On the radial limits of analytic and meromorphic functions

Early in the fifties, A. J. Lohwater proved that if $f(z)$ is analytic in $|z| < 1$ and has the radial limit $0$ almost everywhere on $|z| = 1$, then every complex number $\zeta$ is an asymptotic value of $f(z)$ provided the $\zeta$-points satisfy the following Blaschke condition: $\sum _{k = 1}^\infty (1 - |{z_k}|) < \infty$, where $f({z_k}) = \zeta$, $k = 1 ,2, \ldots$. We may, therefore, ask under the hypothesis on $f(z)$ how many complex numbers $\zeta$ are there whose $\zeta$-points can satisfy the Blaschke condition. We show that there is at most one such number and this one number phenomenon can actually occur if the number is zero.