Suppose
T
T
is a one-parameter semigroup of bounded linear operators on a Banach space, strongly continuous on
[
0
,
∞
)
[0,\infty )
. It is known that
lim
sup
x
→
0
|
T
(
x
)
−
I
|
>
2
\lim {\sup _{x \to 0}}|T(x) - I| > 2
implies
T
T
is holomorphic on
(
0
,
∞
)
(0,\infty )
. Theorem I is a generalization of this as follows: Suppose
M
>
0
,
0
>
r
>
s
M > 0,0 > r > s
, and
ρ
\rho
is in (1,2). If
|
(
T
(
h
)
−
I
)
n
|
≤
M
ρ
n
|{(T(h) - I)^n}| \leq M{\rho ^n}
whenever
n
h
nh
is in
[
r
,
s
]
,
n
=
1
,
2
,
⋯
,
h
>
0
[r,s],n = 1,2, \cdots ,h > 0
, then there exists
b
>
0
b > 0
such that
T
T
is holomorphic on
[
b
,
∞
)
[b,\infty )
. Theorem II shows that, in some sense,
b
→
0
b \to 0
as
r
→
0
r \to 0
. Theorem I is an application of Theorem III: Suppose
M
>
0
,
0
>
r
>
s
,
ρ
M > 0,0 > r > s,\rho
is in (1,2), and
f
f
is continuous on
[
−
4
s
,
4
s
]
[ - 4s,4s]
. If
|
∑
q
=
0
n
(
n
q
)
(
−
1
)
n
−
q
f
(
t
+
q
h
)
|
≤
M
ρ
n
|\sum _{q = 0}^n \binom {n}{q}( - 1)^{n - q} f(t + qh)| \leq M \rho ^n
whenever
n
h
nh
is in
[
r
,
s
]
[r,s]
,
n
=
1
n = 1
,
2
2
, …,
h
>
0
h > 0
,
[
t
,
t
+
n
h
]
⊂
[
−
4
s
,
4
s
]
[t,t + nh] \subset [ - 4s,4s]
, then
f
f
has an analytic extension to an ellipse with center zero. Theorem III is a generalization of a theorem of Beurling in which the inequality on the differences is assumed for all
n
h
nh
. An example is given to show the hypothesis of Theorem I does not imply
T
T
holomorphic on
(
0
,
∞
)
(0,\infty )
.