ANOTHER EXTENSION OF VAN DER CORPUT S INEQUALITY. Gabriel STAN 1

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1 Bulleti of the Trasilvaia Uiversity of Braşov Vol 5) - 00 Series III: Mathematics, Iformatics, Physics, -4 ANOTHER EXTENSION OF VAN DER CORPUT S INEQUALITY Gabriel STAN Abstract A extesio ad a refiemet of Va der Corput s iequality was give by Jia Cao, Da-Wei Niu ad Feg Qi. This iequality is furter geeralised i this paper. 000 Mathematics Subject Classificatio: 6D5, 6D0 Key words: Va der Corput s iequality, umeric series, Euler-Maclauri formula, completely mootoic fuctios. Itroductio Let S ad a 0 for N such that 0 < a <. The well ow iequality by Corput 0 states that: Π a/ ) /S < e γ ) a,.) where γ 0, stads for Euler-Mascheroi costat. The costat e γ i.) is the best possible. A improved versio of.) is give by Hu 5 Π a/ ) /S < e γ l ) a,.) 4 A relatio betwee Carlema s iequality ad Va der Corput s iequality is metioed by Yag 4. Π a α where S α) ad α 0, α ) Sα) < e e αα S α) a,.) Aother extesio has bee obtaied by Yag 5 Faculty of Mathematics ad Iformatics, Trasilvaia Uiversity of Braşov, Romaia, gabis75@yahoo.com

2 4 Gabriel Sta Π a α where α, ), S α) ) Sα) α ad γ α) lim Applyig α 0 i.4) leads to Π a < e γ α) l α) α ) S < e γ α ) a.4).5) ) a.6) which is a improvemet of iequality.). The purpose of this paper is to further exted ad refie Va der Corput s iequality which has bee exteded i 6 by Jia Cao, Da Wei Niu ad Feg Qi here give with a small correctio of a misprit) Π a λ) where λ 0, ), ) Sλ) < e λ )γλ) S λ) ) λ l ) 4 λ ) a.7),.8) λ) γ λ) lim S λ) l The Euler Maclauri formula, 6,, 4 states λ λ.9) f ) f x) dx f ) f ) ρ x) f x) dx.0) where ρ x) x x is Beroulli s fuctio ad f C, ). Furthermore, if ) i f i) x) > 0 ad lim f i) x) 0 for i,, the x ρ x) f x) dx f ) ɛ, 0 < ɛ <.)

3 Aother extesio of Va der Corput s iequality 5. Lemmas Some lemmas are eeded. A fuctio f is said to be completely mootoic o a iterval I if f has derivatives of all orders o I ad 0 < ) f ) x) <, ) N, x I 4, 7, 8, 9 Lemma. Fuctio f x) for α, β 0, ) is completely mootoic i xα)xβ) 0, ) ad lim x f i) x) 0 for ) i N. Proof. Fuctios xα ad xβ are completely mootoic o 0, ), a fact very easy to verify. Sice the product of ay completely mootoic fuctios is also completely mootoic, the fuctio f x) is mootoic o 0, ). It ca be proved by iductio that lim f i) x) 0, ) i N. The proof of Lemma has eded. x Lemma. For N ad α, β 0, ) S α, β) < l ) γ α, β).) where S α, β),.) α) β) α β γ α, β) lim S α, β) l x α β.) Proof. It is obvious that Lemma permits to apply the Euler-Maclauri formulas.0 ad. to f x) xα)xβ). Hece it follows that: S α, β) x α) x β) dx α) β) α β l α β α) β) α) β) α) β) ρ x) dx. x α) x β) ρ x) f x) dx

4 6 Gabriel Sta Also, we have ρ x) dx x α) x β) ɛ x α) x β) x where ɛ 0, ), ad γ α, β) lim x Therefore, { α) β) ɛ 4 α) β) ρ x) dx x α) x β) α) β) α β α) β),.4) ρ x) dx.5) x α) x β) S α, β) l α) β) γ α, β) α) β) α) β) l ɛ 4 ρ x) dx x α) x β) α β α β γ α, β) α β α) β) α) β). Hece S α, β) < l γ α, β) α) β).6) Iequality: l α β α β l is used. The S α, β) S α, β) γ α, β) α ) β ) < l ) α ) β ) α ) β ).

5 Aother extesio of Va der Corput s iequality 7 It follows This proof of Lemma is complete. S α, β) < l ) γ α, β)..7) Lemma. For N ad α, β 0, ) α) β) α ) β ) α β α ) β ) α) β) Proof. Iequality.8) is equivalet to.8).9) α β ) α) β) α β ) α ) β ) ad the this is equivalet with the obvious iequality α β) α β) α β ) 4 Iequality.9) ca be deduced directly from 0. α ) β ) α) β) α β α ) β ) α) β) α β The proof of Lemma is complete. α β Lemma 4. For x 0, ) ad α, β 0, ) x lx) l x ) < ).0) 4 x Proof. If we deote λ α β Lemma 4 becomes Lemma.4 from 7. Lemma 5. For N ad α, β 0, ), β α, β) α)β)s α,β) α ) β )S α, β) α) β)s α, β) e )γα,β) l ) ) 4 )

6 8 Gabriel Sta Proof. For N we have β α, β) where { } α)β)s α,β) h, α, β) α) β)s α, β) g,α,β) h,α,β) C h,α,β) g, α, β) α) β) S α, β) g, α, β), h, α, β) h, α, β) α ) β ) α) β) S α, β), g,α,β) C g, α, β) It is easy to see that g, α, β) α ) β )S α, β) α ) β ) α) β) S α, β) α ) β ) α ) β ) α) β).5) By usig iequality ) x x < e x) obtaied i, iequlities.5) ad.8) i Lemma it follows that g,α,β) { } C e g, α, β) g, α, β) α ) β ) α) β) e α ) β ) α) β) e e.6) α ) β ) Hece from iequalities.9),.),.6) ad.0) it follows

7 Aother extesio of Va der Corput s iequality 9 h, α, β) α β ) S α, β) α β β α, β) e ) l ) γ α, β) h,α,β) e e )γα,β) ) )l)γα,β) e )γα,β) ) l ) 4 l) The proof of Lemma.5 is complete.. A ew theorem emerges. Theorem. Let a 0 for N such that 0 a < The α)β) Π a where α, β 0, ). Sα,β) < e )γα,β) ) Proof. Fixig c > 0 for ad defiig l ) a 4,.) α ) β )S α, β) α)β)s α,β) c α) β)s α, β) α)β)s α,β)

8 40 Gabriel Sta the α)β) Π c Sα,β) α ) β )S α, β). for Usig the Jese iequality ) u l b l u b u ad Lemmas it follows α)β) Π a Π a c ) ) Sα,β) Sα,β) α)β) α)β) Π c Sα,β) α) β)s α, β) c a α ) β )S α, β) c a α) β) α ) β )S α, β) S α, β) c a α) β) S α, β) S α, β) c a α) β) S α, β) α)β)s α,β) α ) β )S α, β) a. α) β)s α, β) From the above iequality ad iequality.4) we obtai iequality.). Refereces Adrews, G. E., Asey, R. ad Roy, R., Special Fuctios, Ecyclopedia of Mathematics ad Its Applicatios, vol.7, Cambrige Uiversity Press, Cambrige, 999. Aurora, B., Publicatio list: J.G. va der Corput, Acta Arithmetica 6 980), o., Che, Ch. -P. ad Qi, F., O further sharpeig of Carlema s iequality, Daxue Shuxue College Mathematics) 005), o., Chiese).

9 Aother extesio of Va der Corput s iequality 4 4 Grishpa, A. Z. ad Ismail, M. E. H., Completely mootoic fuctios ivolvig the gamma ad q-gamma fuctios, Procedig of the America Mathematical Society 4 006), o.4, Hu, K., O the va der Corput iequality, Joural of Mathematics Shuxue Zazhi) 00), o., 6-8 Chiese). 6 Kuag, J. -Ch., Asymptotic estimatios of fiite sums, Joural of Hexi Uiversity 00), o., -8 Chiese). 7 Qi, F., Certai logarithmically N-alteratig mootoic fuctios ivolvig gamma ad q-gamma fuctios, RGMIA Research Report Collectio 8 005), o.. article 5, available olie at 8 Qi, F. ad Guo, B. -N., Complete mootoicities of fuctios ivolvig the gamma ad digamma fuctios, RGMIA Research Report Collectio 7 004), o., article 8,6-7, available olie at 9 Qi, F. ad Guo, B. -N. ad Che, C.-P., Some Completely mootoic fuctios ivolvig the gamma ad polygamma fuctios, Joural of the Australia Mathematical Society ), o.,8-88, RGMIA Research Report Collectio 7 004), o., article 5, -6, available olie at 0 Va der Corput, J.G., Geeralizatio of Carlema s iequality, Proceedigs of the Sectio of Scieces, Koilije Aademie va Weteschappe te Amsterdam 9 96), Va Haerige, H., Completely mootoic ad related fuctios, Report 9-08, Faculty of Techical Mathematics ad Iformatics, Delft Uiversity of Techology, Delft, 99. Yag, B. -Ch., O a stregtheed versio of the more precise Hardy-Hilbert iequality, Acta Mathematica Siica 4 999),o Chiese)., O Hardy s iequality, Joural of Mathematical Aalysis ad Applicatios 4 999), o., , O a relatio betwee Carlema s iequality ad Va der Corput s iequality, Taiwaese Joural of Mathematics 9 005), o., , O a extesio ad a refiemet of va der Corput s iequality, to appear i Chiese Qarterly Joural of Mathematics. 6 Cao, J., Niu, D-W. ad Qi, F., A extesio ad a refiemet of Va der Corput s iequality, It. J. Math.Math. Sci., 006), Art. ID 70786, 0pp. 7 Qi, F., Cao, J. ad Niu,D.-W., A geeralizatio of Va der Corput s Iequality, Appl., Math. Comput. 0, 008), o.,

10 4 Gabriel Sta

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