Digital Library of Mathematical Functions
The Gamma Function -- Inequalities

### 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 )