Inequalities
Contents
- Real Variables
- Complex Variables
Real Variables
Throughout this subsection
x
>
0
.
1
<
(
2
π
)
-
1
2
x
(
1
2
)
-
x
ⅇ
x
Γ
(
x
)
<
ⅇ
1
(
12
x
)
1
Γ
(
x
)
+
1
Γ
(
1
x
)
≤
2
1
(
Γ
(
x
)
)
2
+
1
(
Γ
(
1
x
)
)
2
≤
2
x
1
-
s
<
Γ
(
x
+
1
)
Γ
(
x
+
s
)
<
(
x
+
1
)
1
-
s
0
<
s
<
1
exp
(
(
1
-
s
)
ψ
(
x
+
s
1
2
)
)
≤
Γ
(
x
+
1
)
Γ
(
x
+
s
)
≤
exp
(
(
1
-
s
)
ψ
(
x
+
1
2
(
s
+
1
)
)
)
0
<
s
<
1
Complex Variables
|
Γ
(
x
+
ⅈ
y
)
|
≤
|
Γ
(
x
)
|
|
Γ
(
x
+
ⅈ
y
)
|
≥
(
sech
(
π
y
)
)
1
2
Γ
(
x
)
x
≥
1
2
For
b
-
a
≥
1
,
a
≥
0
, and
z
=
x
+
ⅈ
y
with
x
>
0
,
∣
Γ
(
z
+
a
)
Γ
(
z
+
b
)
∣
≤
1
|
z
|
b
-
a
For
x
≥
0
,
|
Γ
(
z
)
|
≤
(
2
π
)
1
2
|
z
|
x
-
(
1
2
)
ⅇ
-
π
|
y
|
2
exp
(
1
6
|
z
|
-
1
)
