Concavity of weighted arithmetic means with applications

Transcription

1 Arch. Math. 69 (1997) 120± X/97/ $ 2.90/0 Birkhäuser Verlag, Basel, 1997 Archiv der Mathematik Cocavity of weighted arithmetic meas with applicatios By ARKADY BERENSTEIN ad ALEK VAINSHTEIN*) Abstract. We prove that the followig three coditios together imply the cocavity P of the sequece a i b i = P cocavity of fb g, log-cocavity of f g ad a i oicreasig of f b b 1 = 1 = 2 = 1 g. As a cosequece we get ecessary ad sufficiet coditios for the cocavity of the sequeces fs 1 x =S x g ad fs 0 x =S x g for ay oegative x, where S x is the th partial sum of a power series with arbitrary positive coefficiets f g. 1. Itroductio ad results. Let s x deote the th partial sum of the Taylor series for the expoetial fuctio, s x ˆX x k k! ad let s x ˆe x s x. I his paper [1] Alzer proves that s 1 x s 1 x 1 2 s2 x for ay atural ^ 1 ad ay x 0 (see also [2] for a substatial extesio of this result). I our previous paper [3] we have proved a similar iequality ivolvig sums s x for ay itegers ^ m l 0 ad ay x 0 l m l 1 l s l x s l m x < s x s m x < s l x s l m x I particular, oe has s 1 x s 1 x 1 s2 x for ay ^ 1 ad x 0. Iequality (1) was obtaied as a corollary of the followig result also proved i [3] the sequece fs 1 x =s x g is strictly cocave for ay x 0, that is, 2 s 2 x s 1 x s x s 1 x < 2 s 1 x s x ^ 1 x 0 (here ad i what follows, wheever egative subscript occurs, the correspodig term is assumed to be 0). Mathematics Subject Classificatio (1991) Primary 26D15 Secodary 26A51. *) The research of this author is supported by the Rashi Foudatio.

2 Vol. 69, 1997 Cocavity of weighted arithmetic meas 121 A atural geeralizatio of this result would be a characterizatio of power series P1 a k x k for which a aalog of (2) holds. Such a geeralizatio is provided by the followig theorem. Theorem 1. Let f g 1 ˆ0 be a positive sequece ad S x ˆ P a k x k. The the sequece fs 1 x =S x g 1 ˆ0 is strictly cocave for ay x 0 if ad oly if the sequece f 1= g 1 ˆ0 is cocave. There exists, however, aother atural geeralizatio of (2), which reflects the followig simple idetity s 0 x ˆ s 1 x for ay ^ 0 ad ay x. So, (2) ca be regarded also as a cocavity result for the sequece fs 0 x =s x g. Our secod geeralizatio of (2) is thus as follows. Theorem 2. Let f g 1 ˆ0 be a positive sequece ad S x ˆ P a k x k. The the sequece fs 0 x =S x g 1 ˆ0 is strictly cocave for ay x 0 if ad oly if the sequece f 1= g 1 ˆ0 is covex. Observe that the oly sequece fs x g that satisfies the coditios of both Theorems 1 ad 2 is the sequece fas bx g for arbitrary positive a ad b. Recetly Dilcher [6] geeralized the iequality of Alzer to the case of positive sequeces f g. He proved that if the power series P1 a k x k is coverget to f x i some iterval R R, S x ˆ f x P a k x k, ad the sequece f 1 = g is covex ad odecreasig, the S 1 x S 1 x ^ a 2 a 2 1 S 2 x x 2 0 R Similar but slightly differet results were later discovered by Che [5] ad Merkle ad Vasic [8]. We derive from Theorem 2 the followig aalog of iequality (1). Theorem 3. Let f g 1 ˆ0 be a positive sequece such that the sequece f 1= g 1 ˆ0 is covex. The for ay ^ m l 0 ad x 0 m l l m S l x S l m x < S x S m x < S l x S l m x 2. Mai theorem. A classical result due to Ozeki (see, e.g., [9, ] or [4, II.5]) says that if a sequece f g is covex (cocave) the the sequece of the arithmetic meas of f g is covex (cocave) as well. This result was geeralized to weighted arithmetic meas i [11]. Theorem 2.1 of [11] states that the ecessary ad sufficiet coditio o the weights fw g P that guaratee that the sequece of the weighted arithmetic meas a i w i = P w i is iˆ1 iˆ1 covex for ay choice of the give covex sequece f g is as follows Q 1 w 2 i 1 w 1 iˆ1 w ˆ 1!w 2 1 ^ 3 with arbitrary positive w 1, w 2. This result was substatially exteded i [7] to cover the case of the kth order covexity for both the give sequece ad the sequece of weighted arithmetic meas (see also [10]).

3 122 A. BERENSTEIN ad A. VAINSHTEIN ARCH. MATH. Below we give aother geeralizatio of Ozeki's theorem, which easily implies all the three results stated i the itroductio. Let f g 1 ˆ0 be a positive ad fb g 1 ˆ0 be oegative sequeces, p ˆ 1 =, ^ 0. Theorem 4. Let the sequece fb g 1 ˆ0 be cocave, the sequece fg 1 ˆ0 ad b b 1 p 1 p ^ b 1 b p p 1 ^ 1 be log-cocave, The the sequece m ˆ a0b 0 a 1 b 1 b 1 a 0 a 1 is cocave. Moreover, it is strictly cocave uless fb g is costat. Proof. Let us itroduce sequeces fa g 1 ˆ0, fc g 1 ˆ0, fd g 1 ˆ0, ad fg g 1 ˆ0 followig relatios A ˆ P a k C ˆ P a k b k ^ 0 d ˆ b 1 b g ˆ p 1 p ˆ0 by the The the first coditio of the theorem, the cocavity of the oegative sequece fb g, ca be writte as 3 d 1 ^ d ^ 0 the secod coditio, the log-cocavity of f g, as 4 g ^ 0 ad the third coditio as the iequality 5 d 1 g ^ d g 1 By the defiitio of m oe has m ˆ C =A, ad thus 6 m m 1 ˆ CA 1 C 1 A A A 1 ˆ A 1 C 1 b C 1 A 1 A A 1 ˆ b A 1 C 1 A A 1 Let us itroduce the additioal sequeces fg g 1 ˆ0 ad ff g 1 ˆ0 by G ˆ X 1 d k A k F ˆ X 1 g k A k The, by the defiitios of d ad g, oe has 7 ad G ˆ X 1 b k 1 b k A k ˆ b A 1 X 1 b k A k A k 1 ˆ b A 1 C 1 F ˆ X 1 p k 1 p k A k ˆ X 2 p k 1 A k A k 1 p A 1 ˆ p A 1 A 2

4 Vol. 69, 1997 Cocavity of weighted arithmetic meas 123 (sice p 0 is assumed to be 0). Addig the idetity p A A 1 ˆ A 1 A 2 to the secod of the above relatios we get 8 F ˆ p A A 1 I view of (6) ad (7), the strict cocavity of the sequece fm g reads as follows G G 1 1 A A 1 A A 1 By the defiitio of G oe has G 1 ˆ G d A, ad thus the above iequality is equivalet to G d A 1 9 A p 1 A 1 A 1 (observe that by (4) p 1 A 1 A 1 ˆ P kˆ1 p 1 p k a k p 1 a 0 0). We ow prove (9) by iductio. Suppose that it is valid for ˆ N 1, ad thus G N 1 d N p N 2 ad fails for ˆ N, that is, G N d N 1 11 % p N or, equivaletly, G N d N 12 % 1 p N We may assume that G N 0 (ideed, otherwise by the defiitio of G oe has d ˆ 0 for ˆ N 1, ad thus all m, ˆ N are equal). Therefore, from (12) we get d N 0, ad thus, by (3), d 0 for ˆ N. Together with (5) this yields g 0 for ˆ N (sice g 0 ˆ p 1 0). Hece (5) implies d =d N ^ g =g N for ˆ N, ad we thus obtai G N ˆ XN 1 d k A k ^ dn g N XN 1 g k A k ˆ dn g N F N Combiig the latter iequality with (11) we get g F N % N 1 13 p N Let us ow add d N 1 to both sides of (10) the, takig ito accout the defiitio of G, we get G N d N 1p N 14 1 p N 2 This iequality together with (12) yields d N d N 1p N p N p N 2

5 124 A. BERENSTEIN ad A. VAINSHTEIN ARCH. MATH. Applyig (3) we get 1 p N p N p N 2 Let us ow take ito accout (8) we thus ca rewrite the last iequality as F N a N 1 p N F N g N a N, or F N 1 p N p N g N If p N ^ 1 we are doe sice the right had side is evidetly oegative, ad thus (11) is false. Otherwise, this relatio together with (13) ad p N < 1 gives p N g N g < N 1 1 p N p N or, equivaletly, 15 1 p N p N 1 1 However, by (4) oe has p p 1 A 1 A 1 ˆ X p p 1 p 1 k p 2 k 2 k p p 1 a 0 a 1 0 kˆ1 which cotradicts (15), ad agai (11) is false. Therefore, it remais to verify (9) for ˆ 1. We have to prove that G 1 d 1A 0 A 1 p 2 A 2 A 0 which is equivalet to d 0 d 1 a 0 a 1 a 1 a 1 g 1 a 0 p 2 which is, i tur, equivalet to a 1 d 0 d 1 a 1 d 0 g 1 d 1 g 0 d 0 a 0 p 2 0 The latter iequality follows easily from (3) ± (5) with the oly exclusio whe d 0 ˆ d 1 ˆ 0, ad thus d 0, which meas that fb g is costat. h 3. Proofs of Theorems 1 ± 3. Proof o f Theorem 1. We have to prove the iequality 16 S 2 x S 1 x S x S 1 x < 2 S 1 x S x x 0 which is equivalet to X 1 X 1 a i 1 x i a i x i X 1 X 1 a i 1 x i a i x i < 2 X X a i 1 x i a i x i x 0

6 Vol. 69, 1997 Cocavity of weighted arithmetic meas 125 Applyig Theorem 4 to the sequeces f ˆ x g ad fb ˆ 1 = g we see that the followig coditios are sufficiet for the validity of the above iequality ^ a 1 1 ^ 0 1 x 1 ^ 0 x 0 x 1 a 2 a 1 1 x 1 x a ^ 1 1 a 1 x a 2 x 0 1 x Evidetly, the first coditio implies the other two, ad it is just the cocavity of the oegative sequece f 1 = g 1 ˆ0. O the other had, this coditio is also ecessary sice it follows from (16) as x goes to ifiity. The exceptioal case of Theorem 4 does ot apply sice b 0 ˆ 0 accordig to our agreemet, ad b 1 is strictly positive. h Proof o f Theorem 2. We have to prove the iequality S x S 1 x S0 1 x S 1 x < 2 S0 x S x x 0 which is equivalet to X 1 X 1 ia i x i a i x i X 1 X 1 ia i x i a i x i < 2 X X ia i x i a i x i x 0 Applyig Theorem 4 to the sequeces f ˆ x g ad fb ˆ g we see that the followig coditios are sufficiet for the validity of the above iequality 1 ^ 1 1 x 1 x ^ 0 x 0 1 x 1 x ^ a 1 x 2 1 x x 0 Observe that the iitial term of the sequece f 1 = g equals 0 sice it correspods to the case ˆ 0. Therefore, the third coditio implies the other two, ad it is just the covexity of the oegative sequece f 1 = g 1 ˆ0. O the other had, (17) implies x xs0 1 x 1 S 1 x x xs0 1 x 1 S 1 x ad thus the covexity of f 1 = g follows as x goes to ifiity. < 2x xs0 x S x h x 0 Proof o f Theorem 3. We proceed i the same way as i [3]. By Theorem 2, the sequece fs 0 x =S x g is strictly cocave for ay x 0. Thus, for ay ^ m l 0 ad ay x 0 oe has S 0 x S x S0 m x S m x < S0 l x S l x S0 l m x S l m x

7 126 A. BERENSTEIN ad A. VAINSHTEIN ARCH. MATH. or, equivaletly, d dx l S x S m x S l x S l m x < 0 Thus, the ratio i the above iequality is a strictly decreasig fuctio of x, ad therefore lim x! 1 S x S m x S l x S l m x < Sice S 0 ˆ a 0 for all ˆ ad we are doe. lim x! 1 h S x S m x S l x S l m x < S x S m x S l x S l m x ˆ a m l l m S 0 S m 0 S l 0 S l m 0 x 0 Refereces [1] H. ALZER, A iequality for the expoetial fuctio. Arch. Math. 55, 462 ± 464 (1990). [2] H. ALZER, J. BRENNER ad O. RUEHR, Iequalities for the tails of some elemetary series. J. Math. Aal. Appl. 179, 500 ± 506 (1993). [3] A. BERENSTEIN, A. VAINSHTEIN ad A. KREININ, A covexity property of the Poisso distributio ad its applicatio i queueig theory. I Stability Problems for Stochastic Models, Moscow 1986, pp. 17 ± 22 Eglish traslatio i J. Soviet Math. 47, 2288 ± 2292 (1989). [4] P. BULLEN, D. MITRINOVIC Â ad P. VASIC Â, eds., Meas ad Their Iequalities. Dordrecht [5] W. CHEN, Notes o a iequality for sectios of certai power series. Arch. Math. 62, 528 ± 530 (1994). [6] K. DILCHER, A iequality for sectios of certai power series. Arch. Math. 60, 339 ± 344 (1993). [7] I. LACKOVIC Â ad S. SIMIC Â, O weighted arithmetic meas which are ivariat with respect to k-th order covexity. Uiv. Beograd Publ. Elektroteh. Fak. Ser. Mat. Fiz. 461 ± 497, 159 ± 166 (1974). [8] M. MERKLE ad P. VASIC Â, A iequality for residual Maclauri expasio. Arch. Math. 66, 194 ± 196 (1996). [9] D. S. MITRINOVIC Â, Aalytic Iequalities. Berli-Heidelberg-New York [10] D. MITRINOVIC Â, I. LACKOVIC Â ad M. STANKOVIC Â, Addeda to the moograph ªAalytic Iequalitiesº, part II O some covex sequeces coected with Ozeki's results. Uiv. Beograd Publ. Elektroteh. Fak. Ser. Mat. Fiz. 634 ± 677, 3 ± 24 (1979). [11] P. VASIC Â, J. KEC Ï KIC Â, I. LACKOVIC Â ad ZÏ. MITROVIC Â, Some properties of arithmetic meas of real sequeces. Mat. Vesik 9, 205 ± 212 (1972). Eigegage am Aschrifte der Autore A. Berestei Departmet of Mathematics Northeaster Uiversity Bosto, MA USA Curret address Departmet of Mathematics Corell Uiversity Ithaca, NY USA A. Vaishtei Departmet of Mathematics ad Computer Sciece Uiversity of Haifa Mout Carmel Haifa Israel