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

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

    Quote Originally Posted by fareastmovement View Post
    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.
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    Re: Problem about Complex Analysis (Maximum Modulus Principle and Cauchy's Inequality

    Thank you very much
    Didnt think it was that easy
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