An extension of a multidimensional Hilbert-type inequality

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1 Zhog ad Yag Joural of Ieualities ad Alicatios 27 27:78 DOI.86/s R E S E A R C H Oe Access A extesio of a ultidiesioal Hilbert-tye ieuality Jiahua Zhog ad Bicheg Yag * * Corresodece: bcyag@gdei.edu.c Deartet of Matheatics, Guagdog Uiversity of Educatio, Guagzhou, Guagdog 533, P.R. Chia Abstract I this aer, by the use of weight coefficiets, the trasfer forula ad the techiue of real aalysis, a ew ultidiesioal Hilbert-tye ieuality with ulti-araeters ad a best ossible costat factor is give, which is a extesio of soe ublished results. Moreover, the euivalet fors, the oerator exressios ad a few articular ieualities are cosidered. MSC: 26D5; 47A5 Keywords: Hilbert-tye ieuality; weight coefficiet; euivalet for; oerator; or Itroductio If >, +,a, b, a {a } l, b {b } l, a a >, b >, the we have the followig Hardy-Hilbert ieuality with the best ossible costat π siπ/ : a b + < π siπ/ a b, ad the followig Hilbert-tye ieuality: a b ax{, } < a b 2 with the best ossible costat factor cf. [], Theore 35, Theore 34. Ieualities ad2 are iortat i the aalysis ad its alicatios cf. [ 3]. Assuig that {μ }, {ν } are ositive seueces, U μ i, V i ν j, N {,2,...}, j we have the followig Hardy-Hilbert-tye ieuality cf. [],Theore 32: a b U + V < π siπ/ a b. 3 The Authors 27. This article is distributed uder the ters of the Creative Coos Attributio 4. Iteratioal Licese htt://creativecoos.org/liceses/by/4./, which erits urestricted use, distributio, ad reroductio i ay ediu, rovided you give aroriate credit to the origial authors ad the source, rovide a lik to the Creative Coos licese, ad idicate if chages were ade.

2 ZhogadYag Joural of Ieualities ad Alicatios 27 27:78 Page 2 of 2 For μ i ν j i, j N, ieuality 3reducesto. I 24, Yag ad Che [4] gave the followig ultidiesioal Hilbert-tye ieuality: For i, j N,, >, i x : x k x x,...,x i R i, j y : y k y y,...,y j R j, <λ + η i,<λ 2 + η j, λ + λ 2 λ, a, b, we have i{, } η ax{, } a b λ+η < K K 2 [ i λ i a ] [ where i, j ositive, ad the best ossible costat factor K [ K K Ɣ j 2 j Ɣ j ] i i Ɣ i j λ 2 j b ], 4 K 2 ] λ +2η λ + ηλ 2 + η., the series o the right-had side are is idicated by For i j λ,η,λ, λ 2,ieuality4reducesto2. The other results o this tye of ieualities were rovided by [5 7]. I 25, Shi ad Yag [8] gave aother extesio of 2 as follows: a b ax{u, V } < a b. 5 Soe other results o Hardy-Hilbert-tye ieualities were give by [9 25]. I this aer, by the use of weight coefficiets, the trasfer forula ad the techiue of real aalysis, a ew ultidiesioal Hilbert-tye ieuality with ulti-araeters ad a best ossible costat factor is give, which is a extesio of 4 ad5. Moreover, the euivalet fors, the oerator exressios ad a few articular ieualities are cosidered. 2 Soe leas If i >k,...,i ; i,...,, ν l j U k : i U U,...,Ui i k,...,i, V l, V V >l,...,j ; j,...,, the we set : j,...,v j ν l j l,...,j, 6, N. We also set fuctios μ k t:, t,] N; ν l t:ν l, t,] N, ad

3 ZhogadYag Joural of Ieualities ad Alicatios 27 27:78 Page 3 of 2 U k x: V l y: x y μ k t dt k,...,i, 7 ν l t dt l,...,j, 8 Ux: U x,...,u i x, Vy: V y,...,v j y x, y. 9 It follows that U k U k k,...,i ; N, V l V l l,...,j ; N, ad for x,, U k xμ kx k,...,i ; N; for y,, V l yν lyν l l,...,j ; N. Lea cf. [2] Suose that gt >is decreasig i R + ad strictly decreasig i [, N, satisfyig gt dt R +. We have gt dt < g< gt dt. Lea 2 If i N,, M >, u is a o-egative easurable fuctio i, ], ad { D M : x R i + ; u i xi i M }, the we have the followig trasfer forula cf.[26]: i D M i xi dx dx s Mi Ɣ i M i Ɣ i uu i du. 2 Lea 3 For i, j N, + N, k,...,i, ν l ν l + N; l,...,j,, >,ε >,we have U i ε V j ε j Proof For M > i /,weset Ɣ i εi ε/ i Ɣ i + O, 3 ν k Ɣ j εj ε/ j Ɣ j + Õ, < u < i M u, i M u. Mu / i +ε, By 2, it follows that {x R i + ;x i } dx x i +ε li M i D M i M i Ɣ i li M i Ɣ i i /M xi ε +. 4 dx dx i M i u Mu / du Ɣ i i +ε εi ε/ i Ɣ i.

4 ZhogadYag Joural of Ieualities ad Alicatios 27 27:78 Page 4 of 2 The by ad the above result, we fid < < { N i ; i 2} { N i ; i 2} { N i ; i 2} U i ε {x N i ; x i <} {x N i ; x i <} Ux i ε {x N i ;x i } U i ε Ux i ε ν i ε {ν R i dν + O i + ;ν i } For i,< { N i ; i } U i ε i i μ i, i the sae way, we fid < { N i ; i, i } U i ε U i ε + dx x dx x dx νux ν i ε {ν R i + ;ν i μ i } dν i μ i i { N i ; j 2j i} Ɣ i εi ε/ i Ɣ i + O i. i < ; fori 2, μ i ax μ i, b U i ε+ bɣ i O + + εi +ε/ i 2 Ɣ i + bo i <. The we have U i ε { N i ; i, i } + { N i ; j 2} U i ε U i ε Ɣ i εi ε/ i Ɣ i + O ε +. k i Hece, we have 3. I the sae way, we have4. Defiitio For, >,<λ + η i,<λ 2 + η j, λ + λ 2 λ, we defie two weight coefficiets wλ, adwλ 2, as follows: wλ, : i{ U, V } η ax{ U, V } λ+η λ V 2 U i λ, 5

5 ZhogadYag Joural of Ieualities ad Alicatios 27 27:78 Page 5 of 2 Wλ 2, : i{ U, V } η U λ ax{ U, V } λ+η V j λ 2 j l ν l. 6 Exale With regard to the assutios of Defiitio,we set k λ x, y The, i for fixed y >, i{x, y}η x, y >. ax{x, y} λ+η, <x < y, y k λ x, y λ+η x i λ η x i λ y η, x y, x i +λ 2 +η is decreasig i R + ad strictly decreasig i [y]+,.ithesaeway,forfixedx >, k λ x, y is decreasig i R + ad strictly decreasig i [x]+,. We still have y j λ 2 kλ : ii For b >,wehave k λ u, du u λ u η u du + λ u λ+η i{u,} η ax{u,} λ+η d b + x b + x x > x >. dx du u λ du u λ +2η λ λ + ηλ 2 + η. 7 Hece, for <x i < i,...,i ; N, we have U > Ux ad i{ U, V } η ax{ U, V } λ+η U i λ < i{ Ux, V } η ax{ Ux, V } λ+η Ux i λ for < x i < +i,...,i ; N, we have U < Ux ad i{ U, V } η ax{ U, V } λ+η U i λ > i{ Ux, V } η ax{ Ux, V } λ+η Ux i λ Lea 4 With regard to the assutios of Defiitio,iwe have ;. wλ, <K 2 λ Wλ 2, <K λ N j, 8 N i, 9 where K λ Ɣj j Ɣ j kλ, K 2 λ Ɣi i Ɣ i kλ ; 2

6 ZhogadYag Joural of Ieualities ad Alicatios 27 27:78 Page 6 of 2 ii for + N, νl ν l + N, Uk V l k,...,i, l,...,j, < λ + η i, λ 2 + η >,<ε < λ >,we have <K 2 λ θ λ < wλ, N j, 2 where, for c : ax k i { } >, θ λ : ci / / V i{v,} η v λ dv O kλ ax{v,} λ+η V λ +η. 22 Proof i By, 2 ad Exale ii, for < λ + η i, λ >, it follows that wλ, {x N i ; i x i i } V λ 2 U i λ < {x N i ; i x i i } V λ 2 Ux i λ i{ U, V } η ax{ U, V } λ+η dx i{ Ux, V } η ax{ Ux, V } λ+η x dx i{ Ux, V } Ri+ η ax{ Ux, V } λ+η λ V 2 Ux i λ uux i{ u, V } Ri+ η V ax{ u, V } λ+η u i λ du i i{m[ li i u i M ] /, V } η M D M ax{m[ i i u i M ] /, V } λ+η M i Ɣ i li M i Ɣ i M i Ɣ i li M i Ɣ i v Mu/ V Ɣ i i Ɣ i λ 2 i{mu /, V } η V λ 2 ax{mu /, V } λ+η x dx M λ i V λ 2 du [ i i u i M ] i λ / i u du M i λ u i λ / i{mu /, V } η V λ 2 u λ du ax{mu /, V } λ+η i{v,} η v λ ax{v,} λ+η Ɣi λ +2η i Ɣ i λ + ηλ 2 + η K 2λ. Hece, we have 8. Ithesaeway, wehave9. ii By ad i the sae way, for c ax k i { } >, we have wλ, {x N i ; i x i i +} V λ 2 U i λ dv i{ U, V } η ax{ U, V } λ+η + dx

7 ZhogadYag Joural of Ieualities ad Alicatios 27 27:78 Page 7 of 2 > {x N i ; i x i i +} i{ Ux, V } η ax{ Ux, V } λ+η V λ 2 Ux i λ [, i vux [c, i x dx i{ Ux, V } η ax{ Ux, V } λ+η λ V 2 Ux i λ λ 2 i{ v, V } η V ax{ v, V } λ+η v i dv. λ x dx For M > ci /,weset, < u < c i M u, i{mu /, V } η λ V 2, c i ax{mu /, V } λ+η Mu / i λ M u. By 2, it follows that {x R i + ;x i c} li M λ 2 i{ x, V } η V ax{ x, V } λ+η x i dx λ i xi dx dx i D M M i M i Ɣ i li i{mu, V } η λ V 2 u i du M i Ɣ i c i /M ax{mu, V } λ+η Mu i λ v Mu/ V Ɣ i i Ɣ i ci / / V i{v,}η vλ ax{v,} λ+η dv. Hece, we have wλ, > Ɣi i{v,}ηvλ i Ɣ i dv ci / / V ax{v,} λ+η K 2 λ θ λ >. For V ci /,weobtai ci / / V <θ λ kλ ci / / V v λ+η dv kλ i{v,} η v λ ax{v,} λ+η λ + ηkλ dv ci / λ +η, V ad the 2ad22 follow.

8 ZhogadYag Joural of Ieualities ad Alicatios 27 27:78 Page 8 of 2 3 Mai results Settig fuctios : U i λ i i N i, : V j λ 2 j j N j, l νl ad the followig ored saces: { { } l, : a {a }; a, : a < }, { l, : b {b }; b, : { l, : c {c }; c, : we have the followig. { } b < }, { } c < }, Theore If >, +,, >,λ >,<λ + η i,<λ 2 + η j, λ + λ 2 λ, the for a, b, a {a } l,, b {b } l,, a,, b, >,we have the followig euivalet ieualities: where i{ U, V } η ax{ U, V } a b λ+η < K λ K2 λ a, b,, 23 I : { j J : vk V j λ 2 < K λ K [ i{ U, V } η ] a } ax{ U, V } λ+η 2 λ a,, 24 [ λ Ɣ j K2 λ j Ɣ j K ] i i Ɣ i Proof By Hölder s ieuality with weight cf. [27], we have I [ i{ U, V } η ax{ U, V } λ+η i [ λ U j λ 2 V j l νl a i Wλ 2, U i λ i i j ][ λ 2 V a i λ U ] [ ] kλ. 25 i b j l νl ] b wλ, V j λ 2 j j l νl ].

9 ZhogadYag Joural of Ieualities ad Alicatios 27 27:78 Page 9 of 2 The by 8ad9, we have 23. We set b : j [ l νl i{ U, V } η ] a V j λ 2, N j. ax{ U, V } λ+η The we have J b,. Sice the right-had side of 24 is fiite, it follows J <.IfJ, the 24 is trivially valid; if J >,theby23, we have b, J I < K λ K b, J < K λ K2 λ a,, 2 λ a, b,, aely 24 follows. O the other had, assuig that 24 is valid, by Hölder s ieuality cf. [27], we have I j l vl / i{ U, V } η V j / λ 2 ax{ U, V } a λ+η V j / λ 2 j l νl b / J b,. 26 The by 24wehave23, which is euivalet to 24. Theore 2 With regard to the assutios of Theore, if + N, ν l ν l + N, Uk V l k,...,i, l,...,j, the the costat factor K λ K2 λ i 23 ad 24 is the best ossible. Proof For < ε < λ + η, λ λ ε η, η + i, λ 2 λ 2 + ε > η, we set ã {ã }, ã : U i + λ b { b }, b : V j + λ 2 ε j l ν l N i, N j. The by 3ad4, we obtai [ ã, b, U i λ i i U i ε ã ] [ V j λ 2 j j l νl V j ε j l b ] Ɣ i ε i ε/ i Ɣ i + εo Ɣ j j ε/ j Ɣ j + εõ. ν l

10 ZhogadYag Joural of Ieualities ad Alicatios 27 27:78 Page of 2 By 2ad22, we fid Ĩ : [ i{ U, V } η ] ax{ U, V } λ+η ã b > K 2 λ K 2 λ w λ, V j ε O Ɣ j j l ν l V λ +η V j ε εj ε/ j Ɣ j + Õ O If there exists a costat K K λ K2 λ suchthat23 is valid whe relacig K λ K2 λ byk,thewehaveεĩ < εk ã, b,,aely. K 2 λ ε Ɣ j j ε/ j Ɣ j + εõ εo Ɣ i < K i ε/ i Ɣ i + εo Ɣ j j ε/ j Ɣ j + εõ. For ε +,wefid j l ν l Ɣ j Ɣi kλ [ Ɣ i j Ɣ j i Ɣ i K ] j ] i Ɣ i j Ɣ j, ad the K λ K2 λ K.Hece,K K λ K2 λ is the best ossible costat factor of 23. The costat factor i 24 is still the best ossible. Otherwise, we would reach a cotradictio by 26 that the costat factor i 23isotthebestossible. 4 Oerator exressios With regard to the assutios of Theore 2,i view of c : c {c }, j [ νk i{ U, V } η ] V j λ 2 ax{ U, V } a λ+η, N j, c, J < K λ K we ca set the followig defiitio. 2 λ a, <, Defiitio 2 Defie a ultidiesioal Hilbert s oerator T : l, l, as follows: For ay a l,, there exists a uiue reresetatio Ta c l,,satisfyig Ta: i{ U, V } η ax{ U, V } λ+η a N j. 27

11 ZhogadYag Joural of Ieualities ad Alicatios 27 27:78 Page of 2 For b l,, we defie the followig foral ier roduct of Ta ad b as follows: Ta, b: [ i{ U, V } η ] ax{ U, V } a λ+η b. 28 The by Theores ad2, we have the followig euivalet ieualities: Ta, b <K λ K2 λ a, b,, 29 Ta, < K λ K2 λ a,. 3 It follows that T is bouded with Ta T : su, a θ l, a, K λ K2 λ. 3 Sice the costat factor K λ K 2 λ i3 is the best ossible, we have [ T K λ Ɣ j K2 λ ] i ] kλ j Ɣ j a i Ɣ i. 32 Reark i For μ i ν j i, j N, 23reducesto4. Hece, 23 is a extesio of 4. ii For η,<λ i,<λ 2 j,23 reduces to the followig ieuality: ax{ U, V } a b λ j < ] i ] λ j Ɣ j a i Ɣ i a, b,. 33 λ λ 2 I articular, for i j λ,λ, λ 2,33reducesto5. Hece, 33isalsoa extesio of 5; so is 23. iii For η λ, λ, λ 2 <,23 reduces to the followig ieuality: i{ U, V } a b λ j < ] i ] λ j Ɣ j a i Ɣ i a, b,. 34 λ λ 2 iv For λ,λ 2 λ η < λ < η, 23 reduces to the followig ieuality: i{ U, V } η a b ax{ U, V } j < ] i ] 2η j Ɣ j a i Ɣ i η 2 λ 2 a, b,. 35 The above articular ieualities are also with the best ossible costat factors.

12 ZhogadYag Joural of Ieualities ad Alicatios 27 27:78 Page 2 of 2 Coetig iterests The authors declare that they have o coetig iterests. Authors cotributios BY carried out the atheatical studies, articiated i the seuece aliget ad drafted the auscrit. JZ articiated i the desig of the study ad erfored the uerical aalysis. All authors read ad aroved the fial auscrit. Ackowledgeets This work is suorted by the Natioal Natural Sciece Foudatio No , No , ad Aroriative Researchig Fud for Professors ad Doctors, Guagdog Uiversity of Educatio No. 25ARF25. We are grateful for their hel. Publisher s Note Sriger Nature reais eutral with regard to jurisdictioal clais i ublished as ad istitutioal affiliatios. Received: 2 Jauary 27 Acceted: 23 March 27 Refereces. Hardy, GH, Littlewood, JE, Pólya, G: Ieualities. Cabridge Uiversity Press, Cabridge Mitriović, DS, Pečarić, JE, Fik, AM: Ieualities Ivolvig Fuctios ad Their Itegrals ad Derivatives. Kluwer Acadeic, Bosto Yag, BC: The Nor of Oerator ad Hilbert-Tye Ieualities. Sciece Press, Beiji 29 i Chiese 4. Yag, BC, Che, Q: A ultidiesioal discrete Hilbert-tye ieuality. J. Math. Ieual. 82, Hog, Y: O Hardy-Hilbert itegral ieualities with soe araeters. J. Ieual. Pure Al. Math. 64, Article ID Zhog, WY, Yag, BC: O ultile Hardy-Hilbert s itegral ieuality with kerel. J. Ieual. Al. 27, Article ID Yag, BC,Krić, M: O the or of a ulti-diesioal Hilbert-tye oerator. Sarajevo J. Math. 72, Krić, M, Pečarić, JE, Vuković, P: O soe higher-diesioal Hilbert s ad Hardy-Hilbert s tye itegral ieualities with araeters. Math. Ieual. Al., Krić, M, Vuković, P: O a ultidiesioal versio of the Hilbert-tye ieuality. Aal. Math. 38, Rassias, M, Yag, BC: A ultidiesioal half-discrete Hilbert-tye ieuality ad the Riea zeta fuctio. Al. Math. Cout. 225, Yag, BC: A ultidiesioal discrete Hilbert-tye ieuality. It. J. Noliear Aal. Al. 5, Che, Q, Yag, BC: O a ore accurate ultidiesioal Mulhollad-tye ieuality. J. Ieual. Al. 24, Rassias, M, Yag, BC: O a ultidiesioal Hilbert-tye itegral ieuality associated to the gaa fuctio. Al. Math. Cout. 249, Yag, BC: O a ore accurate ultidiesioal Hilbert-tye ieuality with araeters. Math. Ieual. Al. 82, Huag, ZX, Yag, BC: A ultidiesioal Hilbert-tye itegral ieuality. J. Ieual. Al. 25, Liu, T, Yag, BC, He, LP: O a ultidiesioal Hilbert-tye itegral ieuality with logarith fuctio. Math. Ieual. Al. 84, Shi, YP, Yag, BC: O a ultidiesioal Hilbert-tye ieuality with araeters. J. Ieual. Al. 25, Shi, YP, Yag, BC: A ew Hardy-Hilbert-tye ieuality with ulti-araeters ad a best ossible costat factor. J. Ieual. Al. 25, Huag, QL: A ew extesio of Hardy-Hilbert-tye ieuality. J. Ieual. Al. 25, Wag, AZ, Huag, QL, Yag, BC: A stregtheed Mulhollad-tye ieuality with araeters. J. Ieual. Al. 25, Yag, BC, Che, Q: O a Hardy-Hilbert-tye ieuality with araeters. J. Ieual. Al. 25, Li, AH, Yag, BC, He, LP: O a ew Hardy-Mulhollad-tye ieuality ad its ore accurate for. J. Ieual. Al. 26, Rassias, M, Yag, BC: O a Hardy-Hilbert-tye ieuality with a geeral hoogeeous kerel. It. J. Noliear Aal. Al. 7, Che, Q, Shi, YP, Yag, BC: A relatio betwee two sile Hardy-Mulhollad-tye ieualities with araeters. J. Ieual. Al. 26, Yag, BC, Che, Q: O a ore accurate Hardy-Mulhollad-tye ieuality. J. Ieual. Al. 26, Yag, BC: Hilbert-tye itegral oerators: ors ad ieualities. I: Pardalos, PM, Georgiev, PG, Srivastava, HM eds. Noliear Aalysis, Stability, Aroxiatio, ad Ieualities, Sriger, New York Kuag, JC: Alied Ieualities. Shagdog Sciece Techic Press, Jia, Chia 24