If
[
e
−
t
A
]
[{e^{ - tA}}]
is a uniformly bounded
C
0
{C_0}
semigroup on a complex Banach space
X
X
, then
−
A
α
,
- {A^\alpha },
,
0
>
α
>
1
0 > \alpha > 1
, generates a holomorphic semigroup on
X
X
, and
[
e
−
t
A
α
]
[{e^{ - t{A^\alpha }}}]
is subordinated to
[
e
−
t
A
]
[{e^{ - tA}}]
through the Lévy stable density function. This was proved by Yosida in 1960, by suitably deforming the contour in an inverse Laplace transform representation. Using other methods, we exhibit a large class of probability measures such that the subordinated semigroups are always holomorphic, and obtain a necessary condition on the measure’s Laplace transform for that to be the case. We then construct probability measures that do not have this property.