# 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 ):

[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  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  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 ). 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. ).

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. ), 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. ) 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 ), 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 ) 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 ), 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.

References

 G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities, Cambridge University Press, Cambridge, USA, 2nd edition, 1988.

 B. C. Yang, The Norm of Operator and Hilbert-Type Inequalities, Science Press, Beijing, China, 2009.

 B. Yang, Hilbert-Type Integral Inequalities, vol. 1, Bentham Science Publishers, The United Arab Emirates, 2009.

 B. Yang, "On the norm of an integral operator and applications," Journal of Mathematical Analysis and Applications, vol. 321, no. 1, pp. 182-192, 2006.

 J. Xu, "Hardy-Hilbert's inequalities with two parameters," Advances in Mathematics, vol. 36, no. 2, pp. 189-202, 2007.

 D. M. Xin, "A Hilbert-type integral inequality with a homogeneous kernel of zero degree," Mathematical Theory and Applications, vol. 30, no. 2, pp. 70-74, 2010.

 L. Debnath and B. Yang, "Recent developments of Hilbert-type discrete and integral inequalities with applications," International Journal of Mathematics and Mathematical Sciences, vol. 2012, Article ID 871845, 29 pages, 2012.

 Y. Hong, "On the structure character of Hilberts type integral inequality with homogeneous kernal and applications," Journal of Jilin University (Science Edition), vol. 55, no. 2, pp. 189-194, 2017.

 M. T. Rassias and B. Yang, "On half-discrete Hilbert's inequality," Applied Mathematics and Computation, vol. 220, pp. 75-93, 2013.

 M. T. Rassias and B. Yang, "A multidimensional half-discrete Hilbert-type inequality and the Riemann zeta function," Applied Mathematics and Computation, vol. 225, pp. 263-277, 2013.

 Q. Huang, "A new extension of a Hardy-Hilbert-type inequality," Journal of Inequalities and Applications, vol. 2015, article 397, 12 pages, 2015.

 B. He and Q. Wang, "A multiple Hilbert-type discrete inequality with a new kernel and best possible constant factor," Journal of Mathematical Analysis and Applications, vol. 431, no. 2, pp. 889-902, 2015.

 M. Krnic' and J. Pecaric', "General Hilberts and Hardys inequalities," Mathematical Inequalities & Applications, vol. 8, no. 1, pp. 29-51, 2005.

 I. Peric and P. Vukovic, "Multiple Hilbert's type inequalities with a homogeneous kernel," Banach Journal of Mathematical Analysis, vol. 5, no. 2, pp. 33-43, 2011.

 V. Adiyasuren, Ts. Batbold, and M. Krnic, "Multiple Hilbert-type inequalities involving some differential operators," Banach Journal of Mathematical Analysis, vol. 10, no. 2, pp. 320-337, 2016.

 Q. Chen and B. Yang, "A survey on the study of Hilbert-type inequalities," Journal of Inequalities and Applications, 2015:302, 29 pages, 2015.

 B. Yang, "A new Hilbert's type integral inequality," Soochow Journal of Mathematics, vol. 33, no. 4, pp. 849-859, 2007

 Z. X. Wang and D. R. Guo, Introduction to Special Functions, Science Press, Beijing, China, 1979.

 B. He and B. C. Yang, "A Hilbert-type integral inequality with a homogeneous kernel of 0-degree and a hypergeometric function," Mathematics in Practice and Theory, vol. 40, no. 18, pp. 203-211, 2010.

 Z. Zeng and Z. T. Xie, "On a new hilbert-type intergral inequality with the intergral in whole plane," Journal of Inequalities and Applications, vol. 2010, Article ID 256796, 8 pages, 2010.

 Z. Xie, Z. Zeng, and Y. Sun, "A new Hilbert-type inequality with the homogeneous kernel of degree-2," Advances and Applications in Mathematical Sciences, vol. 12, no. 7, pp. 391-401, 2013.

 Z. Zhen, K. R. R. Gandhi, and Z. Xie, "A new Hilbert-type inequality with the homogeneous kernel of degree-2 and with the integral," Bulletin of Mathematical Sciences & Applications, vol. 3, no. 1, pp. 11-20, 2014.

 Z. Gu and B. Yang, "A Hilbert-type integral inequality in the whole plane with a non-homogeneous kernel and a few parameters," Journal of Inequalities and Applications, 2015:314, 9 pages, 2015.

 J. C. Kuang, Real and functional analysis (Continuation), vol. second volume, Higher Education Press, Beijing, China, 2015.

 J. C. Kuang, Applied Inequalities, Shangdong Science and Technology Press, Jinan, China, 2004.

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

Title Annotation: Printer friendly Cite/link Email Feedback Research Article Xin, Dongmei; Yang, Bicheng; Wang, Aizhen Journal of Function Spaces 9CHIN Jan 1, 2018 3532 Certain Geometric Properties of Generalized Dini Functions. Generalized Euler-Lagrange Equations for Fuzzy Fractional Variational Problems under gH-Atangana-Baleanu Differentiability. Equality