I ntrnat. J. Mah. Math. Sci.
Vol. 5 No. 2 (1982) 337-343
337
THE POWER MEAN AND THE LOGARITHMIC MEAN
CHRISTOPHER OLUTUNDE IMORU
Department of Mathematics
University of Ife
Ile-Ife
Oyo State
NIGERIA
(Received July 3, 1979
and in revised form, March 24, 1980)
ABSTRACT.
In
a very interesting and recent
note, Tung-Po Lin [I] obtained the
least value q and the greatest value p such that
M
P
<L<M
q
is valid for all distinct positive numbers x and y where
Ms
(x +.y.)s
2
and L
In x-ln Y
The object of this paper is to give a simpler proof than Lin’s of a more
general result.
and
ha,
More precisely, the author obtained the classes of functions f
agR such that
In
f
(t)
KEY WORDS AND PHRASES.
functions.
h
a
(t)
[tl/
+ I]
-
> O, t > I.
Concave Functions, Space of Lebesque -integrable
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES. Primary 26 A 86.
1.
INTRODUCTION.
There is an extensive literature on the extensions and generalizations of
inequalities involving the power mean and the logarithmic mean and their applica-
C.O. IMORU
338
[see 2-5].
tions
Tung-Po Lin [I] considered the least value p and the greatest
value q such that
M
q
<L<M
(1.1)
P
is valid for all distinct positive real numbers x and y, where
L
x-y
L(x,y)
Inx-lny
and H
M.(x,y)
a
(X
+2
"y
c e R
are respectively the logarithmic mean and the power mean of order
of two dis-
tinct positive numbers x and y.
Indeed, Lin showed that q
are the best possible values one can
0 and p
get in order that (I.I) may be satisfied for any distinct positive real numbers x
and y.
This result is sharper than an earlier inequality due to Carlson
[6],
namely:
M
O
< L <
for any distinct x, y > o
M
(I 2)
As observed by Lin (If], Theroems 3 and 4), if 0<p<I/3, there exist x > 0
and y > 0 such that L(x,y) > M (x,y).
On the other hand, there exist x > 0 and
P
y > 0 such that L(x,y) < M (x,y). In fact, if we suppose that x > y > o, which
P
is perfectly in order due to the symmetry of L(x,y) and M (x,y) in x and y, one
P
can readily show that
L(x,y) <
Mp(x,y),
o
< p <
-
.
(1.3)
For
is large enough.
The reversed inequality holds for smaller values of
Y
Y
instance, numerical calculations with the aid of a computer easily show that the
x
result (1.3) holds for p
if
is greater than (9 9) 4
(25 3799) s
I0
Y
if
,,
(56-6188) 6
(119-7628) 7
(246.8528) 8
(501.9099) 9 and (1012.9460) 1 respectively.
.
From these results, we can readily compute the set {(x,y); x > o and y > o}
for which H
set
{(x,y);
P
(x,y) < L(x,y)
X
<
> p >
(I012.9460)I}.
For example, M
It is interesting to
ne
(x,y) < L(x,y)
on the
that the above esti-
POWER MEAN AND LOGARITHMIC MEAN
x
mates show that
339
o, which is compatible with the left-hand side of in-
as p
Y
equality (I.I), namely:
M o (x,y) < L(x,y) for distinct x > y > o.
I/p more than doubles from
p
Furthermore, the critical value of (x/y)
p
I/(5+n), where
(x/y)I/P
values of
o, I, 2,
n
5.
1/(4+n) to
It is our conjecture that the critical
is more than double from p
I/n to I/(n+l), n=l, n,
The purpose of this note is to give a simpler proof than Carlson or Lin of a
Indeed, we obtain in Section 2 the classes of functions f
more general result.
and h
e R, for which the function
ff
F
(t)= In f (t) -ha (t) (t I/ + I)
t >
This leads to the determination of a polynomial of degree
is strictly positive.
n, (n.>l) which is
-
a lower bound of the function
()n In
t.
The compari-
with the ones given by Carlson and Lin show
I, 2,
son of these bounds for n
H(t)
that not only are our results more general in form, but they are somewhat sharper.
We point out that Carlson’s inequalities (1.2) are special cases of an inequality for convex and concave functions.
Indeed, from the Jensen inequality
for concave functions,
t
1
f(x) d(x)
t
fl
where
dp(x)
R,
ff(x)}
t
fl
dp(x)
(1.4)
d(x)
L(R, d), the space of Lebesque p
is concave and fg
tions on
O-
>
we obtain, on setting
f(x)
,
p(x)
x and
integrable func-
(x)
x
I/p
p >
or
p < o, the inequality
In
t
t-1
Since
o
< p < I.
(1.4)
>
is reversed if
Taking p
2 in
tl-1/P-1]p
Z
[_p__
p-1
is
(1.5),
P >
orp < o.
(1.5)
convex, inequality (1.5) is reversed if
we obtain the right-hand side of inequality
(1.2).
2.
MAIN RESULTS.
We pointed out that Lin, with the aid of the Maclaurin series, showed that
340
C.O. IMORU
the best possible value of p for which L < M
A proof of this interesting
"
is
p
Ill
result without the Maclaurin series was indicated in the paper
Following Flander’s approach, we obtain the following result.
Flanders.
F(z)
Let
Theorem 2.1:
in
ha.(1)
(z+l)
(z+l)
o
is differentiable in R.
R,
where
Suppose
fo
is an
f(1)
Then F (z) > 0, for every z > I, provided
0.
o +I
hl (z) f (z)
f
Proof:
ha(z)
f(z)
ho
increasing positive function and
and
-
by Harley
+
(z+l)
a
h(z) f(z)
(z)
> o.
(2.1)
(z)
f
Computing the first derivative of f (z) yields
Fl(z)
Ac(z)
(z+l)
(+1
Gcx(z)]
(2 .2)
where
f
(z)
n (z)
f (z) (z+l)
a+l
and
hcl(z)
Ga(z)
(z+l) -a
h((z) f(x(z)
(z)
f
(z)
f
0 if and only if (z+l) a+l
Observe that F
(z)
Fo(z)
fo(z)
G
Fa(1)
0 and
z > I.
This completes the proof of the theorem.
O(z)
> 0, z
I.
Since
> 0 wherever z > I, we conclude that F (z) > 0, for all
Corollary 2.2:
-
For every
a
E
Inz
k=l
is valid provided
z
c+l
a
+
k R,
8k (zk-1)
5
k=2
+ [{+1
X+lCk
+--a81
+
< k <
< z+l)
k-1
-k-1
k
k=l
,
a,
where a
z > 1,
[k-l+k ]]z
--]z+l
+,
the inequality
(2.3)
k
a
>
0
(2.4)
POWER MEAN AND LOGARITHMIC MEAN
a
5k=l
Let h (z)
Proof"
result follows at once.
13k
(z k_ I) and f (z)
z
a
341
in Theorem 2
- - -
Then the
Observe that for all z > I,
5k=l
satisfies the conditions
13k
130 --> 2
z k_ 1)
O
n
n
z
13 k
u-1
5k=l
and
The inequality (2.5) is strict unless
quently, we have
=> 2c (za-1),
zk
1)
and
k
=>
(13k)l<=k=<
(2.5)
(2.6)
o.
o,
k
z-1) < (z+l )a
5-
z
u 2
13a
provided
< k < u.
Conse-
a.[
k=l
(2.7)
wherever (2.4) and (2.6) are valid.
Hence for all z > I, the polynomial
Pa(z)
{/
(13k)l<k<ot
(2.6),
satisfies
:Y
y
Inx-lny
< k=l
13k (xk/a yk/U)
k
2(In
<
x
(z k- 1
k=l
where
z+l- c
(--)
H(z)
In z,
K(z)
Suppose condition (2.6) holds.
x
a
is a lower bound of the function
as well as an upper bound of the function
Corollary 2.3:
2
For x > y > 0, the inequality
,x I/a
+
2
In y)
yl/x)
e’,
(2.8)
is valid whenever
+I
+
x
5k=2
[u
+ [u +I-
Proof:
+1
f l31
Ck
+
f S_
k=l
Bk-1 k- l+k] xk/ay(a+l-k)/a
a
k-1
_._.]=lk
x
1/ay (o+
The result is immediate on setting z
inequa ity (2.7).
)/c > o
x
_q
y
where x > y > o in
(2.9)
342
C.O. IMORU
The cases a
2 and a
3 with
a 2
k
k
and
son’s result and Lin’s inequality respectively.
o
Suppose
k # a lead to Carl-
a
a2a(1+/2)’
then inequality (2.8) gives an improved lower bound for the power mean
ject to the condition
(2.9).
> o;
H1/a,
sub-
To see this, let us consider the following special
cases.
Suppose a
1.
Then inequality (2.8) yields
x
Inx- Iny
+ + 4(&2+2)];
provided x > y [1
Suppose a
Inx
Iny
’i’nx
Iny
+ y
< x
> o
< (1 +)
z
[1++4(2+2)];
(2.11)
> o.
3, then inequality (2.8) yields
If we finally consider the case when
3
x
y
Inx- Iny
< (1 +
We pointed out in Section
> (9.9) 4
)2
y
x
Inx
Iny
<
(x
3
3
+ y
(2.12)
2
) + 4(z+2) + 412(z+2)+2(I+)4(+2)];
provided x > y(l+
Y
(2 10)
2
2, we have
Inx- Iny
wherever x > y
y
x
< (1 +
> o.
that inequality (1.3) holds for p
%
if
It is pertinent to add here that these values can be obtained by
4. To illustrate the usefulness
in the inequality (2.9) when
Y
of this method, we compute the set on which inequality (1.3), with p
%, is true.
solving for
Take
2
k
4 in inequality (2.8).
and
o, k
x
y
Inx- Iny
provided z s
11 z 4 + 10 z 3 + 10 z
<
(x
11 z +
+
(2.13)
2
> o, where z 4
hand side of this inequality has the factorization
(z + 1) (z-l)
(z-5+246)(z-5-246).
The result is
x/y.
The left
POWER MEAN AND LOGARITHMIC MEAN
343
Consequently, inequality (2.13) is valid provided either
>
()x
L(x,y) <
for a
> 5
256
Mk(x,y)
0.I0 or
on the set
(_Zyx)k
> 5 +
[(x,y);
256
9.9.
> (9.9)4].
Since x > y > o, we have
Similar results can be obtained
5, 6, 7,
Further specializations of the constants
8k’
< k < a, may be of interest
in the determination of improved upper and lower bounds for the logarithmic mean
and the power mean respectively.
REFERENCES
I.
Tung-Po Lin:
2.
E.B. Leach and M.C. Sholander:
The Power Mean and the Logarithmic Mean; Amer. Math. Monthly
81 (1974), 879-883.
Extended Mean Values; Amer. Math. Monthly
85 (1978), 84-90.
3.
E.B. Leach and M.C. Sholander: Corrections to "Extended Mean Values";
Amer. Math. Monthly 85 (1978), 656.
4.
K.B. Stolarsky: Generalizations of the Logarithmic Mean; Amer. Math. Mag.
48 (1975), 87-92.
5.
K.B. Stolarsky:
Monthly,
6.
B.C. Carlson:
The Power and Generalized Logarithmic
to appear.
The Logarithmic
Mean; Amer. Math.
Mean; Amer. Math. Monthly 79 (1972), 615-618.