For a given function
k
k
, positive, continuous, nondecreasing and unbounded on
[
0
,
1
)
[0,1)
, let
A
(
k
)
{A^{(k)}}
denote the class of functions regular in the unit disc for which log
|
f
(
z
)
|
>
k
(
|
z
|
)
|f(z)| > k(|z|)
when
|
z
|
>
1
|z| > 1
. Hayman and Korenblum have shown that a necessary and sufficient condition for the sets of positive zeros of all functions in
A
(
k
)
{A^{(k)}}
to be Blaschke is that
\[
∫
0
1
(
k
(
t
)
/
(
1
−
t
)
)
d
t
\int _0^1 {\sqrt {(k(t)/(1 - t))\,dt} }
\]
is finite. It is shown that the imposition of a further regularity condition on the growth of
k
k
ensures that in some tangential region the zero set of each function in
A
(
k
)
{A^{(k)}}
is also Blaschke.