Internat. J. Math. & Math. S.
Vol. 6 No. 3 (1983) 609-611
6O9
A NOTE ON A PAPER BY $. HABER
A. NcD. NERCER
Department of Mathematics and Statistics
University of Guelph
Guelph, Ontario, Canada NIG 2WI
(Received September 23, 1982 and in revised form February 26, 1983)
ABSTRACT.
A technique used by S. Haber to prove an elementary inequality is applied
here to obtain a more general inequality for convex sequences.
KEY WORDS AND PHRASES. Convex sequences, Hadamard’s inequality for convex functions,
r eaa ng em e
1980 MATHEMATICS SUBJECT CLASSIFICATION C(E. Primary 26D15, Secondary 40G05.
I.
INTRODUCTION.
Let
a
and
b
be non-negative.
Then the following elementary inequality was
proved in [i].
n$I [ a
n
+
an-i
b+.
+
b
n
a+b.n
e
(--)
(n=0,1,2,...)...
Now this inequality can be obtained at once by taking
f(t)
(i.I)
t
n
in the well-
known result
b
which holds whenever
f
(I.I) is interesting and
a+b.
f(t)dt >
b-a
is convex in
(I 2)
f(--)
a
[a,b].
However, the method used in [i] to obtain
it is the purpose of the present note to show that it can be
used to prove a considerably more general result about sequences.
general result will have (1.2) as a consequence.
2.
MAIN RESULTS.
A lemma which we shall use is the following
LEMMA. If
0
>
_>
I 2
>
>
and
m
E
=0
V
0
m
Indeed this more
610
A.M. MERCER
and if the ordering of the
precedes all the nega-
is such that each positive
tive ones, then
m
: vl3v
-> 0
v=O
This lemma, which is easily proved, is not the one stated by Haber but, essentially,
For with
it is what he used.
b
defined as in
i
(i
we do
not
[]:
2,
0
n even)
in fact have
Y.
0
b.
i=0
which is what is needed to apply the lemma quoted there.
Our result is the following.
THEOREM.
Let
n
{u}= 0
be a convex sequence.
n
11 u
1
Then
n
n
r.
u
_(v)
v
v=0
2
v=O
To see that (1.2) is a consequence of (2.1) let the function
and convex (and hence continuous) on [a,b] and take
n-{--"-
(2.1)
_>--
n
f(a
u
f(x) be bounded
(b-a))
+ v__
n
Then (2.1) reads
In+l
On letting
n
n
v
(b-a)) >
E f (a +
n
^n
n--O
z
n
(2.2)
v
f(a + n (b-a))...
n
()
v=O
E
the left-hand side here tends to the left-hand side of (1.2).
And
by virtue of Bernsteln’s result
n
Y.
,im
n-
=0
(3)q (’) x(l-x) n-
C[O,I] we see that the right-hand
I/2
f(a + x(b-a)) and x
Merely take (x)
We now proceed to prove (2.1)
whenever
PROOF.
E
Following Haber let us put
Q
E
v-O
,
{
1
7v
Then
n
r. u
v
n+-- v=O
-
Q
[]
+ 7Q_ I +
side of (2.2) tends to
in
(2.3)
and write
+
70qY +
70+71 +
(2.3)
+(x)
7Q
7Q
Q*
n
r.
r. (v)uv"
v-O
v=O
if
n
Is odd
if
n
is even
n
eEuv + un-v
.a+b.
f[---)
NOTE ON A PAPER BY S. HABER
611
where
n
n+-- ’(X)
Cv
Since {u
}n=0
is convex then
uv+
+ un-v-I
which is to say that the sequence
the sequence {c
}Q=0
< u
+ un-
+ un- }Q=0
u
is non-lncreasing and that
(0<<Q-I)
is non-lncreaslng,
*
c
0.
we
see too that
Appealing to the Lemma
--0
quoted above we find that
Q,
c[u
--0
+un-
and this complets the proof of (2.1).
In conclusion I wish to thank the referee for his helpful advice concerning the
lemma used here.
REFERENCES
I.
HABER, S.
2.
HARDY, G.H., LITTLEWOOD, J.E., POLYA, G.,
An Elementary Inequality, Internat. J. Math. and Math. Sci. Vol. 2
No. 3 (1979) 531-535.
:Inequalities
Cambridge (1973).