If
f
(
z
)
f(z)
is an asymmetric entire function of exponential type
τ
\tau
,
\[
‖
f
‖
=
sup
−
∞
>
x
>
∞
|
f
(
x
)
|
,
\left \| f \right \| = \sup \limits _{ - \infty > x > \infty } |f(x)|,
\]
then according to a well-known result of R. P. Boas,
\[
‖
f
′
‖
≤
τ
2
‖
f
‖
\left \| {f’} \right \| \leq \frac {\tau }{2}\left \| f \right \|
\]
and
\[
|
f
(
x
+
i
y
)
|
≤
(
e
τ
|
y
|
+
1
)
2
‖
f
‖
,
−
∞
>
x
>
∞
,
−
∞
>
y
≤
0.
|f(x + iy)| \leq \frac {{({e^{\tau |y|}} + 1)}}{2}\left \| f \right \|,\quad - \infty > x > \infty , - \infty > y \leq 0.
\]
Both of these inequalities are sharp. In this paper we generalize the above two inequalities of Boas by proving a sharp inequality which, besides giving as special cases the above two inequalities of Boas, yields some other results as well.