Concavity of weighted arithmetic means with applications - Springer Link

Arch. Math. 69 (1997) 120 ± 126 0003-889X/97/020120-07 $ 2.90/0  Birkhäuser Verlag, Basel, 1997

Archiv der Mathematik

Concavity of weighted arithmetic means with applications By ARKADY BERENSTEIN and ALEK VAINSHTEIN *)

Abstract. We prove that the following three conditions together imply the concavity n  n P P ai bi = ai : concavity of fbn g, log-concavity of fan g and of the sequence iˆ0

iˆ0

nonincreasing of f…bn ÿ bnÿ1 †=…anÿ1 =an ÿ anÿ2 =anÿ1 †g. As a consequence we get necessary and sufficient conditions for the concavity of the sequences fSnÿ1 …x†=Sn …x†g and fS0n …x†=Sn …x†g for any nonnegative x, where Sn …x† is the nth partial sum of a power series with arbitrary positive coefficients fan g.

1. Introduction and results. Let sn …x† denote the nth partial sum of the Taylor series for the exponential function, sn …x† ˆ

n X xk kˆ0

k!

;

n‡1 2 s …x† for n‡2 n any natural n ^ 1 and any x > 0 (see also [2] for a substantial extension of this result). In our previous paper [3] we have proved a similar inequality involving sums sn …x†: for any integers n ^ m > l > 0 and any x > 0

and let sn …x† ˆ ex ÿ sn …x†. In his paper [1] Alzer proves that  snÿ1 …x† sn‡1 …x† >

…n‡lÿm l † snÿl …x†sn‡lÿm …x† < sn …x†snÿm …x† < snÿl …x†sn‡lÿm …x† : …nl† n 2 In particular, one has snÿ1 …x†sn‡1 …x† > s …x† for any n ^ 1 and x > 0 . n‡1 n Inequality (1) was obtained as a corollary of the following result also proved in [3]: the sequence fsnÿ1 …x†=sn …x†g is strictly concave for any x > 0, that is,

…1†

…2†

snÿ2 …x† sn …x† snÿ1 …x† ‡ 0 ;

(here and in what follows, whenever a negative subscript occurs, the corresponding term is assumed to be 0).

Mathematics Subject Classification (1991): Primary 26D15; Secondary 26A51. *) The research of this author is supported by the Rashi Foundation.

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A natural generalization of this result would be a characterization of power series

1 P kˆ0

ak xk

for which an analog of (2) holds. Such a generalization is provided by the following theorem. 1 Theorem 1. Let fan gnˆ0 be a positive sequence and Sn …x† ˆ

n P kˆ0

ak xk . Then the sequence

1 1 fSnÿ1 …x†=Sn …x†gnˆ0 is strictly concave for any x > 0 if and only if the sequence fanÿ1 =an gnˆ0 is concave.

There exists, however, another natural generalization of (2), which reflects the following simple identity: s0n …x† ˆ snÿ1 …x† for any n ^ 0 and any x. So, (2) can be regarded also as a concavity result for the sequence fs0n …x†=sn …x†g. Our second generalization of (2) is thus as follows. n P 1 Theorem 2. Let fan gnˆ0 be a positive sequence and Sn …x† ˆ ak xk . Then the sequence kˆ0

1 1 fS0n …x†=Sn …x†gnˆ0 is strictly concave for any x > 0 if and only if the sequence fanÿ1 =an gnˆ0 is convex.

Observe that the only sequence fSn …x†g that satisfies the conditions of both Theorems 1 and 2 is the sequence fasn …bx†g for arbitrary positive a and b. Recently Dilcher [6] generalized the inequality of Alzer to the case of positive sequences 1 P ak xk is convergent to f …x† in some interval fan g. He proved that if the power series kˆ0 n P …ÿR; R†, Sn …x† ˆ f …x†ÿ ak xk , and the sequence fanÿ1 =an g is convex and nondecreasing, then kˆ0

an an‡2 2 S …x†; Snÿ1 …x†Sn‡1 …x† ^ 2 an‡1 n

x 2 …0; R†:

Similar but slightly different results were later discovered by Chen [5] and Merkle and Vasic [8]. We derive from Theorem 2 the following analog of inequality (1). 1 1 Theorem 3. Let fan gnˆ0 be a positive sequence such that the sequence fanÿ1 =an gnˆ0 is convex. Then for any n ^ m > l > 0 and x > 0 an anÿm Snÿl …x†Sn‡lÿm …x† < Sn …x†Snÿm …x† < Snÿl …x†Sn‡lÿm …x†: anÿl an‡lÿm

2. Main theorem. A classical result due to Ozeki (see, e.g., [9, 3.2.21] or [4, II.5]) says that if a sequence fan g is convex (concave) then the sequence of the arithmetic means of fan g is convex (concave) as well. This result was generalized to weighted arithmetic means in [11]. Theorem 2.1 of [11] states that the necessary and sufficient condition on the weights fwn g n  n P P that guarantee that the sequence of the weighted arithmetic means ai wi = wi is iˆ1 convex for any choice of the given convex sequence fan g is as follows: iˆ1 nÿ1 Q

wn ˆ

iˆ1

…w2 ‡ …i ÿ 1†w1 † …n ÿ 1†!wnÿ2 1

;

n ^ 3;

with arbitrary positive w1 , w2 . This result was substantially extended in [7] to cover the case of the kth order convexity for both the given sequence and the sequence of weighted arithmetic means (see also [10]).

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Below we give another generalization of Ozeki's theorem, which easily implies all the three results stated in the introduction. 1 1 be a positive and fbn gnˆ0 be a nonnegative sequences, pn ˆ anÿ1 =an , n ^ 0. Let fan gnˆ0 1 1 Theorem 4. Let the sequence fbn gnˆ0 be concave, the sequence fan gnˆ0 be log-concave, and …bn ÿ bnÿ1 †…pn‡1 ÿ pn † ^ …bn‡1 ÿ bn †…pn ÿ pnÿ1 †; n ^ 1:

Then the sequence   a0 b0 ‡ a1 b1 ‡


‡ an bn 1 mn ˆ a0 ‡ a1 ‡


‡ an nˆ0 is concave. Moreover, it is strictly concave unless fbn g is constant. 1 1 1 1 P r o o f . Let us introduce sequences fAn gnˆ0 , fCn gnˆ0 , fdn gnˆ0 , and fgn gnˆ0 by the following relations:

An ˆ

n P kˆ0

ak ;

dn ˆ bn‡1 ÿ bn ;

Cn ˆ

n P kˆ0

ak bk ;

n ^ 0:

gn ˆ pn‡1 ÿ pn ;

Then the first condition of the theorem, the concavity of the nonnegative sequence fbn g, can be written as …3†

dnÿ1 ^ dn ^ 0;

the second condition, the log-concavity of fan g, as …4†

gn ^ 0;

and the third condition as the inequality …5†

dnÿ1 gn ^ dn gnÿ1 :

By the definition of mn one has mn ˆ Cn =An , and thus …6†

Cn Anÿ1 ÿ Cnÿ1 An Anÿ1 …Cnÿ1 ‡ an bn † ÿ Cnÿ1 …Anÿ1 ‡ an † ˆ An Anÿ1 An Anÿ1 bn Anÿ1 ÿ Cnÿ1 ˆ an : An Anÿ1

mn ÿ mnÿ1 ˆ

1 1 Let us introduce the additional sequences fGn gnˆ0 and fFn gnˆ0 by

Gn ˆ

nÿ1 X kˆ0

dk Ak ;

Fn ˆ

nÿ1 X kˆ0

gk Ak :

Then, by the definitions of dn and gn , one has …7†

Gn ˆ

nÿ1 nÿ1 X X …bk‡1 ÿ bk †Ak ˆ bn Anÿ1 ÿ bk …Ak ÿ Akÿ1 † ˆ bn Anÿ1 ÿ Cnÿ1 kˆ0

and Fn ˆ

kˆ0

nÿ1 nÿ2 X X …pk‡1 ÿ pk †Ak ˆ pk‡1 …Ak ÿ Ak‡1 † ‡ pn Anÿ1 ˆ pn Anÿ1 ÿ Anÿ2 kˆ0

kˆ0

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123

(since p0 is assumed to be 0). Adding the identity pn …An ÿ Anÿ1 † ˆ Anÿ1 ÿ Anÿ2 to the second of the above relations we get …8†

Fn ˆ pn An ÿ Anÿ1 :

In view of (6) and (7), the strict concavity of the sequence fmn g reads as follows: an

Gn Gn‡1 > an‡1 : An Anÿ1 An An‡1

By the definition of Gn one has Gn‡1 ˆ Gn ‡ dn An , and thus the above inequality is equivalent to …9†

Gn dn Anÿ1 > An pn‡1 An‡1 ÿ Anÿ1

(observe that by (4) pn‡1 An‡1 ÿ Anÿ1 ˆ an ‡

n P

…pn‡1 ÿ pk †ak ‡ pn‡1 a0 > 0).

kˆ1

We now prove (9) by induction. Suppose that it is valid for n ˆ 1; 2; . . . ; N ÿ 1, and thus …10†

GNÿ1 dNÿ1 ANÿ2 > ; ANÿ1 pN AN ÿ ANÿ2

and fails for n ˆ N, that is, …11†

GN dN ANÿ1 % ; AN pN‡1 AN‡1 ÿ ANÿ1

or, equivalently, …12†

GN dN AN % : ANÿ1 pN‡1 AN‡1 ÿ ANÿ1

We may assume that GN > 0; (indeed, otherwise by the definition of Gn one has dn ˆ 0 for n ˆ 0; 1; . . . ; N ÿ 1, and thus all mn , n ˆ 0; 1; . . . ; N are equal). Therefore, from (12) we get dN > 0, and thus, by (3), dn > 0 for n ˆ 0; 1; . . . ; N. Together with (5) this yields gn > 0 for n ˆ 0; 1; . . . ; N (since g0 ˆ p1 > 0). Hence (5) implies dn =dN ^ gn =gN for n ˆ 0; 1; . . . ; N, and we thus obtain Nÿ1 X

GN ˆ AN

kˆ0

Nÿ1 X

dk Ak

AN

^

dN
gN

kˆ0

gk Ak

AN

ˆ

dN FN
: gN AN

Combining the latter inequality with (11) we get …13†

FN %

gN ANÿ1 AN : pN‡1 AN‡1 ÿ ANÿ1

Let us now add dNÿ1 to both sides of (10); then, taking into account the definition of Gn , we get …14†

GN dNÿ1 pN AN > : ANÿ1 pN AN ÿ ANÿ2

This inequality together with (12) yields dN AN dNÿ1 pN AN > : pN‡1 AN‡1 ÿ ANÿ1 pN AN ÿ ANÿ2

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Applying (3) we get 1 pN‡1 AN‡1 ÿ ANÿ1

>

pN : pN AN ÿ ANÿ2

Let us now take into account (8); we thus can rewrite the last inequality as FN ‡ aNÿ1 > pN …FN ‡ gN AN ‡ aN †, or FN …1 ÿ pN † > pN gN AN : If pN ^ 1 we are done since the right hand side is evidently nonnegative, and thus (11) is false. Otherwise, this relation together with (13) and pN < 1 gives pN gN AN gN ANÿ1 AN < ; 1 ÿ pN pN‡1 AN‡1 ÿ ANÿ1 or, equivalently, …15†

ANÿ1 > pN pN‡1 AN‡1 :

However, by (4) one has pn pn‡1 An‡1 ÿAnÿ1 ˆ

n X …pn pn‡1 ÿ pn‡1ÿk pn‡2ÿk †an‡2ÿk ‡ pn pn‡1 …a0 ‡ a1 † > 0; kˆ1

which contradicts (15), and again (11) is false. Therefore, it remains to verify (9) for n ˆ 1. We have to prove that G1 d1 A0 > ; A1 p2 A2 ÿ A0 which is equivalent to d0 d1 > ; a0 ‡ a1 a1 ‡ a1 g1 ‡ a0 p2 which is, in turn, equivalent to a1 …d0 ÿ d1 † ‡ a1 …d0 g1 ÿ d1 g0 † ‡ d0 a0 p2 > 0: The latter inequality follows easily from (3) ± (5) with the only exclusion when d0 ˆ d1 ˆ 0, h and thus dn  0, which means that fbn g is constant. 3. Proofs of Theorems 1 ± 3. P r o o f o f T h e o r e m 1 . We have to prove the inequality …16†

Snÿ2 …x† Sn …x† Snÿ1 …x† ‡ 0;

which is equivalent to nÿ1 X

n‡1 X

aiÿ1 xi

iˆ0 nÿ1 X iˆ0

‡ i

ai x

n X

aiÿ1 xi

iˆ0 n‡1 X iˆ0

0:

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Concavity of weighted arithmetic means

Applying Theorem 4 to the sequences fan ˆ an xn g and fbn ˆ anÿ1 =an g we see that the following conditions are sufficient for the validity of the above inequality: anÿ1 anÿ2 an anÿ1 ÿ ^ ÿ ^ 0; an anÿ1 an‡1 an an anÿ1 ÿ ^ 0; x > 0; an‡1 x an x       anÿ1 anÿ2 an anÿ1 an anÿ1 anÿ1 anÿ2 ÿ ^ ÿ ; x > 0: ÿ ÿ an anÿ1 an‡1 x an x an‡1 an an x anÿ1 x Evidently, the first condition implies the other two, and it is just the concavity of the 1 . On the other hand, this condition is also necessary since nonnegative sequence fanÿ1 =an gnˆ0 it follows from (16) as x goes to infinity. The exceptional case of Theorem 4 does not apply h since b0 ˆ 0 according to our agreement, and b1 is strictly positive. P r o o f o f T h e o r e m 2 . We have to prove the inequality …17†

S0nÿ1 …x† S0n‡1 …x† S0 …x† ‡ 0;

which is equivalent to nÿ1 X iˆ0 nÿ1 X iˆ0

n‡1 X

iai xi ‡ i

ai x

iˆ0 n‡1 X iˆ0

n X

iai xi 0:

i

ai x

iˆ0

Applying Theorem 4 to the sequences fan ˆ an xn g and fbn ˆ ng we see that the following conditions are sufficient for the validity of the above inequality: 1 ^ 1; an anÿ1 ÿ ^ 0; x > 0 an‡1 x an x an anÿ1 anÿ1 anÿ2 ÿ ^ ÿ ; an‡1 x an x an x anÿ1 x

x > 0:

Observe that the initial term of the sequence fanÿ1 =an g equals 0 since it corresponds to the case n ˆ 0. Therefore, the third condition implies the other two, and it is just the convexity 1 . On the other hand, (17) implies of the nonnegative sequence fanÿ1 =an gnˆ0  0   0   0  xSn‡1 …x† xSnÿ1 …x† xSn …x† ÿ …n ÿ 1† ‡ x ÿ …n ‡ 1† < 2x ÿ n ; x > 0; x Snÿ1 …x† Sn …x† Sn‡1 …x† and thus the convexity of fanÿ1 =an g follows as x goes to infinity.

h

P r o o f o f T h e o r e m 3 . We proceed in the same way as in [3]. By Theorem 2, the sequence fS0n …x†=Sn …x†g is strictly concave for any x > 0. Thus, for any n ^ m > l > 0 and any x > 0 one has S0n …x† S0nÿm …x† S0nÿl …x† S0n‡lÿm …x† ‡ < ‡ ; Sn …x† Snÿm …x† Snÿl …x† Sn‡lÿm …x†

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or, equivalently, d Sn …x†Snÿm …x† < 0: ln dx Snÿl …x†Sn‡lÿm …x† Thus, the ratio in the above inequality is a strictly decreasing function of x, and therefore lim

x! 1

Sn …x†Snÿm …x† Sn …x†Snÿm …x† Sn …0†Snÿm …0† < < ; Snÿl …x†Sn‡lÿm …x† Snÿl …x†Sn‡lÿm …x† Snÿl …0†Sn‡lÿm …0†

x > 0:

Since Sn …0† ˆ a0 for all n ˆ 0; 1; . . . and lim

x! 1

we are done.

Sn …x†Snÿm …x† an anÿm ˆ ; Snÿl …x†Sn‡lÿm …x† anÿl an‡lÿm

h References

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A. Vainshtein Department of Mathematics and Computer Science University of Haifa Mount Carmel 31905 Haifa Israel