An entire function is identically zero?

Solution 1:

Since $f$ is entire it can be expressed as a power series that converges everywhere: $f(z) = \sum_{n=0}^\infty a_n z^n$. From $|f(0)| \leq 0$ we know that $f(0)=0$, hence $a_0 = 0$. So $g(z) := f(z)/z$ can be continued to an entire function that satisfies $$|g(z)| \leq \frac{\log(1+|z|)}{|z|} \quad \text{for all } z\neq 0.$$ The right side converges to zero for $|z| \to \infty$, in particular it is bounded. By Liouville's theorem, $g$ is constantly zero and so is $f$.

Solution 2:

We will prove a slightly more genaral fact. The proof is based on this answer

Theorem. Let $f\in\mathcal{O}(\mathbb{C})$ and for all $z\in\mathbb{C}$ we have $|f(z)|\leq\varphi(|z|)$. Assume that $$ \lim\limits_{R\to+\infty}\frac{\varphi(R)}{R^{p+1}}=0 $$ then $f$ is a polynimial with $\deg (f)\leq p$.

Proof. Since $f\in\mathcal{O}(\mathbb{C})$, then $$ f(z)=\sum\limits_{n=0}^\infty c_n z^n $$ for all $z\in \mathbb{C}$. Moreover, for all $R>0$ we have integral representation for coefficients $$ c_n=\int\limits_{\partial B(0,R)}\frac{f(z)}{z^{n+1}}dz $$ Then, we get an estiamtion $$ |c_n|\leq \oint\limits_{\partial B(0,R)}\frac{|f(z)|}{|z|^{n+1}}|dz|\leq \oint\limits_{\partial B(0,R)}\frac{\varphi(|z|)}{|z|^{n+1}}|dz|= \frac{2\pi R\varphi(R)}{R^{n+1}}= \frac{2\pi \varphi(R)}{R^{n}} $$ Hence for $n>p$ we obtain $$ |c_n|\leq\lim\limits_{R\to+\infty}\frac{2\pi\varphi(R)}{R^n}= 2\pi\lim\limits_{R\to+\infty}\frac{1}{R^{n-p-1}}\lim\limits_{R\to+\infty}\frac{ R\varphi(R)}{R^{p+1}}=0 $$ which implies $c_n=0$ for $n>p$. Finally we get $$ f(z)=\sum\limits_{n=0}^p c_n z^n+\sum\limits_{n=p+1}^\infty c_n z^n=\sum\limits_{n=0}^p c_n z^n $$ This means that $f$ is a polynimial with $\deg(f)\leq p$.

For your particular problem $\varphi(R)=\log(1+R)$, and it is easy to check that $$ \lim\limits_{R\to+\infty}\frac{\varphi(R)}{R}=0 $$ Hence, $f(z)=c_0$ is a constant function. Moreover, $$ |c_0|=|f(0)|\leq\log(1+|0|)=0 $$ so $c_0=0$ and $f(z)=0$ for all $z\in\mathbb{C}$.

Solution 3:

We already know that $f(0) = 0$. Now from $|f(z)| \leq \log(1 + |z|)$ and we get that on every circle of radius $R$ about the origin, $$|f^{(n)}(0)| \leq \frac{\log(1 + |R|) \times n!}{R^n} \hspace{5mm} \forall n \geq 1$$ by the Cauchy estimate. Since $R^n$ grows faster than $\log (1+ |R|)$ when $n \geq 1$, we see that $f^{(n)}(0) = 0$ for all $n \geq 1$. But we also know $f(0) = 0$ so that $f^{(n)}(0) = 0$ for all $n \geq 0$. Since $f$ is entire the identity principle implies $f \equiv 0$.