Suppose that f is analytic in the unit disk D. If its range f(D) is contained in a simply connected proper subdomain of the plane, then the principle of subordination and the distortion theorem for univalent functions show that
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0305004100000268/resource/name/S0305004100000268eqnU3.gif?pub-status=live)
where M(r, f) denotes the maximum modulus of f. Cartwright (2) studied functions which, instead of omitting all values on a continuum stretching to infinity, omit only a sequence of values. She assumed that the sequence {wn} satisfies
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0305004100000268/resource/name/S0305004100000268eqn1.gif?pub-status=live)
and
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0305004100000268/resource/name/S0305004100000268eqn2.gif?pub-status=live)
and proved that if f(D) contains none of the points {wn} thenm
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0305004100000268/resource/name/S0305004100000268eqnU4.gif?pub-status=live)
for every ε > 0. Cartwright's proof was based on the Ahlfors Distortion Theorem, and is quite complicated. A much simpler proof was given by Pommerenke in (10). The key idea in his proof will also be used in the present paper.