Problem 7

Exercise 7.7 (Coercive estimate on entire functions)   If $f(z)$ is an entire function such that $\vert f(z)\vert\to\infty$ as $\vert z\vert\to\infty$, then find constants $c>0$ and $R>0$ such that $\vert f(z)\vert>c\vert z\vert$ for all $\vert z\vert>R$.

Proof. First we see that $f$ is a polynomial, because $f$ is entire and $\vert f(z)\vert\to\infty$ as $\vert z\vert\to\infty$. Set

$$f(z) := a_0 + a_1z + \cdots + a_nz^n$$ (7.27)

The following reverse triangle inequality needs no absolute value

$$\vert f(z)\vert = \vert a_nz^n + a_{n-1}z^{n-1} + \cdots + a_0\vert$$ (7.28)

$$\geq \vert\vert a_nz^n\vert - \vert a_{n-1}z^{n-1} + \cdots + a_0\vert\vert$$ (7.29)

when $z$ satisfies

$$\vert a_nz\vert > R > \vert a_{n-1}\vert + \vert a_{n-2}\vert + \cdots + \vert a_0\vert$$ (7.30)

because

$$\vert a_{n-1}z^{n-1} + \cdots + a_0\vert \leq \sum_{k=n-1}^0 \vert a_k z^k\vert \leq \sum_{k=n-1}^0 \vert a_k z^{n-1}\vert$$ (7.31)

$$= \vert z^{n-1}\vert \sum_{k=0}^{n-1} \vert a_k\vert$$ (7.32)

$$\leq \vert z^{n-1}\vert\vert a_nz\vert$$ (7.33)

$$=\vert a_nz^n\vert$$ (7.34)

Therefore,

$$\vert f(z)\vert \geq \vert a_nz^n\vert - \vert a_{n-1}z^{n-1} + \cdots + a_0\vert$$ (7.35)

$$\geq \vert a_nz^n\vert - \sum_{k=0}^{n-1} \vert a_kz^k\vert$$ (7.36)

$$= \sum_{k=0}^{n-1}\frac{\vert a_nz^n\vert}{n} - \vert a_kz^k\vert$$ (7.37)

$$= \sum_{k=0}^{n-1}\vert z^n\vert\left(\frac{\vert a_n\vert}{n} - \frac{\vert a_k\vert}{\vert z^{n-k}\vert}\right)$$ (7.38)

For each $k=0,\dots,n-1$, select $R_k>R$ such that $\vert z\vert>R$ implies

$$\frac{\vert a_n\vert}{n} - \frac{\vert a_k\vert}{\vert z^{n-k}\vert} > \frac{\vert a_n\vert}{2n}$$ (7.39)

Then if $\vert z\vert>\max\{R_k\}$, we know

$$\vert f(z)\vert > \sum_{k=0}^{n-1} \vert z^n\vert \frac{\vert a_n\vert}{2n} > \vert z\vert \frac{\vert a_n\vert}{2}$$ (7.40)

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