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†
‡
<2
;
snÿ1 …x† sn‡1 …x†
sn …x†
n ^ 1; 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.
Vol. 69, 1997
121
Concavity of weighted arithmetic means
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 VasicÂ
[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]).
122
A. BERENSTEIN and A. VAINSHTEIN
ARCH. MATH.
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
Vol. 69, 1997
Concavity of weighted arithmetic means
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
124
A. BERENSTEIN and A. VAINSHTEIN
ARCH. MATH.
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†
‡
<2
;
Snÿ1 …x† Sn‡1 …x†
Sn …x†
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
<2
i
ai x
aiÿ1 xi
iˆ0
n
X
iˆ0
;
ai xi
x > 0:
Vol. 69, 1997
125
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†
‡
<2 n ;
Snÿ1 …x† Sn‡1 …x†
Sn …x†
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
<2
i
ai x
iˆ0
n
X
iai xi
;
x > 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†
126
A. BERENSTEIN and A. VAINSHTEIN
ARCH. MATH.
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
[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).
 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