A Further Refinement of Van Der Corput s Inequality

Transcription

1 IOSR Joural of Mathematics (IOSR-JM) e-issn: , p-issn:9-75x Volume 0, Issue Ver V (Mar-Apr 04), PP 7- wwwiosrjouralsorg A Further Refiemet of Va Der Corput s Iequality Amusa I S Mogbademu A A Baiyeri J Fuso ad Mohammed M A 4 Eaitie ET 5 Departmet Of Mathematics, Uiversity Of Lagos &, Departmet Of Mathematics, Yaba College Of Techology & 4 Abstract: I this paper, we obtai a further refiemet of Va der Corput s iequality usig a aalytical techique Key words ad phrases: Refiemet, Va der Corput s iequality, Harmoic umber I Itroductio Let S be the harmoic umber ad 0 The Va der Corput s iequality states that S / 0 a a for such that, a e a where deotes the Euler-Mascheroi s costat The costat I 00, Hu [5], gave the followig versio of (): S / l a e a 4 This iequality is a refiemet of () I 005, YANG [] obtaied a better result tha Hu s iequality () as S / e is the best possible l a e a This iequality is also a refiemet of () He further exteded the origial Va der Corput s iequality () as follows l Let a 0 for such that 0 a ad T The T a e a, 4 where lim l for, ad T0 S Settig 0 i (4), it becomes S / a e a 5 Clearly, iequality (5) improves iequality () I 00, Cao et al [] established aother versio of () as follows wwwiosrjouralsorg 7 Page

2 Let a 0 for N such that 0 a The U / / l 4 A Further Refiemet of Va Der Corput s Iequality a e a, where 0,, U ad lim U l Cosequetly, they established a sharper iequality that further refies (), (), () ad (5), which is give by s / l a e a 7 / 4 The aim of this paper is to further refie iequality (7) to obtai sharper iequality tha that of (7) Our mai results are the followig II Mai Results Theorem : Let a 0 ad S such that for ad l 0 a l / The s / l a e a, 9 l / () where is the Euler-Mascheroi s costat ad e is the best possible Remar Let l l U e ad V e 9 l / / 4 For, the umerical computatios of U ad U V V gives the followig table of values: U V, for Clearly Also, we cosider l / ad / 4 l l For 4: l 4 l 4 Clearly, iequality ( ) is a improvemet ad the refiemet of the (7) Theorem : Let a 0 ad S such that for ad l 0 a l / The wwwiosrjouralsorg 8 Page

3 A Further Refiemet of Va Der Corput s Iequality s / l a e a, l / () where is the Euler-Mascheroi s costat ad Remar 4: Let e is the best possible l l T e ad V e l / / 4 For 4: Numerical computatios of T adv give the followig table of values T V Clearly, iequality ( ) is a refiemet of (7) sice T V I order to prove our mai results, we cosider the followig lemmas Lemma 5 [] For, S l, where the costat ad are the best possible Lemma [] If x 0, Lemma 7 x the e x x S For N, let A, the S l A e 9 l / Proof of Lemma 7: We have that () S S S A S S S S S S S S S S S Suppose we set B S S S The applyig Lemma i (0), we get S S S S S B e e e S S wwwiosrjouralsorg 9 Page

4 A Further Refiemet of Va Der Corput s Iequality usig Lemma 5, i view of (), we obtai S S S A e e 4 ad S l S l Hece, Equatio 4 yields l l A e 5 To proceed from here, we cosider the followig iequalities: m If m, the e m m If m, the e 7 m m Observe from that e m l l l Thus, e 8 Usig 7 i 8, we have l l e l / l l l l l Therefore, 9 e l Similarly, 9 e e 0 9 Ad, from 7, Settig m, iequality 0 becomes e 9 Combiig iequalities (5), (9) ad (), we obtai wwwiosrjouralsorg 0 Page

5 A Further Refiemet of Va Der Corput s Iequality A e l l l l e e e l Thus, A e 9 l Hece Lemma is proved Iequality () ca further be writte as: l l A e e l l I III Proof Of Theorems Ad We ow give proofs II of our mai results Proof of Theorem : Let c S S S S / c S, where S o is assumed zero ie S o = 0 The s s S S / / / / a a c c s S / a c c By usig arithmetic mea geometric mea iequality ad iterchagig the order of the iequality (), we have ac s s S S / / / a ac c c s / S / ac ac a c S S S Lettig S S S s / ac S a S S Applyig iequality () i (), we get S a S wwwiosrjouralsorg Page

6 s / a S S Replacig by S a A Further Refiemet of Va Der Corput s Iequality l e a 9 l i the right had side of (), this becomes s / l 9 l a e a This completes the proof of theorem Proof of Theorem : Substitutig () ito (), we obtai s / l l a e a 4 This proves Theorem 4 () Refereces [] B Ch Yag, (005), O a extesio ad a refiemet of Va der Corput s iquality, Chiese Quart J Math,, 5 pages [] Da-Wei Niu, J Cao ad Feg Qi, A refiemet of va der Corput's iequality, J Iequal Pure Appl Math 7 (00), o 4, Article 7; Available olie at [] J Cao, Da-Wei Niu ad Feg Qi, A extesio ad a refiemet of va der Corput's iequality, Iterat J Math Math Sci 00 (00), Article ID 7078, 0 pages; Available olie at [4] F Qi, J Cao, ad D-W Niu, A geeralizatio of va der Corput's iequality, RGMIA Res Rep Coll 0 (007), Suppl, Article 5; Available olie at [5] K Hu, O Va der Corput s iequality, Shuxue Zazhi, (J Maths (Wuha)), (00) No, -8 (Chiese) wwwiosrjouralsorg Page