6 Gauge Freedom of Entropies on q-Gaussian Measures
129
Taking account into the negativity of ln q in (0, 1), we introduce a refinement of the
q-logarithmic function and the q-exponential function. For q ∈ R and a ∈ R \ {0},
define two functions χ q,a : (0, 1) → (0, ∞) and ln q,a : (0, 1) → R by
χ q,a (s) := χ q (s) · (− ln q (s))
1−a
,
ln q,a (t) := −
1
a
− ln q (t)
a ,
respectively. It turns out that
d
ds
χ q,a (s) = χ
q (s)(− ln q (s))
1−a
− (1 − a)(− ln q (s))
−a
for s ∈ (0, 1),
d
dt
ln q,a (t) =
1
χ q,a (t)
> 0
f o r t ∈ (0, 1). (6.1)
Hence the function ln q,a : (0, 1) → R is strictly increasing. We denote by exp q,a the
inverse function of ln q,a : (0, 1) → ln q,a (0, 1), which is given by
exp q,a (τ ) = exp q
− (−aτ )
1
a
for τ ∈ ln q,a (0, 1).
(6.2)
The functions ln q,a and exp q,a are called the a-refined q-logarithmic function and
the a-refined q-exponential function, respectively.
On one hand, it holds for q ≥ 1 that
ln q,a (0, 1) =
(−∞, 0) if a > 0,
(0, ∞)
if a < 0.
On the other hand, it holds for q < 1 that
ln q,a (0, 1) =
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
−
1
a
(1 − q)
−a
, 0
if a > 0,
−
1
a
(1 − q)
−a
, ∞
if a < 0.
Remark 2 (1) The refinement of the ordinary logarithmic function, that is the case
q = 1, was introduced by Ishige, Salani and the second named author [2], where
they studied the preservation of concavity by the heat flow in Euclidean space.
(2) For a positive function χ : (0, ∞) → (0, ∞) and a ∈ R \ {0}, the χ -logarithmic
function ln χ : (0, ∞) → R and its refinement ln χ,a : (0, 1) → R are defined in
the same way as those of χ q , respectively.