Abstract
For a continuous and positive function w (λ), λ> 0 and µ a positive measure on [0, ∞) we consider the following 𝒟-logarithmic integral transform
𝒟
ℒ
o
g
(
w
,
μ
)
(
T
)
:
=
∫
0
∞
w
(
λ
)
1
n
(
λ
+
T
λ
)
d
μ
(
λ
)
,
\mathcal{D}\mathcal{L}og\left( {w,\mu } \right)\left( T \right): = \int_0^\infty {w\left( \lambda \right)1{\rm{n}}\left( {{{\lambda + T} \over \lambda }} \right)d\mu \left( \lambda \right),}
where the integral is assumed to exist for T a positive operator on a complex Hilbert space H.
We show among others that, if A, B > 0 with BA + AB ≥ 0, then
𝒟
ℒ
o
g
(
w
,
μ
)
(
A
)
+
𝒟
ℒ
o
g
(
w
,
μ
)
(
B
)
≥
𝒟
ℒ
o
g
(
w
,
μ
)
(
A
+
B
)
.
\mathcal{D}\mathcal{L}og\left( {w,\mu } \right)\left( A \right) + \mathcal{D}\mathcal{L}og\left( {w,\mu } \right)\left( B \right) \ge \mathcal{D}\mathcal{L}og\left( {w,\mu } \right)\left( {A + B} \right).
In particular we have
1
6
π
2
+
di
log
(
A
+
B
)
≥
di
log
(
A
)
+
di
log
(
B
)
,
{1 \over 6}{\pi ^2} + {\rm{di}}\log \left( {A + B} \right) \ge {\rm{di}}\log \left( A \right) + {\rm{di}}\log \left( B \right),
where the dilogarithmic function dilog : [0, ∞) → ℝ is defined by
di
log
(
t
)
:
=
∫
1
t
1
n
s
1
-
s
d
s
,
t
≥
0.
{\rm{di}}\log \left( t \right): = \int_1^t {{{1ns} \over {1 - s}}ds,} \,\,\,\,t \ge 0.
Some examples for integral transform 𝒟Log (·, ·) related to the operator monotone functions are also provided.