ON A MORE ACCURATE MULTIDIMENSIONAL HILBERT TYPE INEQUALITY WITH PARAMETERS BICHENG YANG. 1. Introduction. dxdy < x + y. sin(π/p) f p g q, (1)
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1 M atheatical I eualities & A licatios Volue 8, Nuber 2 (25, doi:.753/ia-8-32 ON A MORE ACCURATE MULTIDIMENSIONAL HILBERT TYPE INEQUALITY WITH PARAMETERS BICHENG YANG (Couicated by J. Pečarić Abstract. I this aer, by usig the way of weight coefficiets ad techiue of real aalysis ad colex aalysis, a ore accurate ultidiesioal discrete Hilbert-tye ieuality with a best ossible costat factor ad soe araeters is give. The euivalet for, the oerator exressio with the or are also cosidered.. Itroductio Assuig that >, + =, f (x, g(y, f L (R +, g L (R +, f = f (xdx} >, g >, we have the followig Hardy-Hilbert s itegral ieuality (cf. []: f (xg(y dxdy < x + y π si(π/ f g, ( π with the best ossible costat factor si(π/.ifa,b, a = a } = l, b = b } = l, a = = a } >, b >, the we have the followig Hardyπ Hilbert s ieuality with the sae best costat si(π/ (cf.[]: = = a b + < π si(π/ a b. (2 Ieualities ( ad(2 are iortat i aalysis ad its alicatios (cf. [], [2], [3], [4], [6], [7]. I 998, by itroducig a ideedet araeter λ (,], Yag[5] gavea extesio of ( for = = 2. Followig the results of [5], Yag [6] gave soe best extesios of (ad(2 as follows: Matheatics subject classificatio (2: 26D5, 47A7. Keywords ad hrases: Hilbert-tye ieuality, weight coefficiet, euivalet for, oerator, or. This work is suorted by the Natioal Natural Sciece Foudatio of Chia (No , ad 23 Kowledge Costructio Secial Foudatio Ite of Guagdog Istitutio of Higher Learig College ad Uiversity (No. 23KJCX4. c D l,zagreb Paer MIA
2 43 BICHENG YANG If λ,λ 2,λ R, λ + λ 2 = λ,k λ (x,y is a o-egative hoogeeous fuctio of degree λ, with k(λ = k λ (t,t λ dt R +, φ(x=x ( λ, ψ(x=x ( λ2, f (x, g(y, } f L,φ (R + = f ; f,φ := φ(x f (x dx} <, g L,ψ (R +, f,φ, g,ψ >, the k λ (x,y f (xg(ydxdy < k(λ f,φ g,ψ, (3 where the costat factor k(λ is the best ossible. Moreover, if k λ (x,y is fiite ad k λ (x,yx λ (k λ (x,yy λ2 is decreasig with resect to x > (y >, the for a, b, } a l,φ = a; a,φ := φ( a } <, = b = b } = l,ψ, a,φ, b,ψ >, it follows = = k λ (,a b < k(λ a,φ b,ψ, (4 where, the costat factor k(λ is still the best ossible. Clearly, for λ =, k (x,y= x+y, λ =, λ 2 =, ieuality (3 reduces to (, while (4 reduces to (2. Soe other results icludig the ultidiesioal Hilberttye itegral ieualities are rovided by [8] [22]. About half-discrete Hilbert-tye ieualities with the o-hoogeeous kerels, Hardy et al. rovided a few results i Theore 35 of []. But they did ot rove that the the costat factors are the best ossible. However, Yag [23] gave a result with the kerel ( < λ 2 by itroducig a variable ad roved that the costat factor (+x λ is the best ossible. I 2 Yag [24] gave the followig half-discrete Hardy-Hilbert s ieuality with the best ossible costat factor B(λ,λ 2 : f (x = a (x + λ dx < B(λ,λ 2 f,φ a,ψ, (5 where, λ >, < λ 2, λ + λ 2 = λ, B(u,v= ( + t u+vtu dt(u,v > is the beta fuctio. Zhog et al ([25] [7] ivestigated several half-discrete Hilberttye ieualities with articular kerels.
3 ON A MORE ACCURATE MULTIDIMENSIONAL HILBERT-TYPE INEQUALITY 43 Usig the way of weight fuctios ad the techiues of discrete ad itegral Hilbert-tye ieualities with soe additioal coditios o the kerel, a half-discrete Hilbert-tye ieuality with a geeral hoogeeous kerel of degree λ R ad a best costat factor k (λ is obtaied as follows: f (x = k λ (x,a dx < k(λ f,φ a,ψ, (6 which is a extesio of (5 (see Yag ad Che [33]. At the sae tie, a half-discrete Hilbert-tye ieuality with a geeral o-hoogeeous kerel ad a best costat factor is give by Yag [34]. I this aer, by usig the way of weight coefficiets ad techiue of real aalysis, a ultidiesioal discrete Hilbert s ieuality with araeters ad a best ossible costat factor is give, which is a extesio of (4 fork λ (,= s k= (λ/s +c k λ/s. The euivalet for, the oerator exressio with the or are also cosidered. 2. Soe leas If i, j N(N is the set of ositive itegers,, >, we ut ( i x := x k (x =(x,,x i R i, (7 k= ( j y := y k (y =(y,,y j R j. (8 k= LEMMA. If s N, γ,m>, Ψ(u is a o-egative easurable fuctio i (,], ad D M := x R s s + ; x γ i M }, γ the we have (cf. [35] i= ( s ( xi γ Ψ D M dx dx s i= M = Ms Γ s ( γ γ s Γ( s γ Ψ(uu s γ du. (9 LEMMA 2. If s N,γ >, ε >, c =(c,,c s [, s, the we have Γ s ( c s ε γ γ = εs ε/γ γ s Γ( s γ + O((ε +. (
4 432 BICHENG YANG Proof. For M > s /γ, we set, < u < s Ψ(u= M γ, (Mu /γ s ε s, M γ u. The by (9, it follows c s ε γ = x R s + ;x i +c i } u R s + ;u i } = li M x c s ε γ dx u s ε γ du D M Ψ ( s ( xi γ dx dx s i= M M s Γ s ( γ = li M γ s Γ( s γ (Mu/γ s ε u s γ du = s/m γ For s =, it follows < 2 = c ε γ < ; fors 2, Γ s ( γ εs ε/γ γ s Γ( s γ. < c s ε γ a + c (s (+ε γ N s ; i, i =,2} N s ; i 3} The we have Γ s ( γ a + ( + ε(s (+ε/γ γ s 2 Γ( s γ < (a R +. < c s ε γ N s ; i } = c s ε γ + c s ε γ N s ; i, i =,2} N s ; i 3} Õ(+ Γ s ( γ εs ε/γ γ s Γ( s γ (ε +. Hecewehave(. LEMMA 3. If C is the set of colex ubers ad C = C }, z k C\z Rez, Iz = } (k =,2,, are differet oits, the fuctio f (z is aalytic i C excet for z i (i =,2,,, ad z = is a zero oit of f (z whose order is ot less tha, the for R, we have f (xx dx = 2πi e 2πi k= Res[ f (zz,z k ], (
5 ON A MORE ACCURATE MULTIDIMENSIONAL HILBERT-TYPE INEQUALITY 433 where, < Ilz = argz < 2π. I articular, if z k (k =,, are all oles of order, settig ϕ k (z=(z z k f (z(ϕ k (z k, the f (xx dx = π siπ k= ( z k ϕ k (z k. (2 Proof. By [36] (P. 8, we have (. We fid e 2πi = cos2π isi2π = 2isiπ(cosπ+ isiπ= 2ie iπ siπ. I articular, sice f (zz = z z k (ϕ k (zz, it is obvious that Res[ f (zz, a k ]=z k ϕ k (z k = e iπ ( z k ϕ k (z k. The by (, we obtai (2. EXAMPLE. For s N, we set k λ (x,y= s k= (x λ /s + c k y λ /s ( < c < < c s, < λ s. For < λ i,< λ 2 j, λ + λ 2 = λ, by (2, we fid k s (λ := u=t λ/s = s λ = s k= s πs t λ /s t λ dt + c k k= λ si( πsλ I articular, for s =, we obtai k (λ = λ u sλ u + c k s λ k= c λ du u (λ /λ u + c du = sλ λ s k j=( j k π λ si( πλ λ c R +. (3 c j c k λ λ. LEMMA 4. If ( i h (i (t > (t > ; i=,2, the for b >, <, we have ( i di dx i h((b + x > (x > ;i =,2. (4
6 434 BICHENG YANG Proof. We fid d dx h((b + x =h ((b + x (b + x x <, d 2 dx 2 h((b + x = d dx [h ((b + x (b + x x ] = h ((b + x (b + x 2 2 x 2 2 ( + h ((b + x (b + x 2 x 2 2 +( h ((b + x (b + x x 2 = h ((b + x (b + x 2 2 x 2 2 +b( h ((b + x (b + x 2 x 2 >. The lea is roved. DEFINITION. For s N, <,, < c < < c s,< λ s, < λ i, < λ 2 j, λ + λ 2 = λ,τ =(τ,,τ i (, 2 ]i, σ =(σ,,σ j (, 2 ] j, τ =( τ,, i τ i R i +, σ =( σ,, j σ j R j +, defie two weight coefficiets w λ (λ 2, ad W λ (λ, as follows: w λ (λ 2, := W λ (λ, := σ λ 2 τ λ i k= ( τ λ + c k σ λ /s (5, τ λ σ λ 2 j k= ( τ λ where, = i = = ad = j = =. where, + c k σ λ /s (6, LEMMA 5. Let the assutios as i Defiitio are fulfilled. The, we have (i w λ (λ 2, < K 2 ( N j, (7 W λ (λ, < K ( N i, (8 K := Γ j ( j Γ( j k s(λ, K 2 := Γi( i Γ( i k s(λ, (9 where, k s (λ is idicated by (3; (ii for >, < ε < iλ, λ 2 }, settig λ = λ ε, λ 2 = λ 2 + ε, we have < K 2 ( θ λ ( < w λ ( λ 2,, (2
7 ON A MORE ACCURATE MULTIDIMENSIONAL HILBERT-TYPE INEQUALITY 435 where, θ λ ( = k s ( λ i λ/(s / σ λ/s v sλ λ dv s k= (v + c k = O σ λ, (2 K 2 = Γi ( k s( λ i Γ( i R +. (22 Proof. By Lea 4, Herite-Hadaard s ieuality (cf. [37], (9 ad(3, it follows w λ (λ 2, < = ( 2, i σ λ 2 x τ λ i k= ( x τ λ + c k σ λ /s dx u R i + ;u i > 2 τ i} σ Ri+ λ 2 u λ i σ λ 2 u λ i k= ( u λ + c k σ λ /s du k= ( u λ + c k σ λ /s du σ = li M DM λ 2 Mλ i [ j i= ( u i M ] (λ i / du s k= M s λ [ i i= ( u i M ] s λ s + c k σ λ } = li M = li M M i Γ i ( i Γ( i M λ Γ i ( i Γ( i t= σ M v s/λ = σ λ 2 Mλ i t (λ i / dt s k= (M λ s t λ s + c k σ λ s σ λ 2 t λ dt s k= (M λ s t λ s + c k σ λ s sγ i ( λ i Γ( i v sλ λ s k= (v + c k dv = Γi ( i Γ( i k s(λ =K 2. Hece, we have (7.Bythesaeway,wehave(8. By the decreasig roerty ad the sae way of obtaiig (, we have w λ ( λ 2, > x R i + ;x i +τ i } = σ λ 2 u R i + ;u i } σ λ 2 x τ λ i dx k= ( x τ λ + c k σ λ /s u λ i du k= ( u λ + c k σ λ /s
8 436 BICHENG YANG = sγi ( v sλ λ λ i Γ( i i λ/(s / σ λ/s s k= (v + c k dv = K 2 ( θ λ ( >, < θ s λ (= λ k s ( λ The lea is roved. s λ k s ( λ s k= c k = λ k s ( λ s k= c k i λ/(s / σ λ/s λ/(s i / σ λ/s i λ / σ λ. v sλ λ s k= (v + c k dv v s λ λ dv 3. Mai results ad oerator exressios Settig Φ( := τ (i λ i ( N i ad Ψ( := σ ( j λ 2 j ( N j, we have THEOREM. If s N, <,, < c < < c s, < λ s, < λ i, < λ 2 j, λ + λ 2 = λ, τ (, 2 ]i, σ (, 2 ] j, the for >, + =, a,b, < a,φ, b,ψ <, we have the followig ieuality a b I := k= ( τ λ + c k σ λ /s < K K 2 a,φ b,ψ, (23 where the costat factor K K 2 = [ Γ j ( j Γ( j is the best ossible (k s (λ is idicated by (3. ] [ Γ i ( i Γ( i Proof. By Hölder s ieuality (cf. [37], we have I = s k= /s ( τ λ + c k σ λ /s τ (i λ / σ ( j λ 2 / a ] k s (λ (24 σ ( j λ 2 / τ (i λ / b
9 ON A MORE ACCURATE MULTIDIMENSIONAL HILBERT-TYPE INEQUALITY 437 } W λ (λ, τ (i λ i a w λ (λ 2, σ ( j λ 2 j b }. The by (7ad(8, we have (23. For < ε < iλ, λ 2 }, λ = λ ε, λ 2 = λ 2 + ε, we set ã = τ i +λ ε, b = σ j +λ 2 ε ( N i, N j. The by (ad(2, we obtai ã,φ b,ψ = τ (i λ i ã = τ i ε [ = Γ i ( ε Ĩ := } i ε/ i Γ( i + εo( } } σ ( j λ 2 j b } σ j ε ] Γ j ( j ε/ j Γ( j + εõ( ã k= ( τ λ + c k σ λ /s = w λ ( λ 2, σ j ε > K 2 O( σ λ σ j ε b, (25 Γ j ( = K 2 ε j ε/ + Õ( O(. (26 j Γ( j If there exists a costat K K K 2, such that (23 is valid as we relace K by K, the we have Γ j ( (K 2 + o( j ε/ j Γ( j [ Γ i ( < εk ã,ϕ b,ψ = K ] i ε/ i Γ( i + εo( + εõ( εo( < εĩ Γ j ( j ε/ j Γ( j + εõ( K 2.
10 438 BICHENG YANG For ε +, we fid Γ j ( j Γ( j ad the K K 2 K. Hece, K = K K 2 Γ i ( [ Γ i i Γ( i k ( s(λ K ] [ Γ j ( ] i Γ( i j Γ( j, is the best ossible costat factor of (23. THEOREM 2. With the assutios of Theore, for < a,φ <, we have the followig ieuality with the best costat factor K J := σ λ 2 j K 2 : a k= ( τ λ + c k σ λ /s < K K 2 a,φ, which is euivalet to (23. (27 Proof. We set b as follows: b := σ λ 2 j a k= ( τ λ + c k σ λ /s The it follows J = b,ψ. If J =, the (27 is trivially valid sice < a,φ < ; if J =, the it is a cotradictio sice the right had side of (27 isfiite. Suose that < J <. The by (23, we fid b,ψ = J = I < K K 2 a,φ b,ψ, K 2 aely, b,ψ = J < K K 2 a,φ, ad the (27 follows. O the other had, assuig that (27 is valid, by Hölder s ieuality, we have I = (Ψ( a [(Ψ( k= ( τ λ + c k σ λ /s b ] J b,ψ. (28 The by (27, we have (23. Hece (27 ad(23 are euivalet. By the euivalecy, the costat factor K i (27 is the best ossible. Otherwise, we would reach a cotradictio by (28 that the costat factor K ot the best ossible. K 2. i (23 is
11 ON A MORE ACCURATE MULTIDIMENSIONAL HILBERT-TYPE INEQUALITY 439 For >, we defie two real weight oral discrete saces l,ϕ ad l,ψ as follows: } l,ϕ := a = a }; a,φ = Φ(a } <, } l,ψ :=. b = b }; b,ψ = Ψ(b } < With the assutios of Theore, i view of J < K followig defiitio: K 2 a,φ, we have the DEFINITION 2. Defie a ultidiesioal Hilbert-tye oerator T : l,φ l,ψ as follows: For a l,φ, there exists a uiue reresetatio Ta l,ψ, satisfyig for N j, (Ta( := k= ( τ λ + c k σ λ /s (29. For b l,ψ, we defie the followig foral ier roduct of Ta ad b as follows: (Ta,b := a k= ( τ λ a b + c k σ λ /s (3. The by Theore ad Theore 2, for < a,ϕ, b,ψ <, we have the followig euivalet ieualities: (Ta,b < K K 2 a,φ b,ψ, (3 Ta,Ψ < K K 2 a,φ. (32 It follows that T is bouded sice Ta T := su,ψ a( θ l,φ a,φ K K 2. (33 Sice the costat factor K K 2 i (32 is the best ossible, we have COROLLARY. With the assutios of Theore 2, T is defied by Defiitio 2, it follows [ T = K K Γ j 2 = ( ] [ Γ i ( ] j Γ( j i Γ( i k s (λ. (34 REMARK. (i Settig Φ ( := (i λ i ( N i ad Ψ ( := ( j λ 2 j ( N j,
12 44 BICHENG YANG the uttig τ = σ = i(23 ad(27, we have the followig euivalet ieualities with the best costat factor K λ 2 j K 2 : a b k= ( λ + c k λ /s a < K k= ( λ + c k λ /s K 2 a,φ b,ψ, (35 Hece, (23ad(27 are ore accurate ieualities of (35ad(36. (ii Puttig i = j = i(32, we have ieuality = = Hece, (35 is a extesio of (4 for < K K 2 a,φ. (36 a b s k= (λ /s + c k λ /s < k s(λ a,φ b,ψ. (37 k λ (,= s k= (λ /s + c k λ /s. REFERENCES [] G. H. HARDY, J. E. LITTLEWOOG, G. PÓLYA, Ieualities, Cabridge Uiversity Press, Cabridge, 934. [2] D. S. MITRINOVIĆ, J. E. PEČARIĆ, A. M. FINK, Ieualities ivolvig fuctios ad their itegrals ad derivatives, Kluwer Acareic Publishers, Bosto, 99. [3] B. YANG, Hilbert-tye itegral ieualities, Betha Sciece Publishers Ltd., Dubai, 29. [4] B. YANG, Discrete Hilbert-tye ieualities, Betha Sciece Publishers Ltd., Dubai, 2. [5] B. YANG, O Hilbert s itegral ieuality, Joural of Matheatical Aalysis ad Alicatios, 22 (998, [6] B. YANG, The or of oerator ad Hilbert-tye ieualities, Sciece Press, Beiji, 29 (Chia. [7] B. YANG, Two tyes of ultile half-discrete Hilbert-tye ieualities, Labert Acadeic Publishig, Berli, 22. [8] B. YANG, I. BRNETIĆ, M. KRNIĆ, J. E. PEČARIĆ, Geeralizatio of Hilbert ad Hardy-Hilbert itegral ieualities, Math. Ie. ad Al., 8, 2 (25, [9] M. KRNIĆ, J. E. PEČARIĆ, Hilbert s ieualities ad their reverses, Publ. Math. Debrece, 67, 3 4 (25, [] B. YANG, TH. M. RASSIAS, O the way of weight coefficiet ad research for Hilbert-tye ieualities, Math. Ie. Al., 6, 4 (23, [] B. YANG, TH. M. RASSIAS, O a Hilbert-tye itegral ieuality i the subiterval ad its oerator exressio, Baach J. Math. Aal., 4, 2 (2,. [2] L. AZAR, O soe extesios of Hardy-Hilbert s ieuality ad Alicatios, Joural of Ieualities ad Alicatios, 29, o [3] B. ARPAD, O. HOONGHONG, Best costat for certai ulti liear itegral oerator, Joural of Ieualities ad Alicatios, 26, o [4] J. KUANG, L. DEBNATH, O Hilbert s tye ieualities o the weighted Orlicz saces, Pacific J. Al. Math.,, (27, [5] W. ZHONG, The Hilbert-tye itegral ieuality with a hoogeeous kerel of Labda-degree, Joural of Ieualities ad Alicatios, 28, o
13 ON A MORE ACCURATE MULTIDIMENSIONAL HILBERT-TYPE INEQUALITY 44 [6] Y. HONG, O Hardy-Hilbert itegral ieualities with soe araeters, J. Ie. i Pure & Alied Math., 6, 4 (25 Art. 92,. [7] W. ZHONG, B. YANG, O ultile Hardy-Hilbert s itegral ieuality with kerel, Joural of Ieualities ad Alicatios, Vol. 27, Art. ID 27962, 7 ages, doi:.55/ 27/27. [8] B. YANG, M. KRNIĆ, O the Nor of a Mult-diesioal Hilbert-tye Oerator, Sarajevo Joural of Matheatics, 7, 2 (2, [9] M. KRNIĆ, J. E. PEČARIĆ, P. VUKOVIĆ, O soe higher-diesioal Hilbert s ad Hardy-Hilbert s tye itegral ieualities with araeters, Math. Ieual. Al., (28, [2] M. KRNIĆ, P. VUKOVIĆ, O a ultidiesioal versio of the Hilbert-tye ieuality, Aalysis Matheatica, 38 (22, [2] M. TH. RASSIAS, B. YANG, A ultidiesioal half-discrete Hilbert-tye ieuality ad the Riea zeta fuctio, Alied Matheatics ad Coutatio, 225 (23, [22] Y. LI, B. HE, O ieualities of Hilbert s tye, Bulleti of the Australia Matheatical Society, 76, (27, 3. [23] B. YANG, A ixed Hilbert-tye ieuality with a best costat factor, Iteratioal Joural of Pure ad Alied Matheatics, 2, 3 (25, [24] B. YANG, A half-discrete Hilbert-tye ieuality, Joural of Guagdog Uiversity of Educatio, 3, 3 (2, 7. [25] W. ZHONG, A ixed Hilbert-tye ieuality ad its euivalet fors, Joural of Guagdog Uiversity of Educatio, 3, 5 (2, [26] W. ZHONG, A half discrete Hilbert-tye ieuality ad its euivalet fors, Joural of Guagdog Uiversity of Educatio, 32, 5 (22, 8 2. [27] J. ZHONG, B. YANG, O a extesio of a ore accurate Hilbert-tye ieuality, Joural of Zhejiag Uiversity (Sciece Editio, 35, 2 (28, [28] J. ZHONG, Two classes of half-discrete reverse Hilbert-tye ieualities with a o-hoogeeous kerel, Joural of Guagdog Uiversity of Educatio, 32, 5 (22, 2. [29] W. ZHONG, B. YANG, A best extesio of Hilbert ieuality ivolvig several araeters, Joural of Jia Uiversity (Natural Sciece, 28, (27, [3] W. ZHONG, B. YANG, A reverse Hilbert s tye itegral ieuality with soe araeters ad the euivalet fors, Pure ad Alied Matheatics, 24, 2 (28, [3] M. TH. RASSIAS,B. YANG, O half-discrete Hilbert s ieuality, Alied Matheatics ad Coutatio, 22 (23, [32] W. ZHONG, B. YANG, O ultile Hardy-Hilbert s itegral ieuality with kerel, Joural of Ieualities ad Alicatios, Vol. 27, Art. ID 27962, 7 ages, doi:.55/27/27. [33] B. YANG, Q. CHEN, A half-discrete Hilbert-tye ieuality with a hoogeeous kerel ad a extesio, Joural of Ieualities ad Alicatios, 24 (2, doi:.86/29-242x [34] B. YANG, A half-discrete Hilbert-tye ieuality with a o-hoogeeous kerel ad two variables, Mediterraea Joural of Metheatics, (23, [35] B. YANG, Hilbert-tye itegral oerators: ors ad ieualities (I Chater 42 of Noliear Aalysis, stability, aroxiatio, ad ieualities (P. M. Paralos et al., Sriger, New York, , 22. [36] Y. PAN, H. WANG, F. WANG, O colex fuctios, Sciece Press, Beijig, 26 (Chia. [37] J. KUANG, Alied ieualities, Shagdog Sciece Techic Press, Jia, 24 (Chia. (Received Deceber 2, 23 Bicheg Yag Deartet of Matheatics Guagdog Uiversity of Educatio Guagzhou, Guagdog 533, P.R. Chia e-ail: bcyag@gdei.edu.c, bcyag88@63.co Matheatical Ieualities & Alicatios ia@ele-ath.co
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