130
H. Matsuzoe and A. Takatsu
6.1.2 Properties
In this section, we give a condition for ln q,a to be concave and compute the higher
order derivatives of exp q,a , which will be used to define information geometric structures.
For q ∈ R and a ∈ R \ {0}, define
t q,a :=
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
0
if either q > 0 or q = 0 with a − 1 > 0,
1
i f q ≤ 0 with a − 1 ≤ 0,
1
exp q
1−a
q
otherwise,
T q,a :=
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
0
i f q > 1 with 1 − a ≥
q
q − 1
,
1
i f q ≤ 0,
1
exp q
max
0,
1−a
q
otherwise,
and set I q,a := (t q,a , T q,a ) ⊂ (0, 1). Note that I q,a is nonempty if and only if one of
the following three conditions holds:
• q > 1 with 1 − a <
q
q − 1
;
• 0 < q ≤ 1;
• q ≤ 0 with a − 1 > 0.
Proposition 1 Fix q ∈ R and a ∈ R \ {0}. For an interval I ⊂ (0, 1), the strict concavity of ln q,a in I is equivalent to the strict convexity of exp q,a in ln q,a (I ). Moreover,
if I q,a = ∅, then ln q,a is strictly concave in I q,a .
Proof Due to Eq. (6.1), ln q,a is strictly increasing in (0, 1) and so is exp q,a in
ln q,a (0, 1). Fix an interval I ⊂ (0, 1). For t i ∈ I, τ i ∈ ln q,a (I ) (i = 0, 1) with
τ i = ln q,a (t i ) or equivalently t i = exp q,a (τ i )
and λ ∈ (0, 1), it follows from the continuity of ln q,a that
(1 − λ)t 0 + λt 1 ∈ I, (1 − λ)τ 0 + λτ 1 ∈ ln q,a (I ).
We observe from the monotonicity of ln q,a and exp q,a that