Abel type inequalities, complex numbers and Gauss Pólya type integral inequalities

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1 Mthemticl Commuictios 31998, Abel tye iequlities, comlex umbers d Guss Póly tye itegrl iequlities S. S. Drgomir, C. E. M. Perce d J. Šude Abstrct. We obti iequlities of Abel tye but for odecresig sequeces rther th the usul oicresig sequeces. Strikig comlex logues re reseted. The iequlities o the rel domi re used to derive e itegrl iequlities relted to those of Guss Póly tye. Key ords: Abel s idetity, Abel s iequlity, Guss Póly iequlity Sžetk. Nejedkosti Abelovog ti, komleksi brojevi i itegrle ejedkosti Guss-Polyiog ti. Ključe riječi: Abelov idetitet, Abelov ejedkost, Guss-Polyi ejedkost AMS subject clssifictios: 26D15, 26D20, 26D99 Received December 24, 1997, Acceted Aril 22, Itroductio The folloig result is ell ko i the literture s Abel s iequlity see [1], Theorem 1. Let be rel tule d oegtive, oicresig tule. The for P k := k i, e hve 1 mi P k 1 k i i 1 mx P k. 1 k This erly result s subsequetly geerlized by Bromich see [1],. 337, ho derived the folloig theorem. Dertmet of Mthemtics, Uiversity of Trskei, Privte Bg X1, Uitr, Umtt 5100, South Afric Dertmet Dertmet of Mthemtics, Uiversity of Adelide, Adelide SA 5005, Austrli Defece Sciece d Techology Orgistio, Commuictio Divisio, PO Box 1500, Slisbury SA 5108, Austrli

2 96 S. S. Drgomir, C. E. M. Perce d J. Šude Theorem 2. For give rel tule d give iteger v 1 v, defie H 1 = h 1 = 0, H v = mxp 1,..., P v 1, h v = mip 1,..., P v 1, H v = mxp v,..., P, h v = mip v,..., P. If is ositive, oicresig tule, the h v 1 v + h v v i i H v 1 v + H v v. These iequlities coti i their roof the idetities 1 i i i = 1 i + k i 1 = i k i 1 here i := i+1 i due to Abel. This est of reltios is surrisigly fertile oe desite its simlicity. Recetly the Abel motif hs bee exloited to effect by Perce, Pečrić d Šude [2] i coectio ith the Chebyshev d Pooviciu iequlities. I this ote e rig the chges d tke s odecresig rther th oicresig tule. I Sectio 2 e reset Abel tye iequlities for this cse. I Sectio 3 e derive some logous results i the comlex domi. These re strikig i tht lthough there re costrits ivolved o the comlex tules z d, the reltios hold for y comlex tules htsoever. Filly, i Sectio 4, e use the results of Sectio 2 to derive itegrl iequlity. Recetly umber of results hve bee derived extedig the clssicl Guss Póly iequlities to yield vrious results coectig eighted mes of set of fuctios d their derivtives see [3], [5 7]. The methods re here logous to some used i tht coectio, but the iequlities foud re e d differet. 2. Iequlities for rel umbers We strt ith the folloig theorem. Theorem 3. Let = 1,..., d = 1,..., be -tule of rel umbers such tht 1... d k 0 for i = 2,...,. The i i 1 P + i i 1 P. 2 Proof. As is odecresig e hve tht i 1 = i i 1 = i i 1 i i 1 = i 1 0 for ll i = 2,..., d 0 k = k

3 Abel tye iequlities 97 for ll i = 2,...,. Thus, by the first equlity i 1, e hve i i 1 P = k i 1 = k i 1 k i 1 = k i 1 k i 1. Thus By Abel s idetity for := 1,...,, e hve lso tht i i 1 i = k i 1. d 2 is roved. i i 1 i i i 1 P 0 The secod result is embodied i the folloig theorem. Theorem 4. Let = 1,..., d = 1,..., be tules of rel umbers such tht 1... d i k 0 i = 1,..., 1. The P i i P i i 0. 3 Sice d Proof. By the secod idetity i 1 e c rite 1 1 i P i i = k i. i = i+1 i = i+1 i i+1 i = i i k 0 for i = 1,..., 1, e hve tht 1 i k i = = 1 i 1 i k i k i =1 1 i 1 k i i k i. =1

4 98 S. S. Drgomir, C. E. M. Perce d J. Šude By Abel s idetity for e hve lso 1 i i i = P k i, hece e derive 3. Remrk 1. The coditio k 0 i = 2,..., is equivlet to P P i 1 0 i = 2,..., or P P i for i = 1,..., 1 d the coditio i k 0, i = 1,..., 1 is equivlet to P i 0 i = 1,..., 1. The folloig corollry lso holds. Corollry 1. Suose is odecresig d IR ith P P i 0 for ll i = 1,..., 1. The P P i i i i 1 P + i i 1 P. Remrk 2. The bove iequlity is similr to Abel s result s it rovides uer d loer boud for i i he the sequece is odecresig d is such tht 0 P i P for ll i = 1,..., Iequlities for comlex umbers We o derive some similr results vlid for comlex umbers. Theorem 5. Suose z = z 1,..., z C d = 1,..., IR re such tht z i z i 1 i i 1 i = 2,...,. 4 The for ll = 1,..., C, e hve { i i 1 i mx i z i z 1 i, i z i z 1 i, } i z i z 1 i, i z i z 1 i. Proof. By Abel s idetity d 4 e hve tht i i 1 i = k i 1 k z i 1 =: A. By the roerties of the modulus mig, e hve further tht k k,

5 Abel tye iequlities 99 d so A k z i 1 k z i 1 = i z i z 1 Also, e c rite z i 1 z i 1 i. for i = 2,..., + 1. Thus A k z i 1 k z i 1 = i z i z 1 i. d I the sme y, e hve A = k z i 1 = k z i 1 k z i 1 = i z i z 1 i A = d e re doe. k z i 1 k z i 1 k z i 1 = i z i z 1 i, I the sme y, the secod rt of Abel s idetity leds to the folloig theorem. Theorem 6. Uder the coditios of Theorem 5., e hve { i i i mx z i i z i, z i z i i, } z i i z i, z i i z i. 4. Alictio to itegrl iequlities Theorem 7. Let f : [, b] IR be oegtive, icresig fuctio d x i : [, b] IR fuctios ith cotiuous first derivtive such tht 1 x 1 t x t, t [, b], 2 x 1t x t, t [, b].

6 100 S. S. Drgomir, C. E. M. Perce d J. Šude Suose lso i 0 d i = 1. The b i x i tftdt x 1tftdt + i x it x 1 t dft fb i [x i b x 1 b] f i [x i x 1 ]. i Proof. By itegrtio by rts, x itftdt = fb i x i b f i x i i x i t dft. 6 We c ly 2 to obti i x itftdt x 1tftdt + d i x itftdt i x i t x 1 t + i x i t x 1 t 5 x itftdt 7 for ll t [, b]. Itegrtig this lst iequlity, e deduce tht i x i t dft x 1 tdft + i x i t x 1 t dft = ftx 1 t b b x 1tftdt b + i x i t dft x 1 t dft. Usig 6 8, e derive x b 1ftdt + i x i ftdt x 1ftdt fb i x i b f i x i fbx 1 b fx 1 + hich is equivlet to 5. x 1ftdt b i x i t b dft x 1 t dft, Remrk 3. Similr results c be obtied from the secod Abel tye iequlity 3. 8

7 Abel tye iequlities 101 Refereces [1] D. S. Mitriović, J. E. Pečrić, A. M. Fik, Clssicl d e iequlities i lysis, Kluer Acd. Publishers, Dordrecht, [2] C. E. M. Perce, J. Pečrić, J. Šude, O refiemet of the Chebyshev d Pooviciu iequlities, Mth. Commu , [3] J. Pečrić, S. Vrošec, A geerliztio of Póly s iequlities, Iequlities d lictios, World Sci. Ser. Al. Al , [4] G. Póly, G. Szegö, Aufgbe ud Lehrsätze us der Alysis I, II, Sriger Verlg, Berli, [5] S. Vrošec, Iequlities of Guss tye i Croti, Doct. Diss., Uiv. of Zgreb, [6] S. Vrošec, J. Pečrić, J. Šude, Some discrete iequlities, Zeit für Al. A , [7] S. Vrošec, J. Pečrić, J. Šude, O Guss Póly s iequlity, submitted to Zeit für Al. A.