Leindler, László

English

Leindler László

University of Szeged

Acta Sei. Math., 54 (1990), 285—289 Further sharpening of inequalities of Hardy and Littlewood L. LEINDLER In a previous paper [2] we generalized some classical and very useful inequalities of HARDY and LITTLEWOOD [1]. One special case of our results states the following inequalities: If A„>0 and fl„s0 (»=1, 2, ...) then we have (1) (2) and (3) (4) fc=l n=l k=n 2 K ( 2" aky S p" 2 *S ( 2" 4)" k=n /i=l k=1 n=l k=n n—1 t=l n = l fc=l 71 = 1 k = n /or /or 0 < p â 1. These inequalities reduce to those of Hardy and Littlewood if X„=n~c with c > 1 in (1) and (3); and with c ^ l in (2) and (4). The factor p" is best possible one, but 8 and 9 are not. In the present note we improve, among others, inequalities (3) and (4) proving that the constants 8 and 9 can be replaced by p~p and this is the best possible one. It is easy to see that p~p^ Se 1 / e<l ,45 holds for any Having these improved inequalities (3) and (4), we can state that inequalities (1) and (2) hold for 1 and their reversed ones hold for 0 < p s l . In [2] it was assumed only that /l„=0 and in this more general form we proved the following theorem. Received June 10, 1988. 286 L. Leindler Theorem A. Let a„sO and A„sO (n= 1, 2, . . . ) be given. Let v 1 <. . .<v„<. . . denote the indices for which A.v >0. Le/ JV denote the number of the positive terms of the sequence A„, provided this number is finite; in the contrary case set Set v0=0, and if °° then vN+1=°°. Using the notations ¿m,n : = 2 a, and Am,„ := 2 (1 ^ m S w S»), j=m i=m we have the following inequalities: (10 n = 1 n=l N n=l n=l (the constant pp being the best possible one) and for p S 1 (3') (4') 2 ^ 8 2 KAin n=1 n=l for 0 < p ^ 1. 2 ¿ i n - p A p , v „ < , v „ + 1 - 3 ^ 9 2 K A U 11 = 1 fl — 1 The aim of this note is to prove Theorem. Under the assumptions of Theorem A the opposite inequalities of (1') and (2') holdfor 0</>= 1, and the constant pp is best possible one, in this case, too. Theorem A and Theorem imply immediately Coro l l a ry 1. If A„>0 and anm 0 then (1) and (2) hold for p^l, and their opposite inequalities for 0</>Sl. If 0</>^ l then we can reduce the restriction A„>0 of Corollary 1 to A„§0, i.e. we can prove Coro l l a ry 2. For any «„SO and A„s0, ¡ / 0 < / ) S l , we have (3") and (4") 2 2 4) pa£ s p - ' 2 4 ( 2 «*)" n=l k=n n=l fe=l 2 4 - " ( 2 4)"a> ^ p-> 2 4 ( 2 aky. Inequalities of Hardy and Littlewood 287 This is an immediate consequence of the facts that ' (5) 2 2 ti-'Af.-af and (6) AJ; ' A l V n ( " 2 1 = X"2 1 i = VM ' = V„ obviously hold — since A ~ 0 if — so by the opposite inequalities of (1') and (2') to be proved in Theorem, regarding inequalities (5) and (6), we get (3") and (4"). Proofs 1. First we prove the opposite of (1') for i.e. ( 1 . 1 ) Zt : = 2 KAU 2 %:pA!w-A>n_1+Uyit =:It. n=1 11=1 We may assume that Zx has a positive finite value. For p= 1 (1.1) is obvious, we have only to interchange the order of the summations. If 0< j ?<1 then we set the following notations: n -a„'.= AVn_1+ltV , PO = 0, P„'= 2 QN'-= k = 1 and R„:=AVn^ for every l ^ n ^ N . If then let •RN+I : = GN+I : : = 0. Then, for any positive integer m ( ^ N ) , we have vm m m m 2 Mi,* = 2 QnPpn = 2(A-K+i)K = 2K(Ppn-Ppn-i)-Rm+1Ppm, fc=i «=1 n=i «=1 whence m vm co (1.2) 2Rn(Ppn-№-1)S Rm+1ppm+ 2hAU s> 2 = ^ 11=1 k=1 fc = 1 follows for any m ^ N . Let fi be the smallest positive integer having the property /?„>0. 288 L. Leindler An easy calculation gives that (1-3) 2 RnW-PS-i) ^ 2 Rnp*nf%-1 = flssjj n=s/l - P 2(QnUp)~1RnVn)(eln~il,p}PZ'1) -PI3-n=H p Now using the classical inequality of Holder with p and we get p- 1 ( 1 . 4 ) I 3 S { 2 E L - P R № ) L L P { 2 E J S Y ~ L L P -n = fx n = fl By (1.2), (1.3) and (1.4) we have 2 h au s P ( 2 Ql-pRpn<yp{2 QnPS)1-1'" = k=1 n=1 n=l = P(2 ¿ L - A ^ J V ( f ;.v„ ( 2 ak)py-vp, n = 1 n=l fc =1 whence (1.1) obviously follows. 2. Secondly we prove the opposite of (2') for i.e. (1.5) Z4.:= 2 S / 2 =•• n=1 n=l As before thé case p= 1 is trivial and we may assume that I4 has a positive N finite value. Using the previous notations and let a*:=Av v yn:= 2 «Î and n + I k = n n B„:= A, v = 2 Qkl furthermore let a> denote the greatest natural number, if there ' B ii=i exists, for which Av „ > 0 , otherwise let co:—°°. If i2:=min (N, a>), then for any l^n^k^Q yn = Av 0 and we have the estimation fc+i it+i n=l n = l = 2Bn(ypn-yUÙ+Bk^yk+1 ^ 2Bn(;fn--fn+1). n=1 n=l By (1.6) it is clear that for any k^Q (1.7) 2 BM-yUI) ^ 2 BnPiïyr1 = n=l n= l = p 2(Qllp-1Bn<)(e1n-Vpyp-1) B = 1 Inequalities of Hardy and Littlewood 289 k k ¿4 ^ p ( 2 e i - ' B s W ( 2 e-tf)1-1" = n=1 n=l whence (1.5) follows immediately. 3. To verify that the constant pp in the opposite inequalities of (1') and (2') for 1 is best possible it is enough to consider e.g. the following special case: a„ := (log(w+ l))_ 2 / p and A„ = n'1-" for and everything is zero for n < N . The proof of Theorem is completed. [1] G. H. HARDY—J. E. LITTLEWOOD, Elementary theorems concerning power series with positive coefficients and moment constants of positive functions, J. reine angew. Math., 157 (1927), [2] L. LEINDLER, Generalization of inequalities of Hardy and Littlewood, Acta Sci. Math., 31 (1970), 279—285. References 141—158. BOLYAI INSTITUTE ARADI VÉRTANÚK TERE 1 6720 SZEGED, HUNGARY 

Further sharpening of inequalities of Hardy and Littlewood

http://acta.bibl.u-szeged.hu/15280/1/math_054_fasc_003_004_285-289.pdf

Further sharpening of inequalities of Hardy and Littlewood

Abstract

Similar works

Full text

Available Versions

University of Szeged