Phragmén-Lindêlöf Theorem

Let $f(z)$ be an Analytic Function in an angular domain $W:\vert\arg z\vert<\alpha\pi/2$. Suppose there is a constant $M$ such that for each $\epsilon>0$, each finite boundary point has a Neighborhood such that $\vert f(z)\vert<M+\epsilon$ on the intersection of $D$ with this Neighborhood, and that for some Positive number $\beta>\alpha$ for sufficiently large $\vert z\vert$, the Inequality $\vert f(z)\vert<\mathop{\rm exp}\nolimits (\vert z\vert^{1/\beta})$ holds. Then $\vert f(z)\vert\leq M$ in $D$.

References

Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 160, 1980.