Let f be an entire function. Suppose there exist constants A, B such that $| f(z)| ≤ A + B|z|^\frac{1}{7}$ for every z ∈ C. Prove that f is constant.
I am unsure whether I have proved this correctly so any feedback would be greatly appreciated. My proof is as follows:
Here $| f(z)| ≤ A + B|z|^\frac{1}{7}$ for every z ∈ C so $A + B|z|^\frac{1}{7}$ is a bound for f. Liouville's theorem states that if f is entire and bounded then f is a constant, which we will prove now.
Let $a ∈ C$ and let $r>0$. As $| f(z)| ≤ A + B|z|^\frac{1}{7}$ we know that $| f'(a)| ≤ \frac{A + B|z|^\frac{1}{7}}{r}$ for every $r>0$. Hence $f'(a)=0$ for every $a ∈ C$.
Now we will show that any function with zero derivative is constant to complete the proof.
Let $z ∈ C$, then by the fundamental theorem of calculus: $f(z) - f(0)=\int_0^z f'(w)dz=0$ so $f(z)=f(0)$ is constant.