Does there exist a non-constant entire function $f$ such that $|f(z^3)| \leq 1 + |z|$ for all $z$?

In this problem not able to use Liouville's theorem here?

  • $\begingroup$ Use the Cauchy estimates, and rewrite the condition as $\lvert f(w)\rvert \leqslant 1 + \lvert w\rvert^{1/3}$ for all $w\in \mathbb{C}$. $\endgroup$ – Daniel Fischer Oct 18 '16 at 19:18

Since $f$ is entire it has a power series expansion around zero. Let $$f(z)=\sum_{k=0}^\infty C_kz^k$$ be the power series expansion.

On a disc of radius $R$ we have $|f(z)|\leq (1+|z|^{1/3})\leq 1+R^{1/3}$. Thus by Cauchy estimate $$|C_k|\leq\frac{1+R^{1/3}}{R^k}.$$ This is true for any $R$ and hence $C_k=0$ for any $k\geq 1$. Therefore $f$ is constant.

In fact this same argument shows that, on $f$ if we have a bound like $|f(z)|\leq p(|z|)$, where $p$ is a polynomial of degree $n$, then $f$ is of degree at most $n$.


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