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Equivalent Property of a Hilbert-Type Integral Inequality Related to the Beta Function in the Whole Plane.

1. Introduction

Suppose that p > 1,1/p + 1/q = 1, f(x),g(y) [less than or equal to] 0, 0 < [[integral].sup.[infinity].sub.0] [f.sup.p](x)dx < [infinity], and 0 < [[integral].sup.[infinity].sub.0] [g.sup.q](y)dy < [infinity]. We have the following well-known Hardy-Hilbert's integral inequality (see [1]):

[mathematical expression not reproducible], (1)

where the constant factor [pi]/ sin([pi]/p) is the best possible. For p = q = 2, (1) reduces to the well-known Hilbert's integral inequality. By using the weight functions, some extensions of (1) were given by [2, 3]. A few Hilbert-type inequalities with the homogenous and nonhomogenous kernels were provided by [4-7]. In 2017, Hong [8] also gave two equivalent statements between Hilbert-type inequalities with the general homogenous kernel and parameters. Some other kinds of Hilbert-type inequalities were obtained by [9-16].

In 2007, Yang [17] gave a Hilbert-type integral inequality in the whole plane as follows:

[mathematical expression not reproducible], (2)

with the best possible constant factor B([lambda]/2, [lambda]/2) ([lambda] > 0, B(u, v) is the beta function) (see [18]). He et al. [19-23] proved a few Hilbert-type integral inequalities in the whole plane with the best possible constant factors.

In this paper, by means of the technique of real analysis and the weight functions, a few equivalent statements of a Hilbert-type integral inequality with the nonhomogeneous kernel in the whole plane similar to (2) are obtained. The constant factor related to the beta function is proved to be the best possible. As applications, the case of the homogeneous kernel, the operator expressions, and a few corollaries are considered.

2. An Example and Two Lemmas

Example 1. For R = (-[infinity], [infinity]), [R.sub.+] = (0,[infinity]), we set h(u) := [(max{w, 1}).sup.[alpha]+[beta]]/[[absolute value of u - i].sup.[lambda]+[alpha]][(minjw, 1}).sup.[beta]] (u [member of] [R.sub.+]), and then for a,b = 0,

[mathematical expression not reproducible], (3)

For [sigma],[mu] > [beta],[sigma] + [mu] = [lambda] < 1 - [alpha] ([alpha] + 2[beta] < 1), in view of h([v.sup.-1])[v.sup.1-[sigma]] = h(v)[v.sup.u-1] (0 < v < 1), we find

[mathematical expression not reproducible], (4)

where B(u, v) := [[integral].sup.1.sub.0] [(1 - t).sup.u-1][t.sup.v-1]dt (u, v > 0) is the beta function (cf. [18]).

In particular, (i) for [alpha] = 0, we have [sigma],[mu] >[beta], [sigma] + [mu] = [lambda] < 1 ([beta] < 1/2),[h.sub.1](u) = [(maxjw, 1}).sup.[beta]]/[[absolute value of ]u - 1].sup.lambda]][[(min{u,1}).sup.[beta]](u > 0), and

[k.sup.(1).sub.[lambda]] ([sigma]) = B (1 - [lambda], [sigma] - [beta]) + B (1 - [lambda], [mu] - [beta]); (5)

(ii) for [beta] = 0, we have [sigma], [mu] > 0, [sigma] + [mu] = [lambda] < 1 - [alpha] ([alpha] < 1), [h.sub.2](u) = [(max{u, 1}).sup.[alpha]]/[[absolute value of u - 1].sup.[lambda]+[alpha]](u > 0), and

[k.sup.(2).sub.[lambda]]([sigma]) = B (1 - [lambda] - [alpha], [sigma], [alpha]) + B (1 - [lambda] - [mu]; (6)

(iii) for [beta] = -[alpha], we have [sigma], [mu] > -[alpha], [alpha] + [mu] = [lambda] < 1 - [alpha] ([alpha] > -1), [h.sub.3](u) = [(min{u, 1}).sup.[alpha]]/[[absolute value of u - 1].sup.[lambda]+[alpha]](u > 0), and

[k.sup.(3).sub.[lambda]]([sigma]) = B (1 - [lambda] - [alpha], [sigma] + [alpha]) + B (1 - [lambda] - [alpha], Mu + [alpha]). (7)

In the case of (iii), for [alpha] = 0, we have [sigma], [mu] > 0, [sigma] + [mu] = [lambda] < 1, [h.sub.4](u) = 1/[[absolute value of u - 1].sup.[lambda]] (u > 0), and

[k.sup.(4).sub.[lambda]]([sigma]) = B (1 - [lambda], [sigma]) + B (1 - [lambda] - [mu]. (8)

In the following, we assume that p > 1,1/p + 1/q = 1,a,b [not equal to] 0, [[sigma].sub.1], [sigma][member of] R, [sigma], [mu] > [beta], [sigma] + [mu] = [lambda] < 1 - [alpha]([alpha] + 2[beta] < 1), and

[mathematical expression not reproducible]. (9)

For n [member of] N = {1,2, ... }, we define two sets [E.sub.c] := {t [member of] R; ct [greater than or equal to] 0}, [F.sub.c] := R [E.sub.c] = {t [member of] R; ct < 0} (c = a,b), and the following two expressions:

[mathematical expression not reproducible], (10)

[mathematical expression not reproducible]1 (11)

Setting u = [e.sup.ax+by] in (10), in view of Fubini theorem (cf. [24]), it follows that

[mathematical expression not reproducible]. (12)

In the same way, we find that

[mathematical expression not reproducible]. (13)

Lemma 2. If there exists a constant M, such that for any nonnegative measurable functions f(x) and g(y) in R, the following inequality

[mathematical expression not reproducible] (14)

holds true, then we have [[sigma].sub.1] = [sigma].

Proof. (i) If [[sigma].sub.1] < [sigma], then for n > 1/([sigma] - [[sigma].sub.1])(n [member of] N), we set two functions

[mathematical expression not reproducible] (15)

and obtain

[mathematical expression not reproducible], (16)

By (12) and (14), we find

[mathematical expression not reproducible], (17)

For any n > 1/([sigma] - [[sigma].sub.1]) (n [member of] N), [[sigma].sub.1] - [sigma] + 1/n < 0, it follows that [mathematical expression not reproducible]. In view of [mathematical expression not reproducible], by (17), we find that [mathematical expression not reproducible], which is a contradiction.

(ii) If [[sigma].sub.1] > [sigma], then for n > (1/([[sigma].sub.1] - [sigma])) (n [member of] N), we set functions

[mathematical expression not reproducible], (18)

and find

[mathematical expression not reproducible], (19)

By (13) and (14), we obtain

[mathematical expression not reproducible]. (20)

For n > 1/([[sigma].sub.1] - [sigma])(n [member of] N), [[sigma].sub.1] - [sigma] - 1/n > 0, it follows that [mathematical expression not reproducible]. By (20), in view of [mathematical expression not reproducible], we have [mathematical expression not reproducible], which is a contradiction.

Hence, we conclude that [[sigma].sub.1] = [sigma].

The lemma is proved.

For [[sigma].sub.1] = [sigma], we have the following.

Lemma 3. If there exists a constant M, such that for any nonnegative measurable functions f(x) and g(y) in R, the following inequality

[mathematical expression not reproducible] (21)

holds true, then we have M [less than or equal to] [K.sub.[lambda]]([sigma])(> 0

Proof. By (12), for [[sigma].sub.1] = [sigma], we obtain

[mathematical expression not reproducible]. (22)

We use inequality [I.sub.1] [less than or equal to] M[[??].sub.2] (for [[sigma].sub.1] = [sigma]) as follows:

[mathematical expression not reproducible]. (23)

By Fatou lemma (cf. [24]) and (23), it follows that

[mathematical expression not reproducible]. (24)

The lemma is proved.

3. Main Results and Some Corollaries

Theorem 4. If M is a constant, then the following statements (i), (ii), and (iii) are equivalent:

(i) For any nonnegative measurable function f(x) in R, we have the following inequality:

[mathematical expression not reproducible]. (25)

(ii) For any nonnegative measurable functions f(x) and g(y) in R, we have thefollowing inequality:

[mathematical expression not reproducible]. (26)

(iii) [[sigma].sub.1] = [sigma], and M > [K.sub.[lambda]]([sigma])(> 0).

Proof. (i) => (ii). By Holder's inequality (see [25]), we have

[mathematical expression not reproducible]. (27)

Then by (25), we have (26).

(ii) => (iii). By Lemma 2, we have o1 = a. Then by Lemma 3, we have M [greater than or equal to] [K.sub.[lambda]([sigma])(> 0).

(iii) => (i). Setting u = [e.sup.ax+by,] we obtain the following weight functions: for y,x [member of] R,

[mathematical expression not reproducible], (28)

[mathematical expression not reproducible]. (29)

By Holder's inequality with weight and (28), we have

[mathematical expression not reproducible]. (30)

For [[sigma].sub.1] = [sigma], by Fubini theorem (see [24]) and (29), we have

[mathematical expression not reproducible]. (31)

For [K.sub.[lambda]([sigma])[less than or equal to] M, we have (25).

Therefore, the statements (i), (ii), and (iii) are equivalent. The theorem is proved.

Theorem 5. The following statements (i) and (ii) are valid and equivalent:

(i) For any [mathematical expression not reproducible], we have the following inequality:

[mathematical expression not reproducible]. (32)

(ii) For any f(x) [mathematical expression not reproducible], we have the following inequality:

[mathematical expression not reproducible]. (33)

Moreover, the constant factor [K.sub.[lambda]]([sigma]) in (32) and (33) is the best possible.

In particular, for [alpha] = [beta] = 0, [sigma], [mu] > 0, [sigma] + [mu] = [lambda] < 1

[mathematical expression not reproducible], (34)

we have the following equivalent inequalities with the best possible constant factor [[??].sub.[lambda]]([sigma]):

[mathematical expression not reproducible], (35)

[mathematical expression not reproducible]. (36)

Proof. We first prove that (32) is valid. If (30) takes the form of equality for a y [member of] R, then (see [25]), there exst constants A and B, such that they are not all zero, and

[mathematical expression not reproducible]. (37)

We suppose that A [not equal to] 0 (otherwise B ? A = 0). Then it follows that

[mathematical expression not reproducible], (38)

which contradicts the fact that 0 < [[integral].sup. [infinity]].sub.-[infinity]][(f(x)/[e.sup.[sigma]ax).sup.p]dx < [infinity].

Hence, (30) takes the form of strict inequality. For [[sigma].sub.1] = [sigma] by the proof of Theorem 4, we obtain (32).

(i) => (ii). By (27) (for [[sigma].sub.1] = [sigma]) and (32), we have (33).

(ii) => (i). We set the following function:

[mathematical expression not reproducible]. (39)

If [J.sub.1] = [infinity], then it is impossible since (32) is valid; if [J.sub.1] = 0, then (32) is trivially valid. In the following, we suppose that 0< [J.sub.1] < [infinity]. By (33), we have

[mathematical expression not reproducible]; (40)

namely, (32) follows, which is equivalent to (33).

Hence, Statements (i) and (ii) are valid and equivalent.

If there exists a constant M [less than or equal to] [K.sup.[lambda]]([sigma]), such that (33) is valid when replacing [K.sup.[lambda]]([sigma]) by M, then by Lemma 3, we have [K.sup.[lambda]]([sigma]) < M. Hence, the constant factor M = [K.sup.[lambda]]([sigma]) in (33) is the best possible.

The constant factor [K.sup.[lambda]]([sigma]) in (32) is still the best possible. Otherwise, by (27) (for [[sigma].sub.1] = [sigma]), we would reach a contradiction that the constant factor Kx(a) in (33) is not the best possible.

The theorem is proved.

For g(y) = [e.sup.-[lambda]by]G(y), and [[mu].sub.1] = [lambda] - [[sigma].sub.1] in Theorems 4 and 5, then replacing b (G(y)) by -b (g(y)), setting

[mathematical expression not reproducible]. (41)

we have the following corollaries.

Corollary 6. If M is a constant, then the following statements (i), (ii), and (iii) are equivalent:

(i) For any nonnegative measurable function f(x) in R, we have the following inequality:

[mathematical expression not reproducible]. (42)

(ii) For any nonnegative measurable functions f(x) and g(y) in R, we have the following inequality:

[mathematical expression not reproducible]. (43)

(iii) [[mu].sub.1] = [mu], and M [greater than or equal to][K.sub.[lambda]]([sigma])(>0).

Corollary 7. The following statements (i) and (ii) are valid and equivalent:

(i) For any f(x) [greater than or equal to] 0, satisfying 0 < [[integral].sup.[infinity].sub.-[infinity][(f(x)/[e.sup.[sigma]ax]).sup.p]dx < [infinity], we have the following inequality:

[mathematical expression not reproducible]. (44)

(ii) For any f(x) [greater than or equal to] 0, satisfying 0 < [[integral].sup.[infinity].sub.-[infinity][e.sup.[sigma]/2-[sigma])ax](f(x)].sup.p]dx [infinity], and g(y) [greater than or equal to] 0, satisfying 0 < [[integral].sup.[infinity].sub.-[infinity][e.sup.[sigma]/2-[sigma])ax](f(x)].sup.p]dy > [infinity], we have the following inequality:

[mathematical expression not reproducible]. (45)

Moreover, the constant factor [K.sub.[lambda]]([sigma]) in (44) and (45) is the best possible.

In particular, for [alpha] = [beta] = 0, [sigma], [beta] > 0, [sigma] + [mu] = [lambda] < 1, we have the following equivalent inequalities with the best possible constant factor [[??].sub.[lambda]]([sigma])([sigma]):

[mathematical expression not reproducible], (46)

[mathematical expression not reproducible], (47)

In (35) and (36), setting F(x) = [e.sup.([lambda]a/2)x]f(x),G(y)] = [e.sup.([lambda]b/2)y] g(y), then replacing back F(x)(G(y)) by f(x)(g(y)), and introducing the hyperbolic sine function as sinh(s) = ([e.sup.s] - [e.sup.-s])/2, we have

Corollary 8. If [sigma], [mu] > 0, [sigma] + [mu] = [lambda] < 1, then the following statements (i) and (ii) are valid and equivalent:

(i) For any f(x) > [greater than or equal to], satisfying 0 < [[integral].sup.[infinity].sub.-[infinity][e.sup.[lambda]/2-[sigma])ax](f(x)].sup.p]dx < [infinity], we have the following inequality:

[mathematical expression not reproducible]. (48)

(ii) For any f(x) > [greater than or equal to], satisfying 0 < [[integral].sup.[infinity].sub.-[infinity][e.sup.[lambda]/2-[sigma])ax](f(x)].sup.p]dx < [infinity] and g(y) [greater than or equal to] 0, satisfying 0 < [[integral].sup.[infinity].sub.-[infinity][e.sup.[lambda]/2-[sigma])ax](f(x)].sup.p]dy < [infinity], we have the following inequality:

[mathematical expression not reproducible]. (49)

Moreover, the constant factor 2[[??].sub.x]([sigma]) in (48) and (49) is the best possible.

4. Operator Expressions

We set the following functions: [phi](x) := [e.sup.-p[sigma]ax],[psi](y) := [e.sup.-q[sigma]by], [phi](y) := [e.sup.-q[mu]by], wherefrom, [[psi].sup.1-p](y) = [e.sup.-q[sigma]by], [[phi].sup.1-p] (y) = [e.sup.-q[mu]by] (x, y [member of] R), and define the following real normed linear spaces:

[mathematical expression not reproducible], (50)

wherefrom,

[mathematical expression not reproducible]. (50)

(a) In view of Theorem 5, for f[member of] [L.sub.p,[phi]] (R), setting

[mathematical expression not reproducible], (52)

by (34). We have

[mathematical expression not reproducible]. (53)

Definition 9. Define a Hilbert-type integral operator with the nonhomogeneous kernel [mathematical expression not reproducible] as follows: for any f [member of] [L.sub.p]f(R), there exists a unique representation [mathematical expression not reproducible], satisfying for any y[member of]R, [T.sup.(1)]f(y) = [h.sub.1](y).

In view of (53), it follows that

[mathematical expression not reproducible], (54)

and then the operator [T.sup.(1)] is bounded satisfying

[mathematical expression not reproducible]. (55)

If we define the formal inner product of [T.sup.(1)] f and g as follows:

[mathematical expression not reproducible], (56)

then we can rewrite Theorem 5 as follows.

Theorem 10. The following statements (i) and (ii) are valid and equivalent:

(i) For any f(x) [greater than or equal to] 0, f [member of] [L.sub.p,[phi](R), satisfying [[parallel]f[parallel].sub.p,[phi]] > 0, we have the following inequality:

[mathematical expression not reproducible]. (57)

(ii) For any f(x),g(y) [greater than or equal to] 0, f [member of] [L.sub.p,[phi](R), g [member of] [L.sub.p,[psi](R), satisfying [[parallel]f[parallel].sub.p,[phi]] > 0, and [[parallel]g[parallel].sub.q,[psi]] > 0, we have the following inequality:

[mathematical expression not reproducible]. (58)

Moreover, the constant factor [K.sub.[lambda]]([sigma]) in (57) and (58) is the best possible, namely,

[parallel][T.sup.(1)][parallel] = [K.sub.[lambda]]([sigma])([sigma]). (59)

(b) In view of Corollary 7, for f [member of] [L.sub.p,[phi]](R), setting

[mathematical expression not reproducible], (60)

by (44), we have

[mathematical expression not reproducible]. (61)

Definition 11. Define a Hilbert-type integral operator with the homogeneous kernel [mathematical expression not reproducible] as follows: for any f [member of] [L.sub.p,[phi](R), there exists a unique representation [mathematical expression not reproducible], satisfying for any y [member of] R, [T.sup.(2)]f(y) = [h.sub.2](y).

In view of (61), it follows that

[mathematical expression not reproducible], (62)

and then the operator [T.sup.(2)] is bounded satisfying

[mathematical expression not reproducible]. (63)

If we define the formal inner product of [T.sup.(2)] f and g as follows:

[mathematical expression not reproducible], (64)

then we can rewrite Corollary 7 as follows.

Corollary 12. The following statements (i) and (ii) are valid and equivalent:

(i) For any f(x) [greater than or equal to] 0, f [member of] [L.sub.p,[phi]](R), satisfying [[parallel]f[parallel].sub.p,[phi]] > 0, we have the following inequality:

[mathematical expression not reproducible]. (65)

(ii) For any f(x),g(y) [greater than or equal to] 0 , f [member of] [L.sub.p,[phi]](R),g [member of] [L.sub.q,[phi]](R), satisfying [[parallel]f[parallel].sub.p,[phi]] > 0, and [[parallel]g[parallel].sub.q,[phi]] > 0, we have the following inequality:

[mathematical expression not reproducible]. (66)

Moreover, the constant factor [K.sub.[lambda]]([sigma]) in (65) and (66) is the best possible, namely,

[[parallel][T.sup.(2)][parallel] = [K.sub.[lambda]] ([sigma]). (67)

https://doi.org/10.1155/2018/2691816

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by the National Natural Science Foundation (nos. 61370186 and 61640222) and Science and Technology Planning Project Item of Guangzhou City (no. 201707010229). We are grateful for this help.

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Dongmei Xin, Bicheng Yang (iD), and Aizhen Wang

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

Correspondence should be addressed to Bicheng Yang; bcyang818@163.com

Received 22 March 2018; Accepted 16 August 2018; Published 2 September 2018

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