M atheatical I eualities & A licatios Volue 8, Nuber 2 (25, 429 44 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. 63786, ad 23 Kowledge Costructio Secial Foudatio Ite of Guagdog Istitutio of Higher Learig College ad Uiversity (No. 23KJCX4. c D l,zagreb Paer MIA-8-32 429
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.
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 +. (
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 ], (
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
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
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
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
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.
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
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,
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, 778 785. [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, 259 272. [9] M. KRNIĆ, J. E. PEČARIĆ, Hilbert s ieualities ad their reverses, Publ. Math. Debrece, 67, 3 4 (25, 35 33. [] B. YANG, TH. M. RASSIAS, O the way of weight coefficiet ad research for Hilbert-tye ieualities, Math. Ie. Al., 6, 4 (23, 625 658. [] 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. 546829. [3] B. ARPAD, O. HOONGHONG, Best costat for certai ulti liear itegral oerator, Joural of Ieualities ad Alicatios, 26, o. 28582. [4] J. KUANG, L. DEBNATH, O Hilbert s tye ieualities o the weighted Orlicz saces, Pacific J. Al. Math.,, (27, 95 3. [5] W. ZHONG, The Hilbert-tye itegral ieuality with a hoogeeous kerel of Labda-degree, Joural of Ieualities ad Alicatios, 28, o. 97392.
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