Erschienen in:

Open Access 01.12.2013 | Research

On a generalization of a Hilbert-type integral inequality in the whole plane with a hypergeometric function

verfasst von: Vandanjav Adiyasuren, Tserendorj Batbold

Erschienen in: Journal of Inequalities and Applications | Ausgabe 1/2013

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Abstract

By introducing some parameters, we establish a generalization of the Hilbert-type integral inequality in the whole plane with the homogeneous kernel of degree

- 2 λ

and the best constant factor which involves the hypergeometric function.

MSC:26D15.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors contributed equally and significantly in writing this paper. Both authors read and approved the final manuscript.

1 Introduction

p > 1

\frac{1}{p} + \frac{1}{q} = 1

and

f (x), g (x) \geq 0

satisfy

0 < \int_{0}^{\infty} f^{p} (x) d x < \infty and 0 < \int_{0}^{\infty} g^{q} (x) d x < \infty,

then

\int_{0}^{\infty} \int_{0}^{\infty} \frac{f (x) g (y)}{x + y} d x d y < \frac{π}{sin (π / p)} {\int_{0}^{\infty} f^{p} (x) d x}^{\frac{1}{p}} {\int_{0}^{\infty} g^{q} (x) d x}^{\frac{1}{q}},

(1)

where the constant factor

π / (sin π / p)

is the best possible. Inequality (1) is called Hardy-Hilbert’s inequality [1] and is important in analysis and applications [2].

In 2001, Yang gave an extension of (1) involving beta function as (see [3]):

\begin{aligned} \int_{0}^{\infty} \int_{0}^{\infty} \frac{f (x) g (y)}{{(x + y)}^{λ}} d x d y \\ < B (\frac{p + λ - 2}{p}, \frac{q + λ - 2}{q}) {\int_{0}^{\infty} x^{1 - λ} f^{p} (x) d x}^{\frac{1}{p}} {\int_{0}^{\infty} x^{1 - λ} g^{q} (x) d x}^{\frac{1}{q}}, \end{aligned}

(2)

where the constant factor

B (\frac{p + λ - 2}{p}, \frac{q + λ - 2}{q})

(

λ > 2 - min {p, q}

) is the best possible.

Recently, some new Hilbert-type inequalities in the whole plane have been obtained [4, 5]. Xin and Yang in [5] established the following:

p > 1

\frac{1}{p} + \frac{1}{q} = 1

| β | < 1

0 < α_{1} < α_{2} < π

f, g \geq 0

, satisfy

0 < \int_{- \infty}^{\infty} {| x |}^{- p β - 1} f^{p} (x) d x < \infty and 0 < \int_{- \infty}^{\infty} {| y |}^{q β - 1} g^{q} (y) d y < \infty,

then we have

\begin{aligned} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} min_{i \in {1, 2}} {\frac{1}{x^{2} + 2 x y cos α_{i} + y^{2}}} f (x) g (y) d x d y \\ < k (β) {(\int_{- \infty}^{\infty} {| x |}^{- p β - 1} f^{p} (x) d x)}^{\frac{1}{p}} {(\int_{- \infty}^{\infty} {| y |}^{q β - 1} g^{q} (y) d y)}^{\frac{1}{q}}, \end{aligned}

(3)

and

\begin{aligned} \int_{- \infty}^{\infty} {| y |}^{p (1 - β) - 1} {(\int_{- \infty}^{\infty} min_{i \in {1, 2}} {\frac{1}{x^{2} + 2 x y cos α_{i} + y^{2}}} f (x) d x)}^{p} d y \\ < k^{p} (β) \int_{- \infty}^{\infty} {| x |}^{- p β - 1} f^{p} (x) d x, \end{aligned}

(4)

where the constant factors

k (β) = \frac{π}{sin β π} (\frac{sin β α_{1}}{sin α_{1}} + \frac{sin β (π - α_{2})}{sin α_{2}})

and

k^{p} (β)

are the best possible. Inequalities (3) and (4) are equivalent.

By introducing some parameters, we establish generalizations of inequalities (3) and (4) with the homogeneous kernel of degree

- 2 λ

and the best constant factor which involves the hypergeometric function.

2 Preliminary lemmas

In order to prove our assertions, we need the following lemmas.

Recall that the hypergeometric function

F (α, β; γ; x)

is defined [6] by

F (α, β; γ; x) = \sum_{r = 0}^{\infty} \frac{{(α)}_{r} {(β)}_{r}}{{(γ)}_{r}} \frac{x^{r}}{r!},

(5)

where

{(α)}_{r}

is the Pochhammer symbol defined by

{(α)}_{r} = α (α + 1) \dots (α + r - 1) = \frac{Γ (α + r)}{Γ (α)} .

It is known the series (5) converges for

| x | < 1

and diverges for

| x | > 1

. The hypergeometric function satisfies the integral representation

F (α, β; γ; x) = \frac{Γ (γ)}{Γ (β) Γ (γ - β)} \int_{0}^{1} t^{β - 1} {(1 - t)}^{γ - β - 1} {(1 - x t)}^{- α} d t, if γ > β > 0 .

Lemma 2.1 (See [7])

Suppose that

a, c > 0

b^{2} < a c

0 < α < 2 λ

. Then we have

\int_{0}^{\infty} \frac{x^{α - 1}}{{(a x^{2} + 2 b x + c)}^{λ}} d x = a^{- \frac{α}{2}} c^{\frac{α}{2} - λ} B (α, 2 λ - α) F (\frac{α}{2}, λ - \frac{α}{2}; λ + \frac{1}{2}; 1 - \frac{b^{2}}{a c}) .

Lemma 2.2 Let

a, c, λ > 0

b \geq 0

1 - 2 λ < β < 1

and

0 < α_{1} < α_{2} < π

be real parameters such that

b^{2} max {{cos}^{2} α_{1}, {cos}^{2} (π - α_{2})} < a c

. Define the weight functions

ω (x)

and

ϖ (y)

(

x, y \in (- \infty, \infty)

) as follows:

\begin{matrix} ω (x) : = \int_{- \infty}^{\infty} min_{i \in {1, 2}} {\frac{1}{{(a x^{2} + 2 b x y cos α_{i} + c y^{2})}^{λ}}} \frac{{| x |}^{β + 2 λ - 1}}{{| y |}^{β}} d y, \\ ϖ (y) : = \int_{- \infty}^{\infty} min_{i \in {1, 2}} {\frac{1}{{(a x^{2} + 2 b x y cos α_{i} + c y^{2})}^{λ}}} \frac{{| y |}^{1 - β}}{{| x |}^{- β - 2 λ + 2}} d x . \end{matrix}

Then we have

ω (x) = ϖ (y) = C_{λ}

(

x, y \neq 0

), where

\begin{aligned} C_{λ} = & a^{\frac{1 - β}{2} - λ} c^{\frac{1 - β}{2}} B (1 - β, 2 λ + β - 1) \\ \times [F (\frac{1 - β}{2}, λ - \frac{1 - β}{2}; λ + \frac{1}{2}; 1 - \frac{b^{2} {cos}^{2} α_{1}}{a c}) \\ + F (\frac{1 - β}{2}, λ - \frac{1 - β}{2}; λ + \frac{1}{2}; 1 - \frac{b^{2} {cos}^{2} (π - α_{2})}{a c})] . \end{aligned}

Proof For

x \in (- \infty, 0)

, setting

u = y / x

u = - y / x

in the following two integrals, respectively, and using Lemma 2.1, we get

\begin{array}{rcl} ω (x) & = & \int_{- \infty}^{0} \frac{1}{{(a x^{2} + 2 b x y cos α_{1} + c y^{2})}^{λ}} \frac{{(- x)}^{β + 2 λ - 1}}{{(- y)}^{β}} d y \\ + \int_{0}^{\infty} \frac{1}{{(a x^{2} + 2 b x y cos α_{2} + c y^{2})}^{λ}} \frac{{(- x)}^{β + 2 λ - 1}}{y^{β}} d y \\ = & \int_{0}^{\infty} \frac{u^{- β}}{{(c u^{2} + 2 b u cos α_{1} + a)}^{λ}} d u + \int_{0}^{\infty} \frac{u^{- β}}{{(c u^{2} + 2 b u cos (π - α_{2}) + a)}^{λ}} d u \\ = & C_{λ} . \end{array}

For

x \in (0, \infty)

, setting

u = - y / x

u = y / x

in the following two integrals, respectively, and using Lemma 2.1, we get

\begin{array}{rcl} ω (x) & = & \int_{- \infty}^{0} \frac{1}{{(a x^{2} + 2 b x y cos α_{2} + c y^{2})}^{λ}} \frac{x^{β + 2 λ - 1}}{{(- y)}^{β}} d y \\ + \int_{0}^{\infty} \frac{1}{{(a x^{2} + 2 b x y cos α_{1} + c y^{2})}^{λ}} \frac{x^{β + 2 λ - 1}}{y^{β}} d y \\ = & \int_{0}^{\infty} \frac{u^{- β}}{{(c u^{2} + 2 b u cos (π - α_{2}) + a)}^{λ}} d u + \int_{0}^{\infty} \frac{u^{- β}}{{(c u^{2} + 2 b u cos α_{1} + a)}^{λ}} d u \\ = & C_{λ} . \end{array}

By the same way, we still can find that

ω (x) = ϖ (y) = C_{λ}

(

x, y \neq 0

). The lemma is proved. □

Lemma 2.3 Let p and q be conjugate parameters with

p > 1

, and let

a, c, λ > 0

b \geq 0

1 - 2 λ < β < 1

0 < α_{1} < α_{2} < π

and

b^{2} max {{cos}^{2} α_{1}, {cos}^{2} (π - α_{2})} < a c

, and

f (x)

be a nonnegative measurable function in

(- \infty, \infty)

, then we have

\begin{aligned} J & : = \int_{- \infty}^{\infty} {| y |}^{p (1 - β) - 1} {(\int_{- \infty}^{\infty} min_{i \in {1, 2}} {\frac{1}{{(a x^{2} + 2 b x y cos α_{i} + c y^{2})}^{λ}}} f (x) d x)}^{p} d y \\ \leq C_{λ}^{p} \int_{- \infty}^{\infty} {| x |}^{- p (β + 2 λ - 2) - 1} f^{p} (x) d x . \end{aligned}

(6)

Proof By Lemma 2.2 and Hölder’s inequality [8], we have

\begin{aligned} {(\int_{- \infty}^{\infty} min_{i \in {1, 2}} {\frac{1}{{(a x^{2} + 2 b x y cos α_{i} + c y^{2})}^{λ}}} f (x) d x)}^{p} \\ = [\int_{- \infty}^{\infty} min_{i \in {1, 2}} {\frac{1}{{(a x^{2} + 2 b x y cos α_{i} + c y^{2})}^{λ}}} \\ \times {(\frac{{| x |}^{(- β - 2 λ + 2) / q}}{{| y |}^{β / p}} f (x)) (\frac{{| y |}^{β / p}}{{| x |}^{(- β - 2 λ + 2) / q}}) d x]}^{p} \\ \leq \int_{- \infty}^{\infty} min_{i \in {1, 2}} {\frac{1}{{(a x^{2} + 2 b x y cos α_{i} + c y^{2})}^{λ}}} \frac{{| x |}^{(1 - p) (β + 2 λ - 2)}}{{| y |}^{β}} f^{p} (x) d x \\ \times {(\int_{- \infty}^{\infty} min_{i \in {1, 2}} {\frac{1}{{(a x^{2} + 2 b x y cos α_{i} + c y^{2})}^{λ}}} \frac{{| y |}^{(q - 1) β}}{{| x |}^{(- β - 2 λ + 2)}} d x)}^{p - 1} \\ = C_{λ}^{p - 1} {| y |}^{p (β - 1) + 1} \int_{- \infty}^{\infty} min_{i \in {1, 2}} {\frac{1}{{(a x^{2} + 2 b x y cos α_{i} + c y^{2})}^{λ}}} \frac{{| x |}^{(1 - p) (β + 2 λ - 2)}}{{| y |}^{β}} f^{p} (x) d x . \end{aligned}

(7)

Then by the Fubini theorem, it follows that

\begin{aligned} J & \leq C_{λ}^{p - 1} \int_{- \infty}^{\infty} [\int_{- \infty}^{\infty} min_{i \in {1, 2}} {\frac{1}{{(a x^{2} + 2 b x y cos α_{i} + c y^{2})}^{λ}}} \frac{{| x |}^{(1 - p) (β + 2 λ - 2)}}{{| y |}^{β}} f^{p} (x) d x] d y \\ = C_{λ}^{p - 1} \int_{- \infty}^{\infty} ω (x) {| x |}^{- p (β + 2 λ - 2) - 1} f^{p} (x) d x \\ = C_{λ}^{p} \int_{- \infty}^{\infty} {| x |}^{- p (β + 2 λ - 2) - 1} f^{p} (x) d x . \end{aligned}

The lemma is proved. □

3 Main results

Theorem 3.1 Let p and q be conjugate parameters with

p > 1

, and let

a, c, λ > 0

b \geq 0

1 - 2 λ < β < 1

0 < α_{1} < α_{2} < π

and

b^{2} max {{cos}^{2} α_{1}, {cos}^{2} (π - α_{2})} < a c

, and

f, g \geq 0

, satisfy

0 < \int_{- \infty}^{\infty} {| x |}^{- p (β + 2 λ - 2) - 1} f^{p} (x) d x < \infty

and

0 < \int_{- \infty}^{\infty} {| y |}^{q β - 1} g^{q} (y) d y < \infty

. Then

\begin{aligned} I & : = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} min_{i \in {1, 2}} {\frac{1}{{(a x^{2} + 2 b x y cos α_{i} + c y^{2})}^{λ}}} f (x) g (y) d x d y \\ < C_{λ} {(\int_{- \infty}^{\infty} {| x |}^{- p (β + 2 λ - 2) - 1} f^{p} (x) d x)}^{1 / p} {(\int_{- \infty}^{\infty} {| y |}^{q β - 1} g^{q} (y) d y)}^{1 / q}, \end{aligned}

(8)

and

\begin{aligned} J & = \int_{- \infty}^{\infty} {| y |}^{p (1 - β) - 1} {(\int_{- \infty}^{\infty} min_{i \in {1, 2}} {\frac{1}{{(a x^{2} + 2 b x y cos α_{i} + c y^{2})}^{λ}}} f (x) d x)}^{p} d y \\ < C_{λ}^{p} \int_{- \infty}^{\infty} {| x |}^{- p (β + 2 λ - 2) - 1} f^{p} (x) d x, \end{aligned}

(9)

where the constant factors

C_{λ}

and

C_{λ}^{p}

are the best possible and

C_{λ}

is defined in Lemma 2.2. Inequalities (8) and (9) are equivalent.

Proof If (7) takes the form of the equality for a

y \in (- \infty, 0) \cup (0, \infty)

, then there exist constants A and B such that they are not all zero, and

A \frac{{| x |}^{(1 - p) (β + 2 λ - 2)}}{{| y |}^{β}} f^{p} (x) = B \frac{{| y |}^{(q - 1) β}}{{| x |}^{(- β - 2 λ + 2)}} a.e. in (- \infty, \infty) \times (- \infty, \infty) .

Hence, there exists a constant K such that

A {| x |}^{p (β + 2 λ - 2)} f^{p} (x) = B {| y |}^{q β} = K a.e. in (- \infty, \infty) \times (- \infty, \infty) .

We suppose

A \neq 0

(otherwise

B = A = 0

). Then

{| x |}^{p (β + 2 λ - 2) - 1} f^{p} (x) = K / (A | x |)

a.e. in

(- \infty, \infty)

, which contradicts the fact that

0 < \int_{- \infty}^{\infty} {| x |}^{- p (β + 2 λ - 2) - 1} f^{p} (x) d x < \infty

. Hence, (7) takes the form of a strict inequality, so does (6), and we have (9).

By Hölder’s inequality [8], we have

\begin{aligned} I = & \int_{- \infty}^{\infty} ({| y |}^{1 / q - β} \int_{- \infty}^{\infty} min_{i \in {1, 2}} {\frac{1}{{(a x^{2} + 2 b x y cos α_{i} + c y^{2})}^{λ}}} f (x) d x) \\ \times ({| y |}^{β - 1 / q} g (y)) d y \\ \leq & J^{1 / p} {(\int_{- \infty}^{\infty} {| y |}^{q β - 1} g^{q} (y) d y)}^{1 / q} . \end{aligned}

(10)

By (9), we have (8). On the other hand, suppose that (8) is valid. Set

g (y) = {| y |}^{p (1 - β) - 1} {(\int_{- \infty}^{\infty} min_{i \in {1, 2}} {\frac{1}{{(a x^{2} + 2 b x y cos α_{i} + c y^{2})}^{λ}}} f (x) d x)}^{p - 1},

then it follows

J = \int_{- \infty}^{\infty} {| y |}^{q β - 1} g^{q} (y) d y

. By (6), we have

J < \infty

. If

J = 0

, then (9) is obviously valid. If

0 < J < \infty

, then by (8), we obtain

\begin{array}{rcl} 0 & < & \int_{- \infty}^{\infty} {| y |}^{q β - 1} g^{q} (y) d y = J = I \\ < & C_{λ} {(\int_{- \infty}^{\infty} {| x |}^{- p (β + 2 λ - 2) - 1} f^{p} (x) d x)}^{1 / p} {(\int_{- \infty}^{\infty} {| y |}^{q β - 1} g^{q} (y) d y)}^{1 / q}, \end{array}

and

\begin{array}{rcl} J^{1 / p} & = & {(\int_{- \infty}^{\infty} {| y |}^{q β - 1} g^{q} (y) d y)}^{1 / p} \\ < & C_{λ} {(\int_{- \infty}^{\infty} {| x |}^{- p (β + 2 λ - 2) - 1} f^{p} (x) d x)}^{1 / p} . \end{array}

Hence, we have (9), which is equivalent to (8).

For

ε > 0

, define functions

\tilde{f} (x)

\tilde{g} (y)

as follows:

\begin{array}{rcl} \tilde{f} (x) & : = & {\begin{cases} x^{(β + 2 λ - 2) - 2 ε / p}, & x \in (1, \infty), \\ 0, & x \in [- 1, 1], \\ {(- x)}^{(β + 2 λ - 2) - 2 ε / p}, & x \in (- \infty, - 1), \end{cases} \\ \tilde{g} (y) & : = & {\begin{cases} y^{- β - 2 ε / q}, & y \in (1, \infty), \\ 0, & y \in [- 1, 1], \\ {(- y)}^{- β - 2 ε / q}, & y \in (- \infty, - 1) . \end{cases} \end{array}

Then

\tilde{L} : = {(\int_{- \infty}^{\infty} {| x |}^{- p (β + 2 λ - 2) - 1} {\tilde{f}}^{p} (x) d x)}^{1 / p} {(\int_{- \infty}^{\infty} {| y |}^{q β - 1} {\tilde{g}}^{q} (y) d y)}^{1 / q} = \frac{1}{ε},

and

\begin{array}{rcl} \tilde{I} & : = & \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} min_{i \in {1, 2}} {\frac{1}{{(a x^{2} + 2 b x y cos α_{i} + c y^{2})}^{λ}}} \tilde{f} (x) \tilde{g} (y) d x d y \\ = & I_{1} + I_{2} + I_{3} + I_{4}, \end{array}

where

\begin{array}{rcl} I_{1} & : = & \int_{- \infty}^{- 1} {(- x)}^{(β + 2 λ - 2) - 2 ε / p} [\int_{- \infty}^{- 1} \frac{{(- y)}^{- β - 2 ε / q}}{{(a x^{2} + 2 b x y cos α_{1} + c y^{2})}^{λ}} d y] d x, \\ I_{2} & : = & \int_{- \infty}^{- 1} {(- x)}^{(β + 2 λ - 2) - 2 ε / p} [\int_{1}^{\infty} \frac{y^{- β - 2 ε / q}}{{(a x^{2} + 2 b x y cos α_{2} + c y^{2})}^{λ}} d y] d x, \\ I_{3} & : = & \int_{1}^{\infty} x^{(β + 2 λ - 2) - 2 ε / p} [\int_{- \infty}^{- 1} \frac{{(- y)}^{- β - 2 ε / q}}{{(a x^{2} + 2 b x y cos α_{2} + c y^{2})}^{λ}} d y] d x, \end{array}

and

I_{4} : = \int_{1}^{\infty} x^{(β + 2 λ - 2) - 2 ε / p} [\int_{1}^{\infty} \frac{y^{- β - 2 ε / q}}{{(a x^{2} + 2 b x y cos α_{1} + c y^{2})}^{λ}} d y] d x .

Taking

u = y / x

, by the Fubini theorem, we obtain

\begin{aligned} I_{1} = & I_{4} = \int_{1}^{\infty} x^{- 1 - 2 ε} \int_{1 / x}^{\infty} \frac{u^{- β - 2 ε / q}}{{(c u^{2} + 2 b u cos α_{1} + a)}^{λ}} d u d x \\ = & \int_{1}^{\infty} x^{- 1 - 2 ε} (\int_{1 / x}^{1} \frac{u^{- β - 2 ε / q}}{{(c u^{2} + 2 b u cos α_{1} + a)}^{λ}} d u + \int_{1}^{\infty} \frac{u^{- β - 2 ε / q}}{{(c u^{2} + 2 b u cos α_{1} + a)}^{λ}} d u) d x \\ = & \int_{0}^{1} (\int_{1 / u}^{\infty} x^{- 1 - 2 ε} d x) \frac{u^{- β - 2 ε / q}}{{(c u^{2} + 2 b u cos α_{1} + a)}^{λ}} d u \\ + \frac{1}{2 ε} \int_{1}^{\infty} \frac{u^{- β - 2 ε / q}}{{(c u^{2} + 2 b u cos α_{1} + a)}^{λ}} d u \\ = & \frac{1}{2 ε} (\int_{0}^{1} \frac{u^{- β - 2 ε / q}}{{(c u^{2} + 2 b u cos α_{1} + a)}^{λ}} d u + \int_{1}^{\infty} \frac{u^{- β - 2 ε / q}}{{(c u^{2} + 2 b u cos α_{1} + a)}^{λ}} d u), \end{aligned}

and

I_{2} = I_{3} = \frac{1}{2 ε} (\int_{0}^{1} \frac{u^{- β - 2 ε / q}}{{(c u^{2} - 2 b u cos α_{2} + a)}^{λ}} d u + \int_{1}^{\infty} \frac{u^{- β - 2 ε / q}}{{(c u^{2} - 2 b u cos α_{2} + a)}^{λ}} d u) .

In view of the above results, if the constant factor

C_{λ}

in (8) is not the best possible, then there exists a positive number

\tilde{C}

with

\tilde{C} < C_{λ}

such that

\begin{aligned} \int_{0}^{1} \frac{u^{- β - 2 ε / q}}{{(c u^{2} + 2 b u cos α_{1} + a)}^{λ}} d u + \int_{1}^{\infty} \frac{u^{- β - 2 ε / q}}{{(c u^{2} + 2 b u cos α_{1} + a)}^{λ}} d u \\ + \int_{0}^{1} \frac{u^{- β - 2 ε / q}}{{(c u^{2} - 2 b u cos α_{2} + a)}^{λ}} d u + \int_{1}^{\infty} \frac{u^{- β - 2 ε / q}}{{(c u^{2} - 2 b u cos α_{2} + a)}^{λ}} d u \\ = ε \tilde{I} < ε \tilde{C} \cdot \tilde{L} = \tilde{C} . \end{aligned}

(11)

By the Fatou lemma and (11), we have

\begin{array}{rcl} C_{λ} & = & \int_{0}^{\infty} \frac{u^{- β}}{{(c u^{2} + 2 b u cos α_{1} + a)}^{λ}} d u + \int_{0}^{\infty} \frac{u^{- β}}{{(c u^{2} - 2 b u cos α_{2} + a)}^{λ}} d u \\ = & \int_{0}^{1} lim_{ε \to 0^{+}} \frac{u^{- β - 2 ε / q}}{{(c u^{2} + 2 b u cos α_{1} + a)}^{λ}} d u + \int_{1}^{\infty} lim_{ε \to 0^{+}} \frac{u^{- β - 2 ε / q}}{{(c u^{2} + 2 b u cos α_{1} + a)}^{λ}} d u \\ + \int_{0}^{1} lim_{ε \to 0^{+}} \frac{u^{- β - 2 ε / q}}{{(c u^{2} - 2 b u cos α_{2} + a)}^{λ}} d u + \int_{1}^{\infty} lim_{ε \to 0^{+}} \frac{u^{- β - 2 ε / q}}{{(c u^{2} - 2 b u cos α_{2} + a)}^{λ}} d u \\ \leq & lim_{ε \to 0^{+}} [\int_{0}^{1} \frac{u^{- β - 2 ε / q}}{{(c u^{2} + 2 b u cos α_{1} + a)}^{λ}} d u + \int_{1}^{\infty} \frac{u^{- β - 2 ε / q}}{{(c u^{2} + 2 b u cos α_{1} + a)}^{λ}} d u \\ + \int_{0}^{1} \frac{u^{- β - 2 ε / q}}{{(c u^{2} - 2 b u cos α_{2} + a)}^{λ}} d u + \int_{1}^{\infty} \frac{u^{- β - 2 ε / q}}{{(c u^{2} - 2 b u cos α_{2} + a)}^{λ}} d u] \\ \leq & \tilde{C}, \end{array}

which contradicts the fact that

\tilde{C} < C_{λ}

. Hence, the constant factor

C_{λ}

in (8) is the best possible.

If the constant factor in (9) is not the best possible, then by (10), we may get a contradiction that the constant factor in (8) is not the best possible. Thus the theorem is proved. □

Remark 1 Setting

λ = a = b = c = 1

in Theorem 3.1, we have (3) and (4).

Remark 2 Setting

λ = 1 / 2

in Theorem 3.1, we have the following particular results:

\begin{aligned} \int_{- \infty}^{\infty} & \int_{- \infty}^{\infty} min_{i \in {1, 2}} {\frac{1}{\sqrt{a x^{2} + 2 b x y cos α_{i} + c y^{2}}}} f (x) g (y) d x d y \\ < & C_{1 / 2} {(\int_{- \infty}^{\infty} {| x |}^{- p (β - 1) - 1} f^{p} (x) d x)}^{1 / p} {(\int_{- \infty}^{\infty} {| y |}^{q β - 1} g^{q} (y) d y)}^{1 / q}, \end{aligned}

and

\begin{aligned} \int_{- \infty}^{\infty} & {| y |}^{p (1 - β) - 1} {(\int_{- \infty}^{\infty} min_{i \in {1, 2}} {\frac{1}{\sqrt{a x^{2} + 2 b x y cos α_{i} + c y^{2}}}} f (x) d x)}^{p} d y \\ < & C_{1 / 2}^{p} \int_{- \infty}^{\infty} {| x |}^{- p (β - 1) - 1} f^{p} (x) d x . \end{aligned}

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors contributed equally and significantly in writing this paper. Both authors read and approved the final manuscript.

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On a generalization of a Hilbert-type integral inequality in the whole plane with a hypergeometric function

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Abstract

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Weitere Artikel der Ausgabe 1/2013

Some probability inequalities for a class of random variables and their applications

A remark on algebraic curves derived from convolution sums

△-convergence for mixed-type total asymptotically nonexpansive mappings in hyperbolic spaces

Ordering the oriented unicyclic graphs whose skew-spectral radius is bounded by 2

A two-level domain decomposition algorithm for linear complementarity problem

Viscosity iterative method for a new general system of variational inequalities in Banach spaces

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