An extended Hilbert s integral inequality in the whole plane with parameters

Transcription

1 He et al. Journal of Ineualities and Alications 88:6 htts://doi.org/.86/s z R E S E A R C H Oen Access An extended Hilbert s integral ineuality in the whole lane with arameters Leing He *,YinLi and Bicheng Yang * Corresondence: jdheleing@63.com College of Mathematics and Statistics, Jishou University, Jishou, P.R. China Full list of author information is available at the end of the article Abstract By introducing indeendent arameters and interval variables, alying the weight functions and the techniue of real analysis, an extended Hilbert s integral ineuality in the whole lane with arameters and a best ossible constant factor is rovided. The euivalent forms, the reverses, and the related homogeneous forms with articular arameters are considered. Meanwhile, an extended Hilbert s integral oerator in the whole lane is defined, and the oerator exressions for the euivalent ineualities are obtained. MSC: 6D5; 47A5 Keywords: Hilbert s integral ineuality; Weight function; Euivalent form; Oerator; Norm Introduction Assuming that f x) and g y), we have the following wellknown Hilbert s integral ineuality with the best ossible constant factor π [: x + y π f x) g y). ) In 95, Har gave an extension of ) as follows [: If >, +,f x), satisfying f x),andgy), satisfying g y),thenwehave x + y π sin π ) f x) g y) ), ) where the constant factor π/ sin π ) is still the best ossible. We call ) Har Hilbert s integral ineuality, which with )is imortantin analysis andits alications cf. [, 3). In 934, Har et al. gave an extension of ) with the general homogeneous kernel of degree see [, Theorem 39). Meanwhile, a Hilbert-tye integral ineuality with the general nonhomogeneous kernel is rovided see [, Theorem 35): If hx)>, hx)x s φs) R +,), then ) hxy) φ x f x) g y) ). 3) The Authors8. This article is distributed under the terms of the Creative Commons Attribution 4. International License htt://creativecommons.org/licenses/by/4./), which ermits unrestricted use, distribution, and reroduction in any medium, rovided you give aroriate credit to the original authors) and the source, rovide a link to the Creative Commons license, and indicate if changes were made.

2 He et al. Journal of Ineualities and Alications 88:6 Page of By introducing an indeendent arameter, ) and the beta function, in 998, Yang [4 gave an extension of ) as follows: x + y) B, ) x f x) y g y), 4) where the constant factor B, ) is the best ossible, and t v Bu, v): dt u, v >) + t) u+v is the beta function cf. [5). In 7, Li [6 gave an extension of 4) and Yang [7 rovided the following Hilbert-tye integral ineuality with the nonhomogeneous kernel: +xy) π x ) f x) y ) g y) ). 5) Since then, a lot of authors have continued to discuss this toic cf. [8 4). In this aer, by introducing indeendent arameters and interval variables, alying the weight functions and the techniue of real analysis, a Hilbert-tye integral ineuality in the whole lane with arameters and a best ossible constant factor is rovided as follows: x + y ) [ Bμ, σ ) x μ) f x) y σ ) g y) 6) μ, σ >,μ + σ ), which is an extension of 4). The more general form of 6) with arameters, the euivalent ineualities, the reverses, and the related homogeneous form with the articular arameter are considered. Meanwhile, an extended Hilbert s integral oerator in the whole lane is defined, and the oerator exressions for the euivalent ineualities are obtained. Weight functions and an initial ineuality Definition Suose that δ {, }, α, β,μ, σ >,μ + σ. Definethefollowing weight functions: ω δ σ, y): ) σ ϖ δ σ, x): ) δσ ) δσ [ + ) δ ) y R, ) ), 7) ) σ [ + ) δ ) x R). 8)

3 He et al. Journal of Ineualities and Alications 88:6 Page 3 of We find ω δ σ, y) ) { σ ) { σ x + αx) δσ [ + x + αx) δ ) + x αx) δσ [ + x αx) δ ) + x + αx) δσ } [ + x + αx) δ ) x + αx) δσ }. [ + x + αx) δ ) For fixed y ),settingu x αx) δ ) in the above first integral, we obtain that x y +βy) δ u δ, y +βy) δ u δ du,and α δ α) x αx) δσ [ + x αx) δ ) u σ δ u δ + u) ) σ δ α) δ α)) σ u σ + u) du. du Inthesameway,settingu x + αx) δ ) in the above second integral, it follows that Hence, we have x + αx) δσ [ + x + αx) δ ) u σ + α)) σ + u) du. ω δ σ, y) α + ) +α By 8), we find u σ + u) du K ασ ): Bμ, σ ). 9) α ϖ δ σ, x) ) { δσ ) { δσ y + βy) σ [ + ) δ y + βy) + y βy) σ [ + ) δ y βy) + y + βy) σ } [ + ) δ y + βy) y + βy) σ }. [ + ) δ y + βy) For fixed x ),settingu ) δ β)y in the above first integral, we obtain y u β) x +αx) δ,and y βy) σ [ + ) δ y βy) u σ du + u) ) δσ ) β)) δ β)) δσ u σ + u) du. In the same way, setting u ) δ + β)y in the above second integral, we find y + βy) σ [ + ) δ y + βy) u σ + β)) δσ + u) du,

4 He et al. Journal of Ineualities and Alications 88:6 Page 4 of and then ϖ δ σ, x) β + ) +β u σ + u) du K βσ ): Bμ, σ ). ) β Theorem Suose that > ), +,δ {, }, α, β,μ, σ >,μ+σ, and Kσ ):Kβ σ )K Bμ, σ ) α σ ). ) β / α ) / If f x) x R), satisfying ) δσ) f x), then i) for >, we have the following ineuality: { ) σ f x) } J : [ + ) δ ) Bμ, σ ) β / α / [ ) δσ) f x) ; ) ii) for, we have the reverse of ). Proof i) For >, by Hölder s ineuality with weight [5and7), when y,wefind f x) [ + ) δ ) [ + ) δ ) [ ) δσ)/ f x) ) σ )/ { ) δσ) ) f x) [ + ) δ ) σ { } σ ) ) ) [ + ) δ ) δσ ω δ σ, y) σ { } [ ) σ )/ ) δσ)/ ) δσ) ) [ + ) δ ) f x) σ }. 3) We rove that 3) takes the form of strict ineuality. Otherwise, there exists y such that 3) takes the formof euality. Then thereexist constants Aand B such that they are not all zero, and [5 ) δσ) ) f x) A [ + ) δ ) σ B σ ) ) a.e. in R. [ + ) δ ) δσ If A,thenB, which is imossible. We suose that A,namely ) δσ) f ) B) σ x) A) a.e. in R, which contradicts the fact that ) δσ) f x).

5 He et al. Journal of Ineualities and Alications 88:6 Page 5 of Then by ) and Fubini s theorem [6, we find { J Kα σ ) ) δσ) ) } [ + ) δ ) ) f x) σ { Kα σ ) ϖ δ σ, x) δσ) f x) }. 4) In view of )and), we have ). ii) For, by the reverse Hölder s ineuality [5, 7), and 9), we have the reverses of 3) and4). Then, by )and), we obtain thereverse of ). The theorem is roved. 3 Main results Theorem Suose that >, +,δ {, }, α, β,μ, σ >,μ + σ. If f x), gy), satisfying ) δσ) f ) σ ) g x) and y), then we have the following ineuality euivalent to ): I : [ + ) δ ) Bμ, σ ) ) δσ) f x) β / α / [ ) σ ) g y), 5) Bμ,σ ) where the constant Kσ ) in 5) and ) is the best ossible. β / α / In articular, for δ,we have the following euivalent ineualities with the nonhomogeneous kernel and the best ossible constant factor Kσ ) Bμ,σ ) β / α / : { ) σ f x) } [ + )) Bμ, σ ) ) σ ) f x), 6) β / α / [ + )) Bμ, σ ) ) σ ) f x) β / α / [ ) σ ) g y). 7)

6 He et al. Journal of Ineualities and Alications 88:6 Page 6 of Proof By Hölder s ineuality, we find I { y + βy ) σ } f x) [ y [ + )) + βy σ gy) [ ) σ ) g J y), 8) and then by )wehave5). On the other hand, suose that 5) is valid. We set gy): ) { } σ f x) [ + ) δ ) y R). By 4) and the assumtions, we find J.IfJ,then) is trivially valid; if J >,then by 5)weobtain ) σ ) g y) J I ) δσ) f Kσ ) x) ) σ ) g y), ) σ ) g J y) [ ) δσ) f Kσ ) x). Hence, we have ), which is euivalent to 5). For n N {,,...}, n > μ,wedefinethesets : {x R; x δ }, E + δ : { x R + ; x δ }, E δ : { x R + ; x) δ }, and the following functions: ) δσ n ), x, f x):, x R\, ) σ + n, y [,, gy):, y, ), ).

7 He et al. Journal of Ineualities and Alications 88:6 Page 7 of Then we obtain that ) δ E δ n x + αx) n δ + x + αx) n δ + [ α) n δ ++α) n δ x n δ + [ α) n δ ++α) n δ u δ ) δ n δ u δ du u x δ) [ α) n δ ++α) n δ u n du [ α) n δ ++α) n δ n, n y + βy n + y + βy n [ β) n ++β) n y n [ β) n ++β) n n, ) L : δσ) f ) σ x) ) g y) ) δ n n n [ α) δ n ++α) δ n [ β n ++β n. Inthesameway,westillfindthat ) δμ+ n ) [ α) δμ+ n ) ++α) δμ+ ) n μ +, n ) μ n ) [ β) μ n ) ++β) μ ) n R\[, μ. n Hence, we obtain Ĩ : f x) gy) [ + ) δ ) ) δσ n ) ) σ + n ) [ + ) δ ) ) δ n ϖ δ σ + n, x R\[, ) ) δσ n ) ) σ + n [ + ) δ ) ) δ n ϖ δ σ + n, x )

8 He et al. Journal of Ineualities and Alications 88:6 Page 8 of K β σ + R\[, n ) ) δμ+ ) δσ n ) ) σ + n [) δ ) ) δ n n ) R\[, ) μ n ) K β σ + ) [ δ α) n ++α) δ n n O). n If there exists a constant k Kσ )suchthat5) is valid when relacing Kσ )byk,then in articular, we have nĩ n f x) gy) [ + ) δ ) k n L. In view of the above results, it follows that K β σ + ) [ δ α) n ++α) δ n n n O k [ α) δ n ++α) δ n [ β n ++β n. 9) For n,wefind 4Bμ, σ ) α ) β ) k, α β namely Kσ ) k.hence,k Kσ ) is the best ossible constant factor of 5). The constant factor Kσ )in) is also the best ossible. Otherwise, we can conclude a contradiction by 8) that the constant factor in 7) is not the best ossible. The theorem is roved. Theorem 3 With regards to the assumtions of Theorem, relacing >by,we have the euivalent reverses of ) and 5) with the best ossible constant factor Kσ ). Proof We only rove that the constant factor Kσ )inthereverseof5) is the best ossible, and omit the others. If there exists a constant k Kσ ) such that the reverse of 5)isvalid when relacing Kσ )byk,theninarticular,forn N {,,...}, n > σ,wehave k [ α) δ n ++α) δ n [ β n ++β n k n L nĩ n n K β σ + n ) δ n ϖ δ σ + f x) gy) [ + ) δ ) ) n, x ) [ α) δ n ++α) δ n.

9 He et al. Journal of Ineualities and Alications 88:6 Page 9 of For n,weobtainthatk Kσ ). Hence, k Kσ )isthebestossibleconstantfactor of the reverse of 5). The theorem is roved. 4 Oerator exressions and a remark For >, +,δ {, }, α, β,μ, σ >,μ + σ, we set the following functions: ϕx):) δσ), ψy):) σ ),wherefrom ψ y) ) σ x, y R). Define the following real normed linear saces: { L,ϕ R): f ; f,ϕ : ϕx) f x) }, { L,ψ R): h; h,ψ : ψ y) hy) { L,ψ R): g; g,ψ : ψy) gy) }. In view of Theorem, for any f L,ϕ R), we set hy): f x) y R). [ + ) δ ) By )wehave }, h,ψ ψ y) hy) Kσ ) f,ϕ. ) Definition Define an extended Hilbert s integral oerator in the whole lane T : L,ϕ R) L,ψ R) as follows: For any f L,ϕ R), there exists Tf h L,ψ R). In view of ), the oerator T is bounded with T su f θ) L,ϕ R) Tf,ψ f,ϕ Kσ ). Since by Theorem the constant factor in )isthebestossible,wehave T Kσ ) Bμ, σ ). ) β / α ) / If we define the normal inner roduct of Tf and g as follows: Tf, g): { } f x) gy), [ + ) δ )

10 He et al. Journal of Ineualities and Alications 88:6 Page of then we can rewrite 5)and) as the followingeuivalent oerator exressions: Tf, g) T f,ϕ g,ψ, Tf,ψ T f,ϕ. ) Remark i)intheorem,forδ,relacing) f x)byf x), we have ) μ) f x), and the following euivalent ineualities with the homogeneous kernel and the best ossible constant factor Kσ ) Bμ,σ ) β / α / : { ) σ f x) } + ) Bμ, σ ) ) μ) f x), 3) β / α / + ) Bμ, σ ) ) μ) f x) β / α / [ ) σ ) g y). 4) ii) For α β,ineuality4)reducesto6), and 7)reducesto + xy ) [ Bμ, σ ) x σ ) f x) y σ ) g y). 5) iii) For,μ σ, f x)fx), g y)gy)x, y >),ineuality6)reducesto 4). Hence, ineuality 6) is an extended Hilbert s integral ineuality in the whole lane, and ineuality 5)isamoregeneralformof6) with arameters. 5 Conclusions In this aer, by introducing indeendent arameters and interval variables, alying the weight functions and the techniue of real analysis, an extended Hilbert s integral ineuality in the whole lane with arameters and a best ossible constant factor is rovided in Theorem. Theeuivalentforms, thereverses, andtherelatedhomogeneousformswith articular arameters are considered. An extended Hilbert s integral oerator in the whole lane is defined, and the oerator exressions for the euivalent ineualities are obtained. The method of weight functions is very imortant, which hels us to rove the euivalent ineualities with the best ossible constant factor. The lemmas and theorems rovide an extensive account of this tye of ineualities.

11 He et al. Journal of Ineualities and Alications 88:6 Page of Funding This work is suorted by the National Natural Science Foundation No. 6774) and Science and Technology Planning Project of Guangzhou City No. 779). We are grateful for this hel. Cometing interests The authors declare that they have no cometing interests. Authors contributions BY carried out the mathematical studies, articiated in the seuence alignment, and drafted the manuscrit. LH and YL articiated in the design of the stu and erformed the numerical analysis. All authors read and aroved the final manuscrit. Author details College of Mathematics and Statistics, Jishou University, Jishou, P.R. China. Deartment of Mathematics, Guangdong University of Education, Guangzhou, P.R. China. Publisher s Note Sringer Nature remains neutral with regard to jurisdictional claims in ublished mas and institutional affiliations. Received: 3 March 8 Acceted: 7 August 8 References. Har, G.H., Littlewood, J.E. Pólya, G.: Ineualities. Cambridge University Press, Cambridge 934). Har, G.H.: Note on a theorem of Hilbert concerning series of ositive terms. Proc. Lond. Math. Soc. 3),Records of Proc. xlv xlvi 95) 3. Mitrinović, D.S., Pečarić, J.E., Fink, A.M.: Ineualities Involving Functions and Their Integrals and Derivatives. Kluwer Academic, Boston 99) 4. Yang, B.C.: On Hilbert s integral ineuality. J. Math. Anal. Al., ) 5. Wang, D.X., Guo, D.R.: Introduction to Esecial Functions. Science Press, Beijin 979) 6. Li, Y.J., He, B.: On ineualities of Hilbert s tye. Bull. Aust. Math. Soc. 76), 3 7) 7. Yang, B.C.: The Norm of Oerator and Hilbert-Tye Ineualities. Science Press, Beijing 9) 8. Xin, D.M., Yang, B.C.: A Hilbert-tye integral ineuality in the whole lane with the homogeneous kernel of degree. J. Ineual. Al., Article ID 448 ) 9. Wang, A.Z., Yang, B.C.: A new Hilbert-tye integral ineuality in the whole lane with the non-homogeneous kernel. J. Ineual. Al., 3 ). Xie, Z.T., Zeng, Z.: A new Hilbert-tye ineuality in whole lane with the homogeneous kernel of degree. i-manag. J. Math. ), 3 9 3). Rassias, M., Yang, B.C.: A Hilbert-tye integral ineuality in the whole lane related to the hyer geometric function and the beta function. J. Math. Anal. Al. 48), ). Yang, B.C.: On the norm of an integral oerator and alication. J. Math. Anal. Al. 3, 8 9 6) 3. Yang, B.C.: On the norm of a Hilbert s tye linear oerator and alications. J. Math. Anal. Al. 35, ) 4. Arad, B., Choonghong, O.: Best constant for certain multi linear integral oerator. J. Ineual. Al. 6, 858 6) 5. Kuang, J.C.: Alied Ineualities. Shangdong Science Technic Press, Jinan 4) 6. Kuang, J.C.: Real and Functional Analysis Continuation) Second Volume). Higher Education Press, Beijing 5)