Concavity of weighted arithmetic means with applications. By. ARKADY BERENSTEIN and ALEK VAINSHTEIN *). Abstract. We prove that the following three ...
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 i0
i0
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
xg and fS0n
x=Sn
xg 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
n X xk k0
k!
;
n1 2 s
x for n2 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 ex ÿ sn
x. In his paper [1] Alzer proves that snÿ1
x sn1
x >
nlÿm l snÿl
xsnlÿm
x < sn
xsnÿm
x < snÿl
xsnlÿm
x :
nl n 2 In particular, one has snÿ1
xsn1
x > s
x for any n ^ 1 and x > 0 . n1 n Inequality (1) was obtained as a corollary of the following result also proved in [3]: the sequence fsnÿ1
x=sn
xg 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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Concavity of weighted arithmetic means
A natural generalization of this result would be a characterization of power series
1 P k0
ak xk
for which an analog of (2) holds. Such a generalization is provided by the following theorem. 1 Theorem 1. Let fan gn0 be a positive sequence and Sn
x
n P k0
ak xk . Then the sequence
1 1 fSnÿ1
x=Sn
xgn0 is strictly concave for any x > 0 if and only if the sequence fanÿ1 =an gn0 is concave.
There exists, however, another natural generalization of (2), which reflects the following simple identity: s0n
x 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
xg. Our second generalization of (2) is thus as follows. n P 1 Theorem 2. Let fan gn0 be a positive sequence and Sn
x ak xk . Then the sequence k0
1 1 fS0n
x=Sn
xgn0 is strictly concave for any x > 0 if and only if the sequence fanÿ1 =an gn0 is convex.
Observe that the only sequence fSn
xg that satisfies the conditions of both Theorems 1 and 2 is the sequence fasn
bxg 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 k0 n P
ÿR; R, Sn
x f
xÿ ak xk , and the sequence fanÿ1 =an g is convex and nondecreasing, then k0
an an2 2 S
x; Snÿ1
xSn1
x ^ 2 an1 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 gn0 be a positive sequence such that the sequence fanÿ1 =an gn0 is convex. Then for any n ^ m > l > 0 and x > 0 an anÿm Snÿl
xSnlÿm
x < Sn
xSnÿm
x < Snÿl
xSnlÿm
x: anÿl anlÿm
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 i1 convex for any choice of the given convex sequence fan g is as follows: i1 nÿ1 Q
wn
i1
w2
i ÿ 1w1
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 gn0 be a nonnegative sequences, pn anÿ1 =an , n ^ 0. Let fan gn0 1 1 Theorem 4. Let the sequence fbn gn0 be concave, the sequence fan gn0 be log-concave, and
bn ÿ bnÿ1
pn1 ÿ pn ^
bn1 ÿ bn
pn ÿ pnÿ1 ; n ^ 1:
Then the sequence a0 b0 a1 b1 an bn 1 mn a0 a1 an n0 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 gn0 , fCn gn0 , fdn gn0 , and fgn gn0 by the following relations:
An
n P k0
ak ;
dn bn1 ÿ bn ;
Cn
n P k0
ak bk ;
n ^ 0:
gn pn1 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 Cn =An , and thus
6
Cn Anÿ1 ÿ Cnÿ1 An Anÿ1
Cnÿ1 an bn ÿ Cnÿ1
Anÿ1 an An Anÿ1 An Anÿ1 bn Anÿ1 ÿ Cnÿ1 an : An Anÿ1
mn ÿ mnÿ1
1 1 Let us introduce the additional sequences fGn gn0 and fFn gn0 by
Gn
nÿ1 X k0
dk Ak ;
Fn
nÿ1 X k0
gk Ak :
Then, by the definitions of dn and gn , one has
7
Gn
nÿ1 nÿ1 X X
bk1 ÿ bk Ak bn Anÿ1 ÿ bk
Ak ÿ Akÿ1 bn Anÿ1 ÿ Cnÿ1 k0
and Fn
k0
nÿ1 nÿ2 X X
pk1 ÿ pk Ak pk1
Ak ÿ Ak1 pn Anÿ1 pn Anÿ1 ÿ Anÿ2 k0
k0
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Concavity of weighted arithmetic means
123
(since p0 is assumed to be 0). Adding the identity pn
An ÿ Anÿ1 Anÿ1 ÿ Anÿ2 to the second of the above relations we get
8
Fn pn An ÿ Anÿ1 :
In view of (6) and (7), the strict concavity of the sequence fmn g reads as follows: an
Gn Gn1 > an1 : An Anÿ1 An An1
By the definition of Gn one has Gn1 Gn dn An , and thus the above inequality is equivalent to
9
Gn dn Anÿ1 > An pn1 An1 ÿ Anÿ1
(observe that by (4) pn1 An1 ÿ Anÿ1 an
n P
pn1 ÿ pk ak pn1 a0 > 0).
k1
We now prove (9) by induction. Suppose that it is valid for n 1; 2; . . . ; N ÿ 1, and thus
10
GNÿ1 dNÿ1 ANÿ2 > ; ANÿ1 pN AN ÿ ANÿ2
and fails for n N, that is,
11
GN dN ANÿ1 % ; AN pN1 AN1 ÿ ANÿ1
or, equivalently,
12
GN dN AN % : ANÿ1 pN1 AN1 ÿ ANÿ1
We may assume that GN > 0; (indeed, otherwise by the definition of Gn one has dn 0 for n 0; 1; . . . ; N ÿ 1, and thus all mn , n 0; 1; . . . ; N are equal). Therefore, from (12) we get dN > 0, and thus, by (3), dn > 0 for n 0; 1; . . . ; N. Together with (5) this yields gn > 0 for n 0; 1; . . . ; N (since g0 p1 > 0). Hence (5) implies dn =dN ^ gn =gN for n 0; 1; . . . ; N, and we thus obtain Nÿ1 X
GN AN
k0
Nÿ1 X
dk Ak
AN
^
dN gN
k0
gk Ak
AN
dN FN : gN AN
Combining the latter inequality with (11) we get
13
FN %
gN ANÿ1 AN : pN1 AN1 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 > : pN1 AN1 ÿ ANÿ1 pN AN ÿ ANÿ2
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A. BERENSTEIN and A. VAINSHTEIN
ARCH. MATH.
Applying (3) we get 1 pN1 AN1 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 pN1 AN1 ÿ ANÿ1 or, equivalently,
15
ANÿ1 > pN pN1 AN1 :
However, by (4) one has pn pn1 An1 ÿAnÿ1
n X
pn pn1 ÿ pn1ÿk pn2ÿk an2ÿk pn pn1
a0 a1 > 0; k1
which contradicts (15), and again (11) is false. Therefore, it remains to verify (9) for n 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 d1 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
n1 X
aiÿ1 xi
i0 nÿ1 X i0
i
ai x
n X
aiÿ1 xi
i0 n1 X i0
0:
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Concavity of weighted arithmetic means
Applying Theorem 4 to the sequences fan an xn g and fbn 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 an1 an an anÿ1 ÿ ^ 0; x > 0; an1 x an x anÿ1 anÿ2 an anÿ1 an anÿ1 anÿ1 anÿ2 ÿ ^ ÿ ; x > 0: ÿ ÿ an anÿ1 an1 x an x an1 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 gn0 it follows from (16) as x goes to infinity. The exceptional case of Theorem 4 does not apply h since b0 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 S0n1
x S0
x 0;
which is equivalent to nÿ1 X i0 nÿ1 X i0
n1 X
iai xi i
ai x
i0 n1 X i0
n X
iai xi 0:
i
ai x
i0
Applying Theorem 4 to the sequences fan an xn g and fbn ng we see that the following conditions are sufficient for the validity of the above inequality: 1 ^ 1; an anÿ1 ÿ ^ 0; x > 0 an1 x an x an anÿ1 anÿ1 anÿ2 ÿ ^ ÿ ; an1 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 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 gn0 0 0 0 xSn1
x xSnÿ1
x xSn
x ÿ
n ÿ 1 x ÿ
n 1 < 2x ÿ n ; x > 0; x Snÿ1
x Sn
x Sn1
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
xg 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 S0nlÿm
x < ; Sn
x Snÿm
x Snÿl
x Snlÿm
x
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A. BERENSTEIN and A. VAINSHTEIN
ARCH. MATH.
or, equivalently, d Sn
xSnÿm
x < 0: ln dx Snÿl
xSnlÿm
x Thus, the ratio in the above inequality is a strictly decreasing function of x, and therefore lim
x! 1
Sn
xSnÿm
x Sn
xSnÿm
x Sn
0Snÿm
0 < < ; Snÿl
xSnlÿm
x Snÿl
xSnlÿm
x Snÿl
0Snlÿm
0
x > 0:
Since Sn
0 a0 for all n 0; 1; . . . and lim
x! 1
we are done.
Sn
xSnÿm
x an anÿm ; Snÿl
xSnlÿm
x anÿl anlÿm
h References
[1] H. ALZER, An inequality for the exponential function. Arch. Math. 55, 462 ± 464 (1990). [2] H. ALZER, J. BRENNER and O. RUEHR, Inequalities for the tails of some elementary series. J. Math. Anal. Appl. 179, 500 ± 506 (1993). [3] A. BERENSTEIN, A. VAINSHTEIN and A. KREININ, A convexity property of the Poisson distribution and its application in queueing theory. In: Stability Problems for Stochastic Models, Moscow 1986, pp. 17 ± 22; English translation in J. Soviet Math. 47, 2288 ± 2292 (1989). Â and P. VASICÂ, eds., Means and Their Inequalities. Dordrecht 1988. [4] P. BULLEN, D. MITRINOVIC [5] W. CHEN, Notes on an inequality for sections of certain power series. Arch. Math. 62, 528 ± 530 (1994). [6] K. DILCHER, An inequality for sections of certain power series. Arch. Math. 60, 339 ± 344 (1993). [7] I. LACKOVICÂ and S. SIMICÂ, On weighted arithmetic means which are invariant with respect to k-th order convexity. Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 461 ± 497, 159 ± 166 (1974). [8] M. MERKLE and P. VASICÂ, An inequality for residual Maclaurin expansion. Arch. Math. 66, 194 ± 196 (1996). [9] D. S. MITRINOVICÂ, Analytic Inequalities. Berlin-Heidelberg-New York 1970. [10] D. MITRINOVICÂ, I. LACKOVICÂ and M. STANKOVICÂ, Addenda to the monograph ªAnalytic Inequalitiesº, part II: On some convex sequences connected with Ozeki's results. Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 634 ± 677, 3 ± 24 (1979). [11] P. VASICÂ, J. KECÏKICÂ, I. LACKOVICÂ and ZÏ. MITROVICÂ, Some properties of arithmetic means of real sequences. Mat. Vesnik 9, 205 ± 212 (1972). Eingegangen am 15. 1. 1996 Anschriften der Autoren: A. Berenstein Department of Mathematics Northeastern University Boston, MA 02115 USA Current address: Department of Mathematics Cornell University Ithaca, NY 14853 USA
A. Vainshtein Department of Mathematics and Computer Science University of Haifa Mount Carmel 31905 Haifa Israel