Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009, Article ID 820176, 8 pages doi:10.1155/2009/820176 Research Article A Hilbert’s Inequality with a Best Constant Factor Zheng Zeng 1 and Zi-tian Xie 2 1 Department of Mathematics, Shaoguan University, Shaoguan, Guangdong 512005, China 2 Department of Mathematics, Zhaoqing University, Zhaoqing, Guangdong 526061, China Correspondence should be addressed to Zi-tian Xie, gdzqxzt@163.com Received 6 February 2009; Revised 3 May 2009; Accepted 23 July 2009 Recommended by Yong Zhou We give a new Hilbert’s inequality with a best constant factor and some parameters. Copyright q 2009 Z. Zeng and Z t. Xie. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction If p>1, 1/p  1/q  1, a n ,b n > 0 such that ∞ >  ∞ n1 a p n > 0and∞ >  ∞ n1 b q n > 0, then the well-known Hardy-Hilbert’s inequality and its equivalent form are given by ∞  n1 ∞  m1 a m b n m  n < π sin  π/p   ∞  n1 a p n  1/p  ∞  n1 b q n  1/q , 1.1 ∞  n1  ∞  m1 a m m  n  p <  π sin  π/p   p  ∞  n1 a p n  , 1.2 where the constant factors are all the best possible 1. It attracted some attention in the recent years. Actually, inequalities 1.1 and 1.2 have many generalizations and variants. Equation 1.1 has been strengthened by Yang and others  including integral inequalities  2–11. 2 Journal of Inequalities and Applications In 2006, Yang gave an extension of 2 as follows. If p>1, 1/p  1/q  1, r>1, 1/r  1/s  1,t ∈ 0, 1, 2 − min{r, s}t  min{r, s}≥λ> 2 − min{r, s}t, such that ∞ >  ∞ n1 n p1−t2t−λ/r−1 a p n > 0, ∞ >  ∞ n1 n q1−t2t−λ/s−1 b q n > 0, then ∞  n1 ∞  m1 a m b n m  n λ <B   r − 2  t  λ r ,  s − 2  t  λ s   ∞  n1 n p1−t2t−λ/r−1 a p n  1/p  ∞  n1 n q1−t2t−λ/s−1 b q n  1/q . 1.3 Bu, v is the Beta function. In 2007 Xie gave a new Hilbert-type Inequality 3 as follows. If p>1, 1/p1/q  1,a,b,c >0, 2/3 ≥ μ>0, and the right of the following inequalities converges to some positive numbers, then ∞  m1 ∞  n1 a m b n  n μ  a 2 m μ  n μ  b 2 m μ  n μ  a 2 m μ  < π μ  a  b  b  c  c  a   ∞  n1 n 1−3μ/2p−1 a p n  1/p  ∞  n1 n 1−3μ/2q−1 b q n  1/q . 1.4 The main objective of this paper is to build a new Hilbert’s inequality with a best constant factor and some parameters. In the following, we always suppose that 1 1/p  1/q  1,p >1, a ≥ 0, −1 <α<1, 2 both functions ux and vx are differentiable and strict increasing in n 0 − 1, ∞ and m 0 − 1, ∞, respectively, 3 u  x/u α x,v  x/v α x are strictly increasing in n 0 − 1, ∞ and m 0 − 1, ∞, respectively. {u  n v  m /u 2 n  2au n v m  v 2 m u α n v α m } is strict decreasing on n and m, 4 unu n ,un 0 u 0 ,un 0 − 1  vm 0 − 1  0,u∞∞,v∞∞,u  n u  n ,vmv m ,vm 0 v 0 ,v  mv  m . Journal of Inequalities and Applications 3 2. Some Lemmas Lemma 2.1. Define the weight coe fficients as follows: W  p, m  : ∞  nn 0 1 u 2 n  2au n v m  v 2 m · v αp−1 m u α n · u  n v  m  p−1 , 2.1 ω  p, m  :  ∞ n o −1 1 u 2  x   2au  x  v m  v 2 m · v αp−1 m u α  x  · u   x  v  m  p−1 dx, 2.2  W  q, n  : ∞  mm 0 1 u 2 n  2au n v m  v 2 m · u αq−1 n v α m · v  m u  n  q−1 , 2.3 ω  q, n  :  ∞ m 0 −1 1 u 2 n  2au n v  y   v 2  y  · u αq−1 n v α  y  · v   y  u  n  q−1 dy , 2.4 then W  p, m  <ω  p, m   Kv pα−2α−1 m  v  m  p−1 ,  W  q, n  < ω  q, n   Ku qα−2α−1 n  u  n  q−1 , 2.5 where K   ∞ 0 dσ  1  2aσ  σ 2  σ α  ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ π 2 √ a 2 − 1sinαπ  a  √ a 2 − 1 α − 1 a  √ a 2 − 1 α  , if α /  0,a>1, | απ | / sin | απ | , if α /  0,a 1, πcscθcsc  απ  sin  αθ  , if α /  0,a cos θ, 0 <θ<π, 1 √ a 2 − 1 ln  a  √ a 2 − 1  , if α  0,a> 1, θcscθ, if α  0,a cos θ, 0 <θ< π 2 , 1, if α  0,a 1, 2.6 4 Journal of Inequalities and Applications Proof. Let fz1/1  2az  z 2 z α 1/z − z 1 z − z 2 z α  then K 2πi/1 − e −2απi Resf, z 1 Resf, z 2  if a>1 then z 1  −a − √ a 2 − 1,z 2  −a  √ a 2 − 1 K  2πi 1 − e −2απi ⎡ ⎢ ⎣  −a − √ a 2 − 1  −α −2 √ a 2 − 1   −a  √ a 2 − 1  −α 2 √ a 2 − 1 ⎤ ⎥ ⎦  π 2 √ a 2 − 1sinαπ ⎡ ⎢ ⎣  a   a 2 − 1  α − 1  a  √ a 2 − 1  α ⎤ ⎥ ⎦ , 2.7 if a  cos θ 0 <θ<π/2, then z 1  −e iθ ,z 2  −e −iθ K  2πi 1 − e −2απi  1  −2i sin θ  −e iθ  α  1  2i sin θ  −e −iθ  α   πcsc θcsc  απ  sin  αθ  . 2.8 On the other hand, Wp, m <ωp, m. Setting uxv m σ, then ωp, mKv pα−2α−1 m / v  m  p−1 . Similarly,  Wq, n < ωq, nKu qα−2α−1 n /u  n  q−1 . Lemma 2.2. For 0 <ε<min{p, p1 − α} one has  ∞ 0 dσ  1  2aσ  σ 2  σ αε/p  K  o  1  ε −→ 0   . 2.9 Proof.      ∞ 0 1  1  2aσ  σ 2  σ αε/p dσ − K     ≤       1 0 σ −α  1 − σ −ε/p  1  2aσ  σ 2 dσ             ∞ 1 σ −α  1 − σ −ε/p  1  2aσ  σ 2 dσ      ≤       1 0 σ −α  1 − σ −ε/p  dσ            ∞ 1 σ −2−α  1 − σ −ε/p  dσ          1 1 − α − 1 1 − α − ε/p          1 1  α − 1 1  α  ε/p     −→ 0forε −→ 0  . 2.10 The lemma is proved. Journal of Inequalities and Applications 5 Lemma 2.3. Setting w n  u n (or v m  and w 0  n 0 (or m 0 , resp.), then k>0. {τ  w /τ k w } is strictly decreasing, then N  ww 0 τ  w τ k w   N w 0 τ   x  τ k  x  dx  A . 2.11 There A ∈ 0,τ  w 0 /τ k w 0 , for any N). Proof. We have  N w 0 τ   x  τ k  x  dx < N  ww 0 τ  w τ k w  τ  w 0 τ k w 0  N  ww 0 1 τ  w τ k w < τ  w 0 τ k w 0   N w 0 τ   x  τ k  x  dx . 2.12 Easily, A had up bounded when N →∞. 3. Main Results Theorem 3.1. If a n > 0,b n > 0, 0 <  ∞ n1 v pα−2α−1 m /v  m  p−1 a p n < ∞, 0 <  ∞ nn 0 u qα−2α−1 n / u  n  q−1 b q n < ∞,then ∞  nn 0 ∞  mm 0 a m b n u 2 n  2au n v m  v 2 m <K  ∞  mm 0 v pα−2α−1 m v  m  p−1 a p m  1/p  ∞  nn 0 u qα−2α−1 n u  n  q−1 b q n  1/q , 3.1 ∞  nn 0 u pαp−2α−1 n u  n  ∞  mm 0 a m u 2 n  2au n v m  v 2 m  p <K p ∞  mm 0 v pα−2α−1 m v  m  p−1 a p m . 3.2 K is defined by Lemma 2.1. Proof. By H ¨ older’s inequality 12 and 2.5, J :  ∞  nn 0 ∞  mm 0 a m b n u 2 n  2au n v m  v 2 m  ∞  nn 0 ∞  mm 0 1 u 2 n  2au n v m  v 2 m · v α/q m u α/p n · u  n  1/p v  m  1/q a m · u α/p n v α/q m · v  m  1/q u  n  1/p b n ≤  ∞  mm 0 Wp, ma p m  1/p  ∞  nn 0  Wq, nb q n  1/q <K  ∞  mm 0 v pα−2α−1 m v  m  p−1 a p m  1/p  ∞  nn 0 u qα−2α−1 n u  n  q−1 b q n  1/q , 3.3 6 Journal of Inequalities and Applications setting b n  u pα−2αp−1 n u  n   ∞ mm 0 a m /u 2 n  2au n v m  v 2 m  p−1 > 0. By3.1 we have ∞  nn 0 u qα−2α−1 n u  n  q−1 b q n  ∞  nn 0 u pα−2αp−1 n u  n  ∞  mm 0 a m u 2 n  2au n v m  v 2 m  p  J ≤ K  ∞  mm 0 v pα−2α−1 m v  m  p−1 a p m  1/p  ∞  nn 0 u qα−2α−1 n u  n  q−1 b q n  1/q . 3.4 By 0 <  ∞ nn 0 u qα−2α−1 n /u  n  q−1 b q n < ∞ and 3.4 taking the form of strict inequality, we have 3.1.ByH ¨ older’s inequality12, we have J  ∞  nn 0  u −α2α/q1/q n u  n  −11/q ∞  mm 0 a m u 2 n  2au n v m  v 2 m   u α−2α/q−1/q n b n  u  n  1−1/q ≤  ∞  nn 0 u pα−2αp−1 n u  n  ∞  mm 0 a m u 2 n  2au n v m  v 2 m  p  1/p  ∞  nn 0 u qα−2α−1 n u  n  q−1 b q n  1/q . 3.5 as 0 < {  ∞ nn 0 u qα−2α−1 n /u  n  q−1 b q n } 1/q < ∞.By3.2, 3.5 taking the form of strict inequality, we have 3.1. Theorem 3.2. If α  0, then both constant factors, K and K p of 3.1 and 3.2, are the best possible. Proof. We only prove that K is the best possible. If the constant factor K in 3.1 is not the best possible, then there exists a positive H with H<K, such that J<H  ∞  mm 0 v −1 m v  m  p−1 a p m  1/p  ∞  nn 0 u −1 n u  n  q−1 b q n  1/q . 3.6 For 0 <ε<min{p, q}, setting a m  v −ε/p m v  m ,  b n  u −ε/q n u  n , then  ∞  mm 0 v −1 m v  m  p−1 a p m  1/p  ∞  nn 0 u −1 n u  n  q−1  b q n  1/q   ∞  mm 0 v  m v 1ε m  1/p  ∞  nn 0 u  n u 1ε n  1/q . 3.7 Journal of Inequalities and Applications 7 On the other hand uxσvy and vyτ, ∞  mm 0 ∞  nn 0 u −ε/p n u  n v −ε/q m v  m u 2 n  2au n v m  v 2 m >  ∞ m 0   ∞ n 0 u −ε/p  x  u   x  dx u 2  x   2au  x  v  y   v 2  y   vy −ε/q v   y  dy   ∞ m 0   ∞ u 0 /vy σ −ε/p dσ σ 2  2aσ  1  vy −1−ε v   y  dy   ∞ v 0   ∞ 0 σ −ε/p dσ σ 2  2aσ  1  τ −1−ε dτ −  ∞ v 0   u 0 /τ 0 σ −ε/p dσ σ 2  2aσ  1  τ −1−ε dτ ≥  K  o  1   ∞ v 0 τ −1−ε dτ −  ∞ v 0 τ −1  u 0 /τ 0  σ −ε/p dσ  dτ   K  o  1   ∞ v 0 τ −1−ε dτ − u 1−ε/p 0 v −1ε/p 0 1 − ε/p 2   K  o  1   ∞ v 0 τ −1−ε dτ − O  1  . 3.8 By 3.6, 3.7, 3.8,andLemma 2.3, w e have  K  o  1  − O  1   ∞ v 0 τ −1−ε dτ <H   ∞ mm 0  v  m /v 1ε m   ∞ v 0 τ −1−ε dτ  1/p   ∞ nn 0  u  n /u 1ε n   ∞ v 0 τ −1−ε dτ  1/q , 3.9  K  o  1  − O  1   ∞ v 0 τ −1−ε dτ <H  1  O1  ∞ v 0 τ −1−ε dτ  1/p  1   O1  ∞ v 0 τ −1−ε dτ  1/q . 3.10 We have K ≤ H, ε → 0  . This contracts the fact that H<K. References 1 G. H. Hardy, J. E. Littlewood, and G. P ´ olya, Inequalities, Cambridge University Press, Cambridge, UK, 1952. 2 B. C. Yang, “On Hilbert’s inequality with some parameters,” Acta Mathematica Sinica. Chinese Series, vol. 49, no. 5, pp. 1121–1126, 2006. 3 Z. Xie, “A new Hilbert-type inequality with the kernel of -3μ-homogeneous,” Journal of Jilin University. Science Edition, vol. 45, no. 3, pp. 369–373, 2007. 4 Z. Xie and B. Yang, “A new Hilbert-type integral inequality with some parameters and its reverse,” Kyungpook Mathematical Journal, vol. 48, no. 1, pp. 93–100, 2008. 5 B. Yang, “A Hilbert-type inequality with a mixed kernel and extensions,” Journal of Sichuan Normal University. Natural Science, vol. 31, no. 3, pp. 281–284, 2008. 6 Z. Xie and Z. Zeng, “A Hilbert-type integral with parameters,” Journal of Xiangtan University. Natural Science, vol. 29, no. 3, pp. 24–28, 2007. 8 Journal of Inequalities and Applications 7 W. Wenjie, H. Leping, and C. Tieling, “On an improvenment of Hardy-Hilbert’s type inequality with some parameters,” Journal of Xiangtan University. Natural Science, vol. 30, no. 2, pp. 12–14, 2008. 8 Z. Xie, “A new reverse Hilbert-type inequality with a best constant factor,” Journal of Mathematical Analysis and Applications, vol. 343, no. 2, pp. 1154–1160, 2008. 9 B. Yang, “On an extended Hardy-Hilbert’s inequality and some reversed form,” International Mathematical Forum, vol. 1, no. 37–40, pp. 1905–1912, 2006. 10 Z. Xie, “A Hilbert-type inequality with the kernel of irrational expression,” Mathematics in Practice and Theory, vol. 38, no. 16, pp. 128–133, 2008. 11 Z. Xie and J. M. Rong, “A new Hilbert-type inequality with some parameters,” Journal of South China Normal University. Natural Science Edition, vol. 120, no. 2, pp. 38–42, 2008. 12 J. Kang, Applied Inequalities, Shangdong Science and Technology Press, Jinan, China, 2004. . Hilbert-type inequality with a best constant factor,” Journal of Mathematical Analysis and Applications, vol. 343, no. 2, pp. 1154–1160, 2008. 9 B. Yang, “On an extended Hardy -Hilbert’s inequality and. Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009, Article ID 820176, 8 pages doi:10.1155/2009/820176 Research Article A Hilbert’s Inequality with a Best Constant. n  p <  π sin  π/p   p  ∞  n1 a p n  , 1.2 where the constant factors are all the best possible 1. It attracted some attention in the recent years. Actually, inequalities 1.1 and 1.2 have many generalizations and