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

Let$\displaystyle f : \mathbb{C}\rightarrow \mathbb{C} $ be an entire function such that $\displaystyle \lim_{z\rightarrow \infty} \frac{f(z)}{z}=0$

Show that $\displaystyle f$ is a constant.

$\displaystyle \noindent\rule{\textwidth}{0.5pt}$

I don't know if $\displaystyle \frac{f(z)}{z}$ is analytic at $\displaystyle z = 0$ , 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.

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

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**fareastmovement** Let$\displaystyle f : \mathbb{C}\rightarrow \mathbb{C} $ be an entire function such that $\displaystyle \lim_{z\rightarrow \infty} \frac{f(z)}{z}=0$

Show that $\displaystyle f$ is a constant.

$\displaystyle \noindent\rule{\textwidth}{0.5pt}$

I don't know if $\displaystyle \frac{f(z)}{z}$ is analytic at $\displaystyle z = 0$ , 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 \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 \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 \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 \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 \displaystyle \begin{align*} \lim_{z \to \infty}\frac{f(z)}{z} = 0 \end{align*} $ is if $\displaystyle \displaystyle \begin{align*} f(z) \end{align*} $ is constant.

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

Thank you very much(Happy)

Didnt think it was that easy