On a Problem of Littlewood

Transcription

1 Ž. JOURAL OF MATHEMATICAL AALYSIS AD APPLICATIOS 199, ARTICLE O O a Poblem of Littlewood Host Alze Mosbache Stasse 10, Waldbol, Gemay Submitted by J. L. Bee Received May 19, 1995 We pove: If 0 a1 a2 a ad A Ý i 1a i, the Ý 1a A Ým am 2 2Ý a 2 A 4. G. Beett had poved this esult, but with the facto 4 istead of Academic Pess, Ic. I 1967 J. E. Littlewood 3 peseted seveal emakable ope poblems coceig elemetay iequalities fo ifiite seies which seem to be vey fa fom simple 3, p Oe of these poblems states: Does thee exist a absolute costat K such that 2 2 Ý Ý Ý a a a 3 2 i m i 1 m 4 2 K Ý a Ý ai 1 i 1 holds fo all oegative eal umbes a 1, a 2,...? As it was poited out by Littlewood this iequality, as well as elated oes Žwhich ae give i. 3, have a close coectio to the theoy of othogoal seies. A aswe to Littlewood s questio was published i 1987 by G. Beett 1, 2, who showed that Ž. 1 is valid with K 4. Actually, Beett poved the followig much moe geeal esult. PROPOSITIO 1. Let p, q, 1; if a 1,...,a ae oegati e eal umbes with A Ý a, the i 1 i Ý aa Ý am m ž / p q Ý Ž. pž q. q 1 q Ž a A.. Ž 2. p X 96 $18.00 Copyight 1996 by Academic Pess, Ic. All ights of epoductio i ay fom eseved.

2 404 HORST ALZER The special case p 1, q 2 leads to Ý Ý m Ý m a A a 4 a A, Ž 3. which implies that iequality Ž. 1 holds with K 4. It is wothwhile to metio that Beett povided two diffeet poofs fo iequality Ž. 2. A aalysis of the poof give i 1 eveals that Ž. 2 is valid ot oly fo p, q, 1, but eve fo all eal umbes p, q, satisfyig p 1, q 0, 0, ad Žpq Ž. q p 1.. I 2 Beett discussed also the questio whethe thee exists a covese of Ž. 2. This meas: Let p, q, 1; does thee exist a absolute costat C such that the iequality p q 1 q Ý Ž. Ý Ý m m a A C a A a Ž 4. holds fo all oegative eal umbes a 1,..., a? Beett povided a example fo p 2, q 1 which shows that the aswe is o. Howeve, if the a ae deceasig, the the followig coutepat of Ž. i 2 is valid Žsee. 2. PROPOSITIO 2. Let p, q, 1; if a1 a2 a 0 with A Ž. Ž Ž.. 1 Ž 1 Ý a, the 4 holds with C Bq 1, H x q Ž 1 x. 1 dx. 1. i 1 i 0 It emais a ope poblem to detemie the best possible costats i Ž. 2 ad Ž. 4. Util ow the best possible costat is ot eve kow fo ay special case. Motivated by Popositio 2 we have tied to shape iequality Ž. 2 ude the additioal assumptio that the ai ae mootoic. Ideed, if the ai ae iceasig ad if Žpq Ž. q p 2,. the it is possible to eplace the costat ŽŽ pq Ž. q p.. by a smalle oe. As a special case of the followig theoem Žp 1, q k 2 i Ž 6.. we obtai that if 0 a 1 a a, the the facto 4 i Beett s iequality Ž. 2 3 ca be eplaced by 2. We ote that ou appoach has bee ispied vey much by the shot ad elegat poof of Ž. 2 give i 1. THEOREM. Let a 1,...,a be oegati e eal umbes such that a1 a2 a, Ž 5. ad let A Ý i 1 i a. If p 1, q 0, 0 ae eal umbes such that pž q. q d k, p

3 A PROBLEM OF LITTLEWOOD 405 whee k 1 is a itege, the Ý Ý am m a A k p q 1 q Ł Ý Ž. Ž d i. a A. Ž 6. Ž. If k 1, the assumptio 5 ca be dopped. Poof. It suffices to establish Ž. 6 fo p 1 ad d k. We defie ad Ý Ý m m L a A a, q p 1 p q b aa, c Ý a m, m 1 jp q BŽ j. Ý b i. i 1 Fist, we pove: If j is a oegative itege, the BŽ j. a 1 p q b 1 Žj 1.p q. Ž 7. Ž. We may suppose that a 0. The 7 ca be witte as 1 jp q jp q 1 j q p Ý i i 1 b a A. Ž 8. We use iductio o. If, the the sig of equality holds i Ž. 8. Usig the iductio hypothesis as well as a a ad A A,we obtai 1 jp q 1 jp q j q p jp q 1 j q p Ý b i a A a A i 1 jp q j q p 1 j q p a aa A Ž 9. jp q j q p j q p a aa AA a jp q A 1 j q p.

4 406 HORST ALZER We set pž q. q pž q. q, u, ; pž q. q qž p 1. p the we have 0 1, 1, ad 1 u 1 1. Applyig Holde s iequality we get pž1. p Ý 1 u pž1.u p Ý Ý L b b c ž / ž / b b c b pž1 q. Ý d Ý Ž. Ž. 4 q p 1 p q q Ž. 4 p p q q 1 bc. Ž 10. ext we detemie a uppe boud fo Ý bc d. Sice c c we coclude fom the mea-value theoem that the iequality c t c t tc t 1 Ž c c. tc t 1 a 1 p q Ž 11. is valid fo all eal t 1. Let 1 jp q d j Ž. Ý S j b c, whee j is a itege with 0 j d 1. Fom Ž 11. with t d j ad fom Ž. 7 we coclude 1 d j d j d j Ž. Ý Ž. Ž. 1 p q d Žj 1. Ž d j. Ý B Ž j. a c 1 Žj 1.p q d Žj 1. Ž d j. Ý b c S j B j c c B j c d j S j 1.

5 A PROBLEM OF LITTLEWOOD 407 Usig this iequality fo j 0, 1,..., k 1, we obtai We set the we have SŽ 0. dsž 1. dž d 1. SŽ 2. dž d 1. Ž d Ž k 1.. SŽ k.. Ž 12. x d Ž d k., y d k; 1 x d k pž1 q. y Ž. Ý S k b c b with 1 x 1 y 1 ad y 1. Fom Holde s iequality we coclude that 1 x Žd k.x pž1 q. SŽ k. Ý bc Ý b 1 y so that 12 ad 13 imply k d 1 k d pž1 q. S 0 Ý b, 13 d k d pž1 q. Ý Ł Ý bc S Ž 0. Ž d i. b. Ž 14. Fially, iequalities 10 ad 14 lead to which we had to show. Remaks. k p q 1 q Ł Ž. Ý Ž. L d i a A, Ž. 1 If d k 2, the we have Ł Ž. d i k d, so that iequal- ity Ž. 6 povides a efiemet of Ž. 2. Ž. 2 To pove iequality Ž. 6 we have used the mootoicity of the a i oly oce, amely, to establish iequality Ž. 9. If j 0, the Ž. 9 holds fo all oegative a, which implies that if k 1, the Ž. i 6 is valid without the additioal assumptio Ž. 5. This is Beett s Popositio 1. Ž. 3 We coclude by askig: Does iequality Ž. 6 emai valid if assumptio Ž. 5 does ot hold? Ad, is it possible to eplace the costat

6 408 HORST ALZER Ł Ž d i. k by a smalle umbe Ž which does ot deped o the a ad.? i REFERECES 1. G. Beett, A iequality suggested by Littlewood, Poc. Ame. Math. Soc. 100 Ž 1987., G. Beett, Some elemetay iequalities, Quat. J. Math. Oxfod Ž Ž 1987., J. E. Littlewood, Some ew iequalities ad usolved poblems, i Iequalities ŽO. Shisha, Ed.., pp , Academic Pess, ew Yok, 1967.