In Bak-Newman's "Complex Analysis", there are two versions of Liouville's Theorem given:
1) A basic version: An entire function bounded by a constant $M$ is constant.
and
2) An extended version: An entire function $f(z)$ bounded by $A + B|z|^k$, for integer $k \ge 0$, is a polynomial of degree at most k.
I am wondering if in the extended version the exponent $k$ really needs to be an integer? Can the conclusion of the theorem be obtained (even with $k$ real but not an integer) by using the Cauchy Integral Formula and the $ML$ inequality for line integrals to show that the power series for $f(z)$ needs to be a polynomial?
Thanks!
Best Answer
$k$ need not be integer. If $f(z)$ is bounded by $A+B |z|^k$ then $f(z)$ is bounded by $\tilde{A}+\tilde{B} |z|^{ \lceil k \rceil}$ which means $f$ is a polinomial of degree $\leq \lceil k \rceil$