Zhog ad Yag Joural of Ieualities ad Alicatios 27 27:78 DOI.86/s366-7-355-6 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.
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
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.
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
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
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
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.
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 ].
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
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
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.
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. 63786, No. 664222, 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 934 2. Mitriović, DS, Pečarić, JE, Fik, AM: Ieualities Ivolvig Fuctios ad Their Itegrals ad Derivatives. Kluwer Acadeic, Bosto 99 3. 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, 267-277 24 5. Hog, Y: O Hardy-Hilbert itegral ieualities with soe araeters. J. Ieual. Pure Al. Math. 64, Article ID92 25 6. Zhog, WY, Yag, BC: O ultile Hardy-Hilbert s itegral ieuality with kerel. J. Ieual. Al. 27, Article ID 27962 27 7. Yag, BC,Krić, M: O the or of a ulti-diesioal Hilbert-tye oerator. Sarajevo J. Math. 72,223-243 2 8. Krić, M, Pečarić, JE, Vuković, P: O soe higher-diesioal Hilbert s ad Hardy-Hilbert s tye itegral ieualities with araeters. Math. Ieual. Al.,7-76 28 9. Krić, M, Vuković, P: O a ultidiesioal versio of the Hilbert-tye ieuality. Aal. Math. 38, 29-33 22. Rassias, M, Yag, BC: A ultidiesioal half-discrete Hilbert-tye ieuality ad the Riea zeta fuctio. Al. Math. Cout. 225,263-277 23. Yag, BC: A ultidiesioal discrete Hilbert-tye ieuality. It. J. Noliear Aal. Al. 5, 8-88 24 2. Che, Q, Yag, BC: O a ore accurate ultidiesioal Mulhollad-tye ieuality. J. Ieual. Al. 24, 322 24 3. Rassias, M, Yag, BC: O a ultidiesioal Hilbert-tye itegral ieuality associated to the gaa fuctio. Al. Math. Cout. 249,48-48 24 4. Yag, BC: O a ore accurate ultidiesioal Hilbert-tye ieuality with araeters. Math. Ieual. Al. 82, 429-44 25 5. Huag, ZX, Yag, BC: A ultidiesioal Hilbert-tye itegral ieuality. J. Ieual. Al. 25, 5 25 6. Liu, T, Yag, BC, He, LP: O a ultidiesioal Hilbert-tye itegral ieuality with logarith fuctio. Math. Ieual. Al. 84, 29-234 25 7. Shi, YP, Yag, BC: O a ultidiesioal Hilbert-tye ieuality with araeters. J. Ieual. Al. 25, 37 25 8. Shi, YP, Yag, BC: A ew Hardy-Hilbert-tye ieuality with ulti-araeters ad a best ossible costat factor. J. Ieual. Al. 25, 38 25 9. Huag, QL: A ew extesio of Hardy-Hilbert-tye ieuality. J. Ieual. Al. 25, 397 25 2. Wag, AZ, Huag, QL, Yag, BC: A stregtheed Mulhollad-tye ieuality with araeters. J. Ieual. Al. 25, 329 25 2. Yag, BC, Che, Q: O a Hardy-Hilbert-tye ieuality with araeters. J. Ieual. Al. 25, 339 25 22. Li, AH, Yag, BC, He, LP: O a ew Hardy-Mulhollad-tye ieuality ad its ore accurate for. J. Ieual. Al. 26, 69 26 23. Rassias, M, Yag, BC: O a Hardy-Hilbert-tye ieuality with a geeral hoogeeous kerel. It. J. Noliear Aal. Al. 7,249-269 26 24. Che, Q, Shi, YP, Yag, BC: A relatio betwee two sile Hardy-Mulhollad-tye ieualities with araeters. J. Ieual. Al. 26, 75 26 25. Yag, BC, Che, Q: O a ore accurate Hardy-Mulhollad-tye ieuality. J. Ieual. Al. 26, 82 26 26. Yag, BC: Hilbert-tye itegral oerators: ors ad ieualities. I: Pardalos, PM, Georgiev, PG, Srivastava, HM eds. Noliear Aalysis, Stability, Aroxiatio, ad Ieualities,. 77-859. Sriger, New York 22 27. Kuag, JC: Alied Ieualities. Shagdog Sciece Techic Press, Jia, Chia 24