Page 273 - Elements of Distribution Theory

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052184472Xc09 CUNY148/Severini June 2, 2005 12:8

9.2 Some Useful Functions 259

log Γ (x)

x
Figure 9.1. Log-gamma function.

Proof. Let 0 <α < 1. Then, for x 1 > 0, x 2 > 0,

∞ 1
x 2 1−α
x 1 α
log (αx 1 + (1 − α)x 2 ) = log (t ) (t ) exp(−t) dt.
0 t
By the H¨older inequality,
∞ 1 ∞

x 1 α
x 2 1−α
log (t ) (t ) exp(−t) dt ≤ α log t x 1 −1 exp(−t) dt
0 t 0

∞

+ (1 − α) log t x 2 −1 exp(−t) dt .
0
It follows that
log (αx 1 + (1 − α)x 2 ) ≤ α log (x 1 ) + (1 − α) log (x 2 ),
proving the result.
Let
d
ψ(x) = log (x), x > 0
dx
denote the logarithmic derivative of the gamma function. The function ψ inherits a recursion
property from the recursion property of the log-gamma function. This property is given in
the following theorem; the proof is left as an exercise.

Theorem 9.4.
1
ψ(x + 1) = ψ(x) + , x > 0
x