infty} H (m, n) {\tilde{a}}_{m} {\tilde{b}}_{n} = \sum_{|m|}^{= 2 \infty} \sum_{|n| = 2}^{\infty} H (m, n) \frac{\ln^{{\tilde{λ}}_{1} - ε - 1} ρ A_{ξ, α} (m) \ln^{{\tilde{λ}}_{2} - 1} ρ A_{η, β} (n)}{A_{ξ, α} (m) A_{η, β} (n)} = \sum_{|m| = 2}^{\infty} ω ({\tilde{λ}}_{2}, m) \frac{\ln^{- 1 - ε} ρ A_{ξ, α} (m)}{A_{ξ, α} (m)} > h_{β} ({\tilde{λ}}_{1}) \sum_{|m| = 2}^{\infty} (1 - θ ({\tilde{λ}}_{2}, m)) \frac{\ln^{- 1 - ε} ρ A_{ξ, α} (m)}{A_{ξ, α} (m)} = h_{β} ({\tilde{λ}}_{1}) [\sum_{|m| = 2}^{\infty} \frac{\ln^{- 1 - ε} ρ A_{ξ, α} (m)}{A_{ξ, α} (m)} - \sum_{|m| = 2}^{\infty} \frac{O (\ln^{- 1 - (\frac{ε}{p} + λ_{2} + σ)} ρ A_{ξ, α} (m))}{A_{ξ, α} (m)}] = \frac{1}{ε} h_{β} (λ_{1} + \frac{ε}{q}) (2 \csc^{2} α + o (1) - ε O (1)) .$

If there exists a positive number $K ⩽ k_{α, β} (λ_{1})$ , such that (17) is still valid when replacing $k_{α, β} (λ_{1})$ by K, then in particular, we have $ε \tilde{I} = ε \sum_{|m| = 2}^{\infty} \sum_{|n| = 2}^{\infty} H (m, n) {\tilde{a}}_{m} {\tilde{b}}_{n} < ε K {\tilde{I}}_{1} .$

Hence, in view of the above results, it follows that $h_{β} (λ_{1} + \frac{ε}{q}) (2 \csc^{2} α + o (1) - ε O (1)) < K \cdot {(2 \csc^{2} α + o (1))}^{1 / p} {(2 \csc^{2} β + \tilde{o} (1))}^{1 / q},$ and then $\frac{4 (λ + 2 σ) \csc^{2} β}{(λ_{1} + σ) (λ_{2} + σ)} \csc^{2} α ⩽ 2 K \csc^{2 / p} α \csc^{2 / q} β (ε \to 0^{+}),$ namely, $k_{α, β} (λ_{1}) = \frac{2 (λ + 2 σ)}{(λ_{1} + σ) (λ_{2} + σ)} \csc^{2 / p} β \csc^{2 / q} α ⩽ K .$ Hence, $K = k_{α, β} (λ_{1})$ is the best possible constant factor in (17).

The constant factor $k_{α, β} (λ_{1})$ in (18) is still the best possible. Otherwise we would reach a contradiction by (24)’ that the constant factor in (17) is not the best possible. □

4 Operator expressions and a remark

Setting $φ (m) : = \frac{\ln^{p (1 - λ_{1}) - 1} ρ A_{ξ, α} (m)}{{(A_{ξ, α} (m))}^{1 - p}} (|m| \in N ⧹ {1}),$ $ψ (n) : = \frac{\ln^{q (1 - λ_{2}) - 1} ρ A_{η, β} (n)}{{(A_{η, β} (n))}^{1 - q}},$ wherefrom, $ψ^{1 - p} (n) = \frac{\ln^{p λ_{2} - 1} ρ A_{η, β} (n)}{A_{η, β} (n)}$ $(|n| \in N ⧹ {1})$ , we define the following real weighted normed function spaces: $\begin{matrix} l_{p, φ} : = & \{a = {\{a_{m}\}}_{|m|}^{= 2 \infty}; {|a|}_{p, φ} = {(\sum_{|m| = 2}^{\infty} φ (m) {|a_{m}|}^{p})}^{1 / p} < \infty\}, \\ l_{q, ψ} : = & \{b = {\{b_{n}\}}_{|n|}^{= 2 \infty}; {|b|}_{q, ψ} = {(\sum_{|n| = 2}^{\infty} ψ (n) {|b_{n}|}^{q})}^{1 / q} < \infty\}, \\ l_{p, ψ^{1 - p}} : = & \{c = {\{c_{n}\}}_{|n|}^{= 2 \infty}; {|c|}_{p, ψ^{1 - p}} = {(\sum_{|n| = 2}^{\infty} ψ^{1 - p} (n) {|c_{n}|}^{p})}^{1 / p} < \infty\} . \end{matrix}$ For $a = {\{a_{m}\}}_{|m|}^{= 2 \infty} \in l_{p, φ}$ , putting $c_{n} = \sum_{|m|}^{= 2 \infty} H (m, n) a_{m}$ and $c = {\{c_{n}\}}_{|n|}^{= 2 \infty}$ , it follows by (18 ) that ${|c|}_{p, ψ^{1 - p}} < k_{α, β} (λ_{1}) {|a|}_{p, φ}$ , namely $c \in l_{p, ψ^{1 - p}}$ .

Definition 2

Define a Mulholland-type operator T: $l_{p, φ} \to l_{p, ψ^{1 - p}}$ as follows: For $a_{m} ⩾ 0, a = {\{a_{m}\}}_{|m|}^{= 2 \infty} \in l_{p, φ}$ , there exists a unique representation $Ta = c \in l_{p, ψ^{1 - p}}$ . We also define the following formal inner product of Ta and $b = {\{b_{n}\}}_{|n|}^{= 2 \infty} \in l_{q, ψ}$ $(b_{n} ⩾ 0)$ as follows:

(25)

(Ta, b) : = \sum_{|n| = 2}^{\infty} (\sum_{|m| = 2}^{\infty} H (m, n) a_{m}) b_{n} .

Hence, we may rewrite (17) and (18) in the following operator expressions:

(26)

(Ta, b) < k_{α, β} (λ_{1}) {|a|}_{p, φ} {|b|}_{q, ψ},

(27)

{|Ta|}_{p, ψ^{1 - p}} < k_{α, β} (λ_{1}) {|a|}_{p, φ} .

It follows that the operator T is bounded with

(28)

|T| : = \sup_{a (\neq θ) \in l_{p, φ}} \frac{{|Ta|}_{p, ψ^{1 - p}}}{{|a|}_{p, φ}} ⩽ k_{α, β} (λ_{1}) .

Since the constant factor

k_{α, β} (λ_{1})

in (18) is the best possible, we have

(29)

|T| = k_{α, β} (λ_{1}) = \frac{2 (λ + 2 σ)}{(λ_{1} + σ) (λ_{2} + σ)} \csc^{2 / p} β \csc^{2 / q} α .

Remark 2

(i) For $ρ = 1, ξ = η = 0$ in (19), we have the following inequality:

(30)

\sum_{|n| = 2}^{\infty} \sum_{|m| = 2}^{\infty} \frac{{(min {\ln | m |, \ln | n |})}^{σ} a_{m} b_{n}}{{(\max {\ln | m |, \ln | n |})}^{λ + σ}} < \frac{2 (λ + 2 σ)}{(λ_{1} + σ) (λ_{2} + σ)} \times {[\sum_{|m| = 2}^{\infty} \frac{\ln^{p (1 - λ_{1}) - 1} | m |}{{|m|}^{1 - p}} a_{m}^{p}]}^{\frac{1}{p}} {[\sum_{|n| = 2}^{\infty} \frac{\ln^{q (1 - λ_{2}) - 1} |n|}{{|n|}^{1 - q}} b_{n}^{q}]}^{\frac{1}{q}},

It follows that (19) is an extension of (30).

(ii) If $a_{- m} = a_{m}$ and $b_{- n} = b_{n}$ $(m, n \in N ⧹ {1})$ , then (19) reduces to

(31)

\sum_{n = 2}^{\infty} \sum_{m = 2}^{\infty} \{\frac{{[\min {\ln ρ (m - ξ), \ln ρ (n - η)]}^{σ}}{{[\max {\ln ρ (m - ξ), \ln ρ (n - η)]}^{λ + σ}} + \frac{{[min {\ln ρ (m - ξ), \ln ρ (n + η)]}^{σ}}{{[\max {\ln ρ (m - ξ), \ln ρ (n + η)]}^{λ + σ}} + \frac{{[min {\ln ρ (m + ξ), \ln ρ (n - η)]}^{σ}}{{[\max {\ln ρ (m + ξ), \ln ρ (n - η)]}^{λ + σ}} + \frac{{[min {\ln ρ (m + ξ), \ln ρ (n + η)]}^{σ}}{{[\max {\ln ρ (m + ξ), \ln ρ (n + η)]}^{λ + σ}}\} a_{m} b_{n} < \frac{2 (λ + 2 σ)}{(λ_{1} + σ) (λ_{2} + σ)} \times {\{\sum_{m = 2}^{\infty} [\frac{\ln^{p (1 - λ_{1}) - 1} ρ (m - ξ)}{{(m - ξ)}^{1 - p}} + \frac{\ln^{p (1 - λ_{1}) - 1} ρ (m + ξ)}{{(m + ξ)}^{1 - p}}] a_{m}^{p}\}}^{\frac{1}{p}} \times {\{\sum_{n = 2}^{\infty} [\frac{\ln^{q (1 - λ_{2}) - 1} ρ (n - η)}{{(n - η)}^{1 - q}} + \frac{\ln^{q (1 - λ_{2}) - 1} ρ (n + η)}{{(n + η)}^{1 - q}}] b_{n}^{q}\}}^{\frac{1}{q}} .

In particular, for $ρ = 1, ξ = η = 0$ , we have the following new Mulholland-type inequality:

(32)

\sum_{n = 2}^{\infty} \sum_{m = 2}^{\infty} \frac{{(min {ln m,ln n})}^{σ}}{{(\max {ln m,ln n})}^{λ + σ}} a_{m} b_{n} < \frac{λ + 2 σ}{(λ_{1} + σ) (λ_{2} + σ)} \times {[\sum_{m = 2}^{\infty} \frac{\ln^{p (1 - λ_{1}) - 1} m}{m^{1 - p}} a_{m}^{p}]}^{\frac{1}{p}} {[\sum_{n = 2}^{\infty} \frac{\ln^{q (1 - λ_{2}) - 1} n}{n^{1 - q}} b_{n}^{q}]}^{\frac{1}{q}} .

5 Conclusions

In this paper, by introducing independent parameters, applying the weight coefficients, we give a new Mulholland-type inequality in the whole plane with a best possible constant factor in Theorems 1 and 2. Moreover, the equivalent forms, a few particular cases and the operator expressions are considered. The lemmas and theorems provide an extensive account of this type of inequalities.

Acknowledgments

This work is supported by the National Natural Science Foundation (Nos. 61370186, 61640222), and Project for distinctive innovation of Ordinary University of Guangdong Province (No. 2015KTSCX097). We are grateful for this help.

References

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Introduction

Some lemmas

Main results and a few particular cases

Operator expressions and a remark

Conclusions

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

[2] Chen Q., Yang B., . On a more accurate half-discret Hardy-Hilbert-type inequality related to the kernel of arc tangent function. SpringerPlus 2016:1317.
[Google Scholar]

[3] Gao M., Yang B., . On the extended Hilbert’s inequality. Proc. Amer. Math. Soc.. 1998;126(3):751-759.
[Google Scholar]

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

[5] He B., . A multiple Hilbert-type discrete inequality with a new kernel and best possible constant factor. J. Math. Anal. Appl..

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

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

[8] Huang Q., Yang B., . A more accurate half-discrete Hilbert inequality with a nonhomogeneous kernel. J. Funct. Spaces Appl.. 2013;2013 Art.ID 628250
[Google Scholar]

[9] Huang Q., Wang A., Yang B., . A more accurate half-discrete Hilbert-type inequality with a genreal nonhomogeneous kernel and operator expressions. Math. Inequalities Appl.. 2014;17(1):367-388.
[Google Scholar]

[10] Huang Q., Wu S., Yang B., . Parameterized Hilbert-type integral inequalities in the whole plane. Sci. World J. 2014 Art. ID 169061:1-8
[Google Scholar]

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

[12] Krnić M., Vukovic P., . On a multidimensional version of the Hilbert type inequality. Anal. Math.. 2012;38(4):291-303.
[Google Scholar]

[13] Kuang J., . Applied inequalities. Jinan: Shangdong Science Technic Press; 2010. (in Chinese)

[14] Li Y., He B., . On inequalities of Hilbert’s type. Bull. Aust. Math. Soc.. 2007;76(1):1-13.
[Google Scholar]

[15] Mitrinović D.S., Pečarić J.E., Fink A.M., . Inequalities Involving Functions and Their Integrals and Derivatives. Boston, Mass, USA: Kluwer Academic Publishers; 1991.

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

[17] Wang A., Yang B., . A new Hilbert-type integral inequality in the whole plane with the non-homogeneous kernel. J. Inequalities Appl.. 2011;2011:123.
[Google Scholar]

[18] Xin D., Yang B., Chen Q., . A discrete Hilbert-type inequality in the whole plane. J. Inequalities Appl.. 2016;2016:133.
[Google Scholar]

[19] Yang B., . On the norm of a Hilbert’s type linear operator and applications. J. Math. Anal. Appl.. 2007;325:529-541.
[Google Scholar]

[20] Yang B., Rassias G.M., Rassias Th.M., . A new restructured Hardy-Littlewood’s inequality. Int. J. Nonlinear Anal. Appl.. 2011;2(1):11-20.
[Google Scholar]

[21] Zhong Y., Yang B., Chen Q., . A more accurate Mulholland-type inequality in the whole plane. J. Inequalities Appl.. 2017;2017:315.
[Google Scholar]

A Mulholland-type inequality in the whole plane with multi parameters

Abstract

Keywords

26D15

47A07

Mulholland-type inequality

Parameter

Weight coefficient

Equivalent form

Operator expression

1 Introduction

2 Some lemmas

3 Main results and a few particular cases

4 Operator expressions and a remark

5 Conclusions

Acknowledgments

References

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