ON THE CONSTANT IN ME ´ NSHOV-RADEMACHER INEQUALITY SERGEI CHOBANYAN, SHLOMO LEVENTAL, AND HABIB SALEHI Received 26 March 2005; Accepted 7 September 2005 The goal of the paper is twofold: (1) to show that the exact value D 2 in the Me ´ nshov- Rademacher inequality equals 4/3, and (2) to give a new proof of the Me ´ nshov- Rademacher inequality by use of a recurrence relation. The latter gives the asymptotic estimate limsup n D n / log 2 2 n ≤1/4. Copyright © 2006 Sergei Chobanyan et al. 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 The Me ´ nshov-Rademacher inequality deals with the estimation of D n = supE max 1≤k≤n k l=1 α l ϕ l 2 , (1.1) where sup is taken over all probability spaces (Ω,Ᏺ,P), all real orthonormal systems (ϕ 1 , ,ϕ n ) on them, and all real coefficient collections (α 1 , ,α n )with n 1 α 2 i = 1. Rademacher [9]andMe ´ nshov [7] independently proved that there exists an absolute constant C>0suchthatforeachn ≥ 2, D n ≤ C log 2 2 n. (1.2) A tr a ditional proof using a bisection method (see, e.g., Doob [2]andLo ` eve [6]) leads to the inequality D n ≤ log 2 n +2 2 , n ≥2. (1.3) Kounias [4] used a trisection method to get a finer inequality: D n ≤ log 2 n log 2 3 +2 2 , n ≥2. (1.4) Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 68969, Pages 1–7 DOI 10.1155/JIA/2006/68969 2 On the constant in Me ´ nshov-Rademacher inequality The aim of this paper is twofold: to show that the exact starting value D 2 = 4/3andto establish a recurrence relation which leads to a refinement of (1.4) and an asymptotic constant ≤ 1/4. Note that there are several other proofs of the Me ´ nshov-Rademacher in- equality and its generalizations, see, for example, Somogyi [10]andM ´ oricz and Tandori [8]. Section 2 dealswiththeproofofD 2 = 4/3, while Section 3 is devoted to the proof of the Me ´ nshov-Rademacher inequality with the asymptotic constant ≤ 1/4. Section 4 contains alternative proofs to those results using the concept of main triangle projec- tion, a subject which was studied in depth in Gohberg and Kre ˘ ın [3]andKwapie ´ nand “Pełczy ´ nski” [5]. 2. The value of D 2 Theorem 2.1. D 2 = 4/3. The proof of the theorem is based on the following lemma which may be of indepen- dent interest. Lemma 2.2. Let c>0 , p c ≡ c 2 /(1 + c 2 ),anddefine f (p,c) = sup X∈Ꮽ(p,c) E X1 X>−c ), p c ≤ p<1, (2.1) where Ꮽ(p,c) ={X ∈ L 0 (Ω,Ᏺ,P):E(X) =0, E(X 2 ) =1, P(X>−c) = p}. (2.2) Then f (p,c) = p(1 − p). (2.3) Proof of Lemma 2.2. To show that the left-hand side is greater than or equal to right-hand side, we observe that E(X p 1 X p >−c ) = p(1 − p), where the distribution of X p ∈ Ꮽ(p,c)is given by p = P X p = (1 − p) p = 1 −P X p =− p (1 − p) . (2.4) To see that the left-hand side is less than or equal to right-hand side, we define h p (x) = x ·1 x>−c − p ·x − p(1 − p) 4 ·x 2 . (2.5) The maximum of h p (x)isachievedatx = (1 − p)/p and at − p/(1 − p) for the regions x> −c and x ≤−c, respectively. We conclude that for any X ∈ Ꮽ(p,c), 0 ≤ E h p X p − E h p (X) = E X p ·1 X p >−c − E X ·1 X>−c . (2.6) This completes the proof of the lemma. Sergei Chobanyan et al. 3 Let us note also that Ꮽ(p,c)isemptyforp<p c . Indeed, by the Chebyshev inequality, E(X) = 0andE(X 2 ) =1implyP(X ≤−c) ≤1/(1 + c 2 ) =1 − p c . Proof of Theorem 2.1. The result follows by standard calculations from the representation D 2 = sup a 2 +b 2 =1,b 2 /(1+3a 2 )<p<1 a 2 + b 2 p +2ab · p(1 − p) . (2.7) To p rove (2.7) convert an orthonormal pair (ϕ 1 ,ϕ 2 )definedon(Ω,Ᏺ,P)into(X ≡ ϕ 1 / ϕ 2 ,1). The new pair is orthonormal with respect to the measure dP = ϕ 2 2 dP.Also E P max aϕ 1 2 , aϕ 1 + bϕ 2 2 = E P max (aX) 2 ,(aX + b) 2 = a 2 + b 2 P (X>−b/2a) + 2ab ·E P X ·1 X>−b/2a ≤ a 2 + b 2 p +2ab · f p, b 2a , (2.8) where p = P (X>−b/2a). Now (2.7)followsfromLemma 2.2 with c = b/2a. 3. An induction proof of the Me ´ nshov-Rademacher inequality Theorem 3.1. (i) D m ≤ 1 4 3+log 2 m 2 , m ≥2. (3.1) In particular , (ii) limsup m D m log 2 2 m ≤ 1 4 . (3.2) Lemma 3.2. The following recurrence relation holds t rue for any n ∈ N: D 2n ≤ D n + D 1/2 n . (3.3) Proof of Lemma 3.2. We have for any n ∈ N, max k≤2n k 1 α i ϕ i 2 ≤ max max k≤n k 1 α i ϕ i 2 , n 1 α i ϕ i +max n<k≤2n k n+1 α i ϕ i 2 ≤ max k≤n k 1 α i ϕ i 2 +2 n 1 α i ϕ i max n<k≤2n k n+1 α i ϕ i +max n<k≤2n k n+1 α i ϕ i 2 . (3.4) Taking expectations in (3.4) and using the Cauchy-Schwartz inequality, we come to the 4 On the constant in Me ´ nshov-Rademacher inequality desired recurrence relation: D 2n ≤ pD n +2 p(1 − p)D n +(1− p)D n = D n + D n , (3.5) where p = n 1 α 2 i . The lemma is proved. Proof of Theorem 3.1. Lemma 3.2 implies that for any n ∈N, D 1/2 2n ≤ D 1/2 n + 1 2 . (3.6) Since D 1 = 1, this implies that for each n ∈ N, D 1/2 2 n ≤ 1+ n 2 . (3.7) Let us take now 2 n ≤ m<2 n+1 . Then D m ≤ D 2 n+1 ≤ 1+ n +1 2 2 ≤ 1+ log 2 m +1 2 2 . (3.8) This implies the validity of Theorem 3.1. Remark 3.3. (1) The proof of Theorem 3.1 is a refinement of that appeared in Chobanyan [1]. (2) Kounias’s result mentioned in the introduction leads to limsup(D n / log 2 2 n) ≤ (log2/log3) 2 which is larger than 1/4 of Theorem 3.1. 4. An alternative approach: the main triangle projection Consider the space L( R n ) of all linear operators (matrices) acting in R n . The correspon- dence between the operators and matrices is given by a ij = (Ae j ,e i ), i, j = 1, ,n. The main triangle projection T n : L(R n ) →L(R n ) is a linear operator introduced as follows. For an A ∈ L(R n ), the matrix of the operator B = T n A has the form b ij = a ij if i + j ≤ n +1 and b ij = 0 otherwise. We assume that R n is endowed with the Euclidean norm, and the norm in L(R n )isthe usual operator norm. Theorem 4.1. D n =T n 2 , n ∈N. Proof. Let us prove first that T n 2 ≡ sup A≤1 T n A 2 ≤ D n . Since the orthogonal op- erators (and only them) are the extreme points of the unit ball of L( R n ), it suffices to show that for any orthogonal operator u ∈ L(R n ), T n u 2 ≤ D n . Let us relate with u the orthonormal system ϕ 1 , ,ϕ n defined on (Ω, P), where Ω ={1, ,n}, P( j) = 1/n, j = 1, ,n,asfollows: ϕ k ( j) = √ n ue k ,e j , k, j =1, ,n. (4.1) Sergei Chobanyan et al. 5 We have for any vector α = (α 1 , ,α n ) ∈R n with |α|=1, D n ≥ Emax k≤n k i=1 α i ϕ i 2 = n j=1 max k≤n k i=1 α i ue i ,e j 2 ≥ n j=1 n−j+1 i=1 α i ue i ,e j 2 = T n u α 2 . (4.2) Taking supremum over all orthogonal u’s and α’s from the unit ball of R n ,wegetD n ≥ T n 2 . To prove the inverse inequality, consider an orthonormal system (ϕ 1 , ,ϕ n ) ⊂ L 2 (Ω,Ᏺ,P)andanyvectorα =(α 1 , ,α n ) ∈R n with |α|=1. I(α,ϕ) ≡ Emax k≤n k i=1 α i ϕ i 2 = n k=1 E1 S k k i=1 α i ϕ i 2 , (4.3) where S k ={ω ∈ Ω :theminimumofl s at which | l i =1 α i ϕ i (ω)| attains its maximum equals k }.Thenwehave I(α,ϕ) = sup g n k=1 Eg k 1 S k k i=1 α i ϕ i 2 , (4.4) where supremum is taken over all collections g = (g 1 , ,g n )suchthatg k ’s vanish outside of S k and g k 2 = 1, k = 1, ,n. We have further I(α,ϕ) = sup g n k=1 k i, j=1 α i α j Eg k ϕ i ϕ j = sup g n i, j=1 n k=max(i,j) α i α j Eg k ϕ i ϕ j = sup g T n Aα 2 , (4.5) where (Ae j ,e i ) =Eg n−j+1 ·ϕ i , i, j =1, ,n.Wehave A=sup |α|=1 n i=1 n j=1 Eα j g n−j+1 ϕ i 2 = sup |α|=1 n i=1 E fϕ i 2 = sup |α|=1 E f 2 = 1, (4.6) where f = α j g j ,ifω ∈S j , j =1, ,n. Therefore, (4.5) implies D n ≤T n 2 .Thetheorem is proved. The following corollary is our Theorem 2.1. Corollary 4.2. D 2 = 4/3. Proof. We have according to Theorem 4.1, D 2 = T 2 2 = sup u T 2 u 2 = sup ab b 0 2 : a 2 + b 2 = 1 = 4 3 . (4.7) 6 On the constant in Me ´ nshov-Rademacher inequality Remark 4.3. It follows from the proof of Theorem 4.1 that D n = supE[max j ( j l =1 a l ϕ l ) 2 ], where the supremum is over all real orthonormal systems ϕ 1 , ,ϕ n ,whereeachϕ j , j = 1, ,n takes at most n values, and all reals α 1 , ,α n with |α|=1. The following lemma establishes a finer recurrence relation than Lemma 3.2.However, the two lemmas are asymptotically equivalent. Lemma 4.4. D 2n ≤ 4 3 D n if D n ≤ 3, D 2n ≤ D n − 1 2 + D n − 3 4 if D n ≥ 3. (4.8) Proof. We have for any n ∈ N: T 2n = sup AT n B T n C 0 , (4.9) where the supremum runs over all matrices A, B, C,andD in L( R n )suchthat( AB CD )≤ 1. For such matrices A, B, C,andD we check that |uA| 2 + |uT n B| 2 ≤T n 2 |u| 2 and |Ax| 2 + |T n Cx| 2 ≤T n 2 |x| 2 for all u,x ∈ R n . Therefore, T 2n ≤sup{(u,Ax)+(u,Fy)+ (v,Gy):u,v,x, y ∈ R n , |u| 2 + |v| 2 ≤ 1, |x| 2 + |y| 2 ≤ 1, A,F,G ∈L(R n ), A≤1, |wA| 2 + |wF| 2 ≤ D n |w| 2 , |Az| 2 + |Gz| 2 ≤ D n |z| 2 for all w,z ∈ R n . The last supremum can easily be computed and its square equals sup a∈[0,1] (D n −a/2+ D n a −3a 2 /4). Hence, D 2n ≤ 4/3D n if D n ≤ 3andD 2n ≤ D n −1/2+ D n −3/4ifD n ≥ 3. This completes the proof of Lemma 4.4. Finally, it is known that for the Hilbert matrix (H n (i, j)=1/(i−j), if i = j and H n (i,i)= 0, i, j =1, ,n, n ≥ 2), T n H n H n ≥ lnn π . (4.10) This along with Theorem 3.1 implies the following bilateral estimate: 1 π 2 log 2 2 e ≤ liminf D n log 2 2 n ≤ limsup D n log 2 2 n ≤ 1 4 . (4.11) Acknowledgments This work was supported in part by the US Civilian Research and Development Foun- dation Award GEMI-3328-TB-03. We want to express our gratitude to the anonymous referee for bringing to our attention the relationship between D n and the norm of the main triangle projection. Furthemore, the results/proofs in Section 4 are based on ideas, suggestions, and comments made by the referee. References [1] S. Chobanyan, Some remarks on the Men’shov-Rademacher functional, Matematicheskie Zametki 59 (1996), no. 5, 787–790, translation in Mathematical Notes 59 (1996), no. 5-6, 571–574. [2] J.L.Doob,Stochastic Processes, John Wiley & Sons, New York, 1953. Sergei Chobanyan et al. 7 [3] I.C.GohbergandM.G.Kre ˘ ın, Theory and Applications of Volterra Operators in Hilbert Space, Izdat. “Nauka”, Moscow, 1967, translated in Translations of Mathematical Monographs, vol. 24, American Mathematical Society, Province, RI, 1970. [4] E. G. Kounias, A note on Rademacher’s inequality, Acta Mathematica Academiae Scientiarum Hungaricae 21 (1970), no. 3-4, 447–448. [5] S. Kwapie ´ n and A. Pełczy ´ nski, The main triangle projection in mat rix spaces and its applications, Studia Mathematica 34 (1970), 43–68. [6] M. Lo ` eve, Probability Theory, 2nd ed., The University Series in Higher Mathematics, D. Van Nostrand, New Jersey, 1960. [7] D. Me ´ nshov, Sur les s ´ eries de fonctions orthogonales, I, Fundamenta Mathematicae 4 (1923), 82– 105. [8] F. M ´ oricz and K. Tandori, An improved Menshov-Rademacher theorem, Proceedings of the Amer- ican Mathematical Society 124 (1996), no. 3, 877–885. [9] H. Rademacher, Einige S ¨ atze ¨ uber Reihen von allgemeinen Orthogonalfunktionen, Mathematische Annalen 87 (1922), no. 1-2, 112–138. [10] ´ A. Somogyi, Maximal inequalities for not necessarily orthogonal random variables and some ap- plications, Analysis Mathematica 3 (1977), no. 2, 131–139. Sergei Chobanyan: Muskhelishvili Institute of Computational Mathematics, Georgian Academy of Sciences, 8 Akuri Street, Tbilisi 0193, Georgia E-mail address: chobanya@stt.msu.edu Shlomo Levental: Department of Statistics & Probability, Michigan State University, East Lansing, MI 48824, USA E-mail address: levental@stt.msu.edu Habib Salehi: Department of Statistics & Probability, Michigan State University, East Lansing, MI 48824, USA E-mail address: salehi@stt.msu.edu . max max k≤n k 1 α i ϕ i 2 , n 1 α i ϕ i +max n<k≤2n k n+1 α i ϕ i 2 ≤ max k≤n k 1 α i ϕ i 2 +2 n 1 α i ϕ i max n<k≤2n k n+1 α i ϕ i +max n<k≤2n k n+1 α i ϕ i 2 . (3.4) Taking expectations in (3.4) and using the Cauchy-Schwartz inequality, we come to the 4 On the constant in Me ´ nshov-Rademacher inequality desired recurrence relation: D 2n ≤ pD n +2 p(1. Tandori [8]. Section 2 dealswiththeproofofD 2 = 4/3, while Section 3 is devoted to the proof of the Me ´ nshov-Rademacher inequality with the asymptotic constant ≤ 1/4. Section 4 contains alternative proofs. unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction The Me ´ nshov-Rademacher inequality deals with the estimation of D n = supE