# Thread: Problem about Complex Analysis (Maximum Modulus Principle and Cauchy's Inequality)

1. ## Problem about Complex Analysis (Maximum Modulus Principle and Cauchy's Inequality)

Let $f : \mathbb{C}\rightarrow \mathbb{C}$ be an entire function such that $\lim_{z\rightarrow \infty} \frac{f(z)}{z}=0$
Show that $f$ is a constant.
$\noindent\rule{\textwidth}{0.5pt}$

I don't know if $\frac{f(z)}{z}$ is analytic at $z = 0$ , therefore I cannot use the Maximum Modulus Principle.On the other hand, I cannot prove that the maximum value exists, therefore I cannot use the Cauchy's Inequality.

Thank you.

2. ## Re: Problem about Complex Analysis (Maximum Modulus Principle and Cauchy's Inequality

Originally Posted by fareastmovement
Let $f : \mathbb{C}\rightarrow \mathbb{C}$ be an entire function such that $\lim_{z\rightarrow \infty} \frac{f(z)}{z}=0$
Show that $f$ is a constant.
$\noindent\rule{\textwidth}{0.5pt}$

I don't know if $\frac{f(z)}{z}$ is analytic at $z = 0$ , therefore I cannot use the Maximum Modulus Principle.On the other hand, I cannot prove that the maximum value exists, therefore I cannot use the Cauchy's Inequality.

Thank you.
If f is an entire function, then it can be written as a Taylor series \displaystyle \begin{align*} f(z) = C_0 + C_1z + C_2z^2 + C_3z^3 + \dots \end{align*}. If only the first term of the Taylor series is nonzero (i.e. you have a constant), then you will have \displaystyle \begin{align*} \lim_{z \to \infty}\frac{C_0}{z} = 0 \end{align*}.

If the first two terms are nonzero, then you will have

\displaystyle \begin{align*} \lim_{z \to \infty}\frac{C_0 + C_1z}{z} = \lim_{z \to \infty}\frac{C_0}{z} + \lim_{z \to \infty}C_1 = 0 + C_1 = C_1 \end{align*}.

If any other terms are nonzero, then you will have

\displaystyle \begin{align*} \lim_{z \to \infty}\frac{C_0 + C_1z + C_2z^2 + C_3z^3 + \dots}{z} = \lim_{z \to \infty} \frac{C_0}{z} + \lim_{z \to \infty}C_1 + \lim_{z \to \infty}z\left( C_2 + C_3z + C_4z^2 + \dots\right) = 0 + C_1 + \infty = \infty \end{align*}

So the only way for \displaystyle \begin{align*} \lim_{z \to \infty}\frac{f(z)}{z} = 0 \end{align*} is if \displaystyle \begin{align*} f(z) \end{align*} is constant.

3. ## Re: Problem about Complex Analysis (Maximum Modulus Principle and Cauchy's Inequality

Thank you very much
Didnt think it was that easy