Let $f(z)$ be entire. Suppose there exists $M >0$ and sequence $\{R_n\}$ of positive real number tending to $\infty$ such that $f(z) \neq 0$ and $|z|=R_n,$ such that $\begin{align} \int_{|z|=R_n} \left|\dfrac{f'(z)}{f(z)} \right||dz|<M, \forall n \end{align}.$
Could anyone advise me on the correct approach to show $f$ is a polynomial? Thank you very much.
First, show that $f$ has only a finite number of zeroes. This is easy, since the number of zeroes inside the circle of radius $R$ (counting multiplicities) is given by an integral which is easily estimated using the given data. (This is the argument principle, as pointed out in a a comment.)
Now, divide $f$ by the polynomial having the same zeroes as $f$ does. I think you'll find that the quotient $g$ satisfies the same sort of estimate as $f$ did.
Next, since $g$ has no zeroes, it has an entire logarithm, i.e., an entire function $h$ exists so that $g=e^h$. Now note that $g'/g)h'$, and show that $h'=0$ by representing $h'$ using Cauchy's integral theorem.
