WEIGHTED HARDY-HILBERT S INEQUALITY

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1 Bulletin of the Marathwada Mathematical Society Vol. 9, No., June 28, Pages 8 3. WEIGHTED HARDY-HILBERT S INEQUALITY Namita Das P. G. Deartment of Mathematics, Utkal University, Vani Vihar, Bhubaneshwar,75 4, Orissa, INDIA. E.mail : namitadas44@yahoo.co.in and Srinibas Sahoo Deartment of Mathematics, Banki College, Banki,754 8, Orissa, INDIA. E.mail : sahoosrinibas@yahoo.co.in Abstract In this aer, by introducing the weight functions ωx and ωx, we give a weighted Hardy-Hilberts integral ineuality and by introducing the weight coefficients ω n and ω n, we give a weighted Hardy-Hilberts ineuality in discrete form with best constant factors. As alications, we give its euivalent forms and weighted Hardy- Littlewoods ineuality. INTRODUCTION then If >, +, f,g satisfy < f x < and < g x < fxgy x + y < / / f x g x ;. sin / and an euivalent form is where the constant factors fx [ ] x + y dy < sin / [ sin/ and f x;.2 sin/] are the best ossible. The corresonding double series ineuality is: if >,/ + /,a n,b n satisfy < a n < Key words and hrases:weight function;β-function;hardy-hilberts ineuality;holders ineuality;hardy- Littlewoods ineuality. 2 Mathematics Subject Classification. 26D5 8

2 Weighted :Hardy-HilbertS Ineuality 9 and < b n <, then a m b n m + n < { } / { } / a n b n ;.3 sin / and an euivalent form is [ ] a m < a m + n sin / n;.4 where the constant factors /sin / and [/sin /] are the best ossible. Ineualities. and.3 are called Hardy-Hilbert s ineualities see [3], and are imortant in analysis and its alications cf.mitrinovic et al. [5].Recently many generalizations and refinements of these ineualities were also obtained. The main objective of this aer is to build the weighted version of the Hardy-Hilberts ineualities. and.3 with best constant factors, which imroves the corresonding ineualities. The euivalent forms are also considered. First, we need the formula of the β-function as cf.wang et al.[6]: B, + t + t dt B,.5 and the Hőlders ineuality with weightcf. Kuang [4] as: If >,/ + /,ωt >,f,g,f L ωe and g L ωe, then / / ωtftgtdt ωtf tdt ωtg tdt ;.6 E E if <, with the above assumtion, the reverse of.6 holds, where the euality in the above two cases holds if and only if there exists non-negative real numbers c and c 2 such that they are not all zero and 2 MAIN RESULTS c f t c 2 g t, a.e. in E. First we will give the integral form of weighted Hardy-Hilberts ineuality as follows: Theorem 2. If >,/ + /,ωx >, ωx > are continuous functions on,,wx x ωtdt, Wx x ωtdt for x, and f,g satisfy < x ωxf x < and < x ωxg x <, then ωxfx ωygy x / x / Wx + Wy < ωxf x ωxg x sin / 2. where the constant factor /sin/ is the best ossible. E

3 2 Namita Das and Srinibas Sahoo Proof. By.6, we have ωxfx ωygy Wx + Wy ωx ωy Wx Wx + Wy Wy { [ ωy Wx Wx + Wy Wy ωx Wx + Wy fx Wy Wx dy ] Wy Wx gy ωxf x } ωyg ydy 2.2 If 2.2 takes the form of euality, then by.6, there exists non negative numbers c and c 2, such that they are not all zero and It follows that Wx c Wy where c 3 is a constant. f x c 2 Wy Wx g y,a. e. in,,. c Wxf x c 2 Wyg y c 3,a. e. in,,, Without loss of generality, suose that c l. Then we have ωxf x c 3 c ωx Wx c 3 c which gives a contradiction. Thus the ineuality 2.2 is strict. Setting t Wy Wx, we have by.5 Similarly ωy Wx + Wy Wx Wy dy + t t dt B t dt, sin. ωx Wx + Wy Wy Wx sin.

4 Weighted :Hardy-HilbertS Ineuality 2 Hence 2. is valid. For sufficiently small ǫ >, we take f ǫ x g ǫ x { if x,aa W, Wx +ǫ if x [a,. { if x,bb W, +ǫ Wx if x [b,. Then ωxfǫ x ωxg ǫx ǫ. 2.3 For fixed x a,, setting t Wy Wx, we have I : a a a a ωxf ǫ ǫx ωyg ǫ y Wx + Wy b Wx + Wy ωx Wx +ǫ ωx Wx +ǫ b Wx + Wy ωx Wx +ǫ Wx ωx Wx +ǫ > ǫ B + ǫ ǫ B ǫ, + ǫ ǫ B ǫ, + ǫ, + ǫ a +ǫ + t t dt ωy +ǫ Wy ωy +ǫ Wy +ǫ + t t dt a ωx Wx Wx ǫ ǫ dy ωx Wx Wx +ǫ t +ǫ dt +ǫ + t t dt ωx Wx + ǫ a If the constant factor /sin/ in 2. is not the best ossible, then there exists a ositive constant K < /sin/, such that 2. is still valid if we relace /sin/ by K. In

5 22 Namita Das and Srinibas Sahoo articular, by 2.3 and 2.4, we have B ǫ, + ǫ ǫ ǫ 2 ωxf ǫ ǫx ωyg ǫ y < ǫ Wx + Wy < ǫk ωxfǫ x ωxgǫx K, and then sin/ B/,/ Kǫ +. This contradiction leads to the conclusion that the constant factor in 2. is the best ossible.the theorem is roved. Theorem 2.2 If >, +,ωx >, ωx > are continuous functions on,,wx x ωtdt, Wx x ωtdt for x, and f satisfy < ωxf x <, then we obtain an euivalent ineuality of 2. as follows: [ ] ωxfx [ ] ωy dy < ωxf Wx + Wy x 2.5 sin/ where the constant factor [/sin/] P is the best ossible. then [ ] ωxfx Proof. Setting gy, we get by 2. Wx + Wy < < ωyg ydy [ ωy ωxfx Wx + Wy ωxfx ωygy Wx + Wy ωxf x sin / { } ωyg ydy [ ωy ] dy, ] ωxfx dy Wx + Wy ωxg x 2.6

6 Weighted :Hardy-HilbertS Ineuality 23 { } ωxf x sin / < 2.7 Hence < ωyg ydy <. So, 2.6 takes the form of strict ineuality by using 2.;so, does 2.7. Hence we can get 2.5. On the other hand, if 2.5 holds, then by.6, we have ωxfx ωygy Wx + Wy [ ] ωxfx ωy gydy Wx + Wy { [ ωxfx ωy Wx + Wy ] dy } { ωyg ydy Hence by 2.5, 2. yields. Thus it follows that 2. and 2.5 are euivalent. By Theorem- 2., the constant factor in 2. is best ossible, hence the constant factor in 2.5 is best ossible. The theorem is roved. } Remark 2. For ωx ωx, 2. reduces to. and 2.5 reduces to.2. Now we will give the discrete form of weighted Hardy-Hilberts ineuality as follows: Theorem 2.3 If >, +,ω n >, ω n >,W n n k ω k, W n n k ω k and a n,b n, satisfy < ω na n <, < ω nb n <, then ω m a m ω n b n < ω n a n sin / ω n b n The constant factor /sin / is the best ossible, if ω n ω n+, ω n ω n+ for n. Proof We take W W and define fx a m,w m x < W m ; gx b n, W n x < W n ; 2.8 Then < f P x < Wm g x W m a m ω m a m < ; ω n b n <

7 24 Namita Das and Srinibas Sahoo and ω m a m ω n b n < Wm W m Wn W n fxgy x + y. a m b n So, by Hardy-Hilberts ineuality., 2.8 is valid. For sufficiently small ǫ >, we take ã n W +ǫ n, b n Similarly +ǫ W n,n. Then ω m ã m ω W +ǫ If ω n ω n+, ω n ω n+, then ω m W +ǫ m ω + m2 ω W +ǫ + ǫw ǫ Wm ǫ ω m b n ǫ W +ǫ + m2 Wm Wm +ǫ W m W +ǫ x +ǫ ω W m { ǫω W +ǫ + W ǫ { + W x +ǫ }. 2.9 } ǫ ω W +ǫ + W. 2. ǫ : ω m ã m ω n bn Wm+ W m + W ωm+ ω n+ +ǫ +ǫ n Wm W n Wn+ W n W m + W n W m W W x + y x +ǫ y +ǫ W +ǫ m W +ǫ n Setting t y x, we get +ǫ W x +ǫ W + t t dt x W x +ǫ +ǫ + t t dt W x +ǫ W x +ǫ + t t dt

8 Weighted :Hardy-HilbertS Ineuality 25 Now by.5 > ǫw ǫ B ǫw ǫ B ǫw ǫ B + ǫ, + ǫ ǫ, + ǫ ǫ, + ǫ W x +ǫ ǫ + ǫ ǫw ǫ ǫ, + ǫ < ǫ ω m a m ω n b n ǫ W m + W n < ǫk ω n a n ω n b n W W x ǫ ǫ t +ǫ dt W x + + ǫ W + ǫ ǫ W. 2. If the constant factor /sin / in 2.8 is not the best ossible, then there exists a ositive constant K < /sin / such that 2.8 is still valid if we relace /sin / by K. In articular, by 2.9, 2. and 2., we have B ǫ + ǫ ǫ W + ǫ W ǫ K { ǫω W +ǫ + W ǫ } { } ǫ ω W +ǫ + W ǫ and then sin/ B/,/ Kǫ +. This contradiction leads to the conclusion that the constant factor in 2.8 is the best ossible, if ω n ω n+, ω n ω n+ for n. The theorem is roved. Theorem 2.4 If >, +,ω n >, ω n >,W n n k ω k, W n n k ω k and a n, satisfy < ω na n <, then we obtain an euivalent ineuality of 2.8as follows: [ ] [ ] ω m a m ω n W m + W < ω n a n n sin /. 2.2 The constant factor /sin / is the best ossible, if ω n ω n+, ω n ω n+ for n Proof. Setting b n < ω n b n [ ω n [ ω ma m W m+ W n ], we get by 2.8 ω m a m ω m a m ω n b n ] sin/ ω n a n ω n b n ; 2.3

9 26 Namita Das and Srinibas Sahoo then < ω n b n sin / ω n [ ω m a m ω n a n ] <. 2.4 Hence < ω nb n <. So, 2.3 takes the form of strict ineuality by using 2.8; so, does 2.4. Hence we can get 2.2. On the other hand, if 2.2 holds, then by Hőlders ineuality, we have ω m a m ω n b n { ω n [ ω n [ ω m a m ω m a m ] ω nb n ] } { Hence by 2.2, 2.8 yields. Thus it follows that 2.8 and 2.2 are euivalent. By Theorem-2.4, the constant factor in 2.8 is best ossible, if ω n ω n+, ω n ω n+ for n. Hence the constant factor in 2.2 is best ossible, if ω n ω n+, ω n ω n+ for n. The theorem is roved. ω n b n } Remark 2.2 For ω n ω, 2.8 reduces to.3 and.2 reduces to.4. 3 APPLICATIONS In this section, we will give the weighted version of Hardy-Littlewoods ineuality. Let f L 2, and fx. f a n x n fx, n,,2,3, then we have the Hardy-Littlewoods ineuality see [3] of the form a 2 n < f 2 x 3. n where the constant factor is the best ossible. In [, 2], Gao gave the integral version of Hardy-Littlewoods ineuality as follows: Let h L 2, and h. If fx t x ht dt, x [,,

10 Weighted :Hardy-HilbertS Ineuality 27 then and f 2 x < h 2 tdt, 3.2 f 2 x < th 2 tdt, 3.3 Yang [7] gave a generalization of 3. for 2 as n a n + < a n f 2 x. 3.4 sin / n First we give the weighted version of Hardy-Littlewoods integral ineuality as follows: Theorem 3.. Let >, +,ωx > be a continuous function on, and Wx x ωtdt, for x, and h L2,, ht. Define a function by fx t Wx ht dt, x [,. If < ωxf x <, then + ωxf x < ωxf x th 2 tdt. 3.5 sin / Proof. We can write f x f x t Wx ht dt. Now alying, Schwartz ineuality and Theorem-2., we have ωxf x { 2 ωxf x } 2 t Wx ht dt. { ωxf xt Wx 2. 2 ωxf xt 2 Wx dt ωxf xωyf y Wx + Wy } 2 t 2 ht dt th 2 tdt th 2 tdt

11 28 Namita Das and Srinibas Sahoo sin / sin / ωxf x ωxf x ωxf x ωxf x th 2 tdt th 2 tdt 3.6 Since ht, so, fx. Hence it is imossible to get the euality in 3.6 and then we get the ineuality 3.5. This comletes the theorem. Remark 3. For 2, 3.5 becomes ωxf 2 x < th 2 tdt 3.7 which is a generalization of Hardy-Littlewoods integral ineuality 3.3. Theorem 3.2 Let >, +,ω n > and W n n k ω k. Let f L 2, and fx. Define a n x Wn 2fx, n N If < ω na n <, then ω n a n + < ω n a n f 2 x. 3.8 sin / Proof. We can write a n a n x Wn 2fx.

12 Weighted :Hardy-HilbertS Ineuality 29 Now alying, Schwartz ineuality and Theorem - 2.4, we have 2 2 ω n an ω n an x Wn 2 fx { sin / sin / ω n a n x Wn 2 ω n a n x Wn 2 2 ω m am ω n an W m + W n ω n a n ω n a n fx } 2 f 2 x f 2 x ω n a n ω n a n f 2 x f 2 x 3.9 Since fx, so, a n. Hence it is imossible to have euality in 3.9 and then we get the ineuality 3.8. This comletes the theorem. Remark 3.2. For 2, 3.8 becomes ω n a 2 n < f 2 x 3. which is a generalization of Hardy-Littlewood s ineuality 3.. References [] Gao M., On Hilberts Ineuality and Its Alications, J. Math. Anal. Al., , [2] Gao M., Tan LI and Debnath L., Some Imrovements on Hilberts Integral Ineuality, J. Math. Anal. Al., , [3] Hardy G.H., Littlewood J.E. and Polya G., Ineualities, Cambridge University Press, Cambrige, 952. [4] Kuang J., Alied Ineualities, Shangdong Science and Technology Press, Jinan, 24. [5] Mintrinovic D.S., Pecaric J.E. and Fink A.M., Ineualities Involving Functions and Their Integrals and Derivatives, Kluwer Academic Publishers, Boston, 99.

13 3 Namita Das and Srinibas Sahoo [6] Wang Z. and Guo D., An Introduction to Secial Functions, Science Press, Beijing, 979. [7] Yang B., On a refinement of Hardy-Hilberts ineuality and its alications. North. Math. J.,632,