I ntrnat. J. Math.
Vol.
227
Mah. Sci.
(1978)227-233
A NOTE ON AN INEQUALITY FOR THE GAMMA FUNCTION
CHRISTOPHER OLUTUNDE IMORU
Department of Mathematics
University of Ife
lle-lfe, Oyo State, Nigeria
(Received August 23, 1976 and in revised form June 30, 1977)
ABSTRACT.
Some inequalities for the Wallis functions are proved.
The results
of this paper are consequences of some characterization of convex functions.
A generalization of a result of Boyd (i) and an extentlon of an inequality
of Gantschl (3) are obtained.
KEY WORDS AND PHRASES. Gamma function, characterization of convex functions,
Ineq for Gamma functions.
AMS(MOS) SUBJECT CLASSIFICATION ’1970) CODES.
26A51, 33A15.
The aim of this note is to show that some inequalities for the Wallis
function
w(, s)
r( + )
r( + s)
(’ s)
R+ x
(0, ),
()
are natural consequences of the property of convex functions or of differ-
entlable functions.
Indeed, our results are, to some extent, consequences
228
C.S. IMOEU
of the followlng characterization of convex functions.
A real-valued function
THEOREM 1.
is convex on a closed interval
I C R if and ol.y if for every point: x
(x)
X
is
-------+
non-decreaslnE on Z.
e
"",
the function
(xo)
I,
X
X
X
(2)
O
In particular, if
u < x, v < y, for all u, v, x, y
is convex on
I, u # v, x
y,
I, then
v-u
(3)
y-x
The proof of the theorem is well known; see for example,
([3], pp. 15-18).
It is, therefore, omitted.
THEOREM 2.
u
# v, w # z,
Let u, v, x, y w and z be positive real-numbers satisfying
u < x < w, x < y < z and v < y.
Then the following inequality is valid
U
V
i
(U)j
PROOF.
<- Fr z l
r-(x)
Since the function q
/
logT(n),
rl
(4)
R+,
is convex, it follows
from inequality (3) that
ogr(v) -,osr,(u)
v-u
<
,togr(y),- ,ogr(x)
y-x
.-
logr(w)
z-w
< logr(z)
provided u, v, x, y, w and z satisfy the hypothesis of the theorem.
(5)
Since
inequality (5) is equivalent to inequality (4), the proof of the theorem is
complete.
COROLLARY I.
For (, @) e
r(m+
R+ x
+ 1)
[0, i], we have
< (m+
+ e) z-e
mZ.
(6)
INEQUALITY FOR THE GAMMA FUNCTION
PROOF.
w
m+
m+
Set u
+
,
m+. +
+
m+
8 and z
1
+
m+
I, x
229
m+
+ 8, y
+
i,
8.
Then inequalities (5) reduce to inequalities (6).
e
0 and 0 <
The case
< 1 is due to Gautschi
([3],
3. 6. 51).
Inequalltles (6) in the form
i
<
+ 8)i-8
(m +
r (m + %, +. 8,)
r(m +
I
<
+ i)
(m+
)I-8
were obtained by Lazarevi and Lupas [2] who made use of the fact that the
Gamma function is logarithmic convex and an unpublished result of Lupas
on inequalities involving the Gamma function.
We now prove a more general result which contains, as a special case,
an imporved version of Boyd’s result
{m
1
+
+ 32m + 32 }%
1
[i], namely,
r(m + i)
i
r(m + z-w)
<
<
1 2
(m + )
{.
m
(7)
+ y3 + 32m i+ 32
We first obtain the following results on differentiable functions:
THEOREM 3.
Let
i
and
2
on an open interval S in R.
exists
e
be two differentiable real-valued functions
Let x, y, u, v e S, x # y, u
+
F ()
Then there
(0, I) such that for every positive real number
(y)
(x)
y-x
PROOF.
w.
2 (v)
2 (u)
v-u
ana-l[l(X + na(y
x))
2(u
+ na(v
Consider the function
v
u
l(X
+ a(y
x))
y
-a
x
2(u + ka(w
u).
u))].
(8)
230
This function is dlfferentiable on
[0, i].
By the usual Mean Value Theorem
for dlfferentlable functions, we obtain the desired concluslon.
Let
THEOREM 4.
be a differentlable real-valued function on an open
’
interval S in R and let
be non-decreaslng on S.
Suppose u, v, x, y ,S, u # v, x # y and either x > u, v > y or x < u,
v < y.
Then, for some a 0
Z+
u)(x-
u) +
(I
(the set of positive integers) such that
u(y-
v) > 0, 0 < u < i,
>
(9)
O
have
_(y)-(x)
(v)- (u)
>
y-x
(io)
v-u
We note, however, that inequality (i0) is valid if x > u, y > v and a
is an arbitrary positive real number.
PROOF.
imply that
Let
x- u
u
<
<
X
either case
that for all
(i
< i.
a)(x-
_>
ao,
u) +
a
a(y_
Z+,
X
<
I, there exists
U
y<
V-
--x- u
v) > 0, for all u e
Z+,
Hence, in
>
The
concluslon follows by Theorem 3 and the non-decreaslng character of
We remark on passing, that inequality (i0) is strict unless
stant or linear function.
I.
Z+,
Then, for all
X-- U
If, however, 0 < x- u+v-y <
x-u+v-y
Z+ such
U
x- u +v- y
X-- U
The assumptions on x, y, u and v
is an arbitrary real number between 0 and
x-u+v-y
Suppose o <
n
@ in Theorem 3.
i 2
’.
is a con-
Furthermore, inequality (i0) is reversed if
is
non-increaslng.
COROLLARY 2.
Let
be a twice differentiable real-valued convex function
on an open interval S in R. Let x, y, u and v satisfy the conditions of
Theorem 4.
Then inequality (i0) holds if inequality (9) is valid. The
inequality is reversed if
is concave.
INEQUALITY FOR THE GAMMA FUNCTION
PROOF.
Since
#
S,
is convex on
If, however,
non-decreaslng on S.
"
231
Hence
is non-negatlve on S.
concave,
is
’
#’
is
is non-lncreaslng on S.
Consequently, the conclusion of the corollary follows from Theorem 4.
.
An immediate consequence of the above corollary can be obtained by
specializing
R+,
For example, if we take (a), a c
function satisfies the condition of Corollary 2.
as logF(=), then thls
Consequently, if inequality
(9) holds and x, y, u, v satisfy the conditions of Theorem 4, we have
r(y)
>
r(x)
For m
x
m
+_,
>---12,
y
m
,
"r(u)
R- {o} be such that
let y
+ i,
y
x
v- u
rr(v) )
m
u
Since x- u > 0, y
+ S(m)
and v
v < 0 and
+
m
< 8(m)
<
()
o <
i +’S(m) where
,
Put
< i.
< S(m)
<
inequality (ii) holds if
and only if for some positive integer a,
Hence
%
(m + 8 (m)) < r(m + 1)
r(m
+1/2)
if
1/2[ i (..)
1
<_ O(m)
1
1
i @(m) <_
<
c
], 0 <
o
(.) <1
Letting a /-, we get
r(m + 1)
B(m,) ! r(m
(m +
Now write v
m
+
i, u
x- u < 0 and v- y < O.
I
a
<
i
<
v
y
x
u
(m +
m
<
+
I-,2
y
m
+
(12)
i
+ B(m)
and x
m
+ O(m).
Then
Consequently, inequality (ii) holds if and only if
Equivalently,
e(m))1/2 >
provided
<_ 8(m)
if
r(m + 1)
r(m+ )
(13)
.}
.
1/2[ i- ()o ], < (o)
< i;
C.S. IHORU
232
-.
1
a condition which reduces to 8(m)
Combining inequalities (12) and (13), we obtain
(m + 8(m))
r(m+ )
r(m +
<
< O(m) <
1/2)
.
The converse of this result was obtained by Watson
rm + Z)1
1
(m + e(m))
%
<_ e (m)
<_-1
or
For m
>-,i
i <
e(m)
-
m
{
<
for
[’4], namely,
m
>_-
i
if
and
>_ O.
<,i
m+ 2
r(m+ i)
r(m+)
then<S(m)
(14)
we obtain
+1/2) },%
m+1/2+ e(m+2I-)
(m
{,
<
r(m+ )
r(m + I)
Hence, this inequality and inequality (14) combined yield
(m
+
e(m)}
where 1 < e(m)
Taking %(m)
On putting e(m)
r(m + i)
r(m + )
<
(m +
’[
<
m
1
,
8(m
+7+
i
}
(15)
+g)
<1
=1/4
1
+ ’32m +1 3’2’
m
.1.
+
32m+ 8
36
+ 4m-
we obtain inequality (7).
1,2,
we obtain an inequality due to
3
Slavi ([5], inequality (12)).
A result which is better than any one known, except for the formula (15) of
Slavi’s paper [5]
is obtained by putting
-
233
INEQUALITY FOR THE GAMMA FUNCTION
e (m)
1
i
+
32m+ 8
36
+ 4m+
5
It is our conjecture that formula (15) of Slavi’s paper [5] can be obtained
from our general result, namely inequality (15), by appropriate choice of
1
(R)]
REFERENCES
i.
Boyd, A. V.
Note on the Paper of Uppuluri,
9-10.
2.
Pacific
J.
Math.
22 (1967)
Lazarevi6, I. B. and A. Lupas.
G-_--, Functions.
FIz..
Functional Equations for Wallis and
Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat,
461-497 (1974) 245-251.
3.
Mitrinovic, D. S. Analytic Inequalities. Grundlehren der Alath. Wiss.,
New York, 1970, 400 pp.
Band 165, Berlin
Heidelberg
4.
Watson, G. N. A Note on Gamma Functions, Proc. Edlnburg Math. Soc. 2
(1958/59) 11.
5.
Slavi, D. V.
On Inequality for r(x + l)Ir(x +
Elektrotechn Fak. Set. Mat. Fiz 499, 17-20.
1
),
Univ. Beograd Publ.