Bound for Analytic Function and Derivatives

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Lemma

Let $f$ be a complex function.

Let $z_0$ be a point in $\C$.

Let $r$ be a real number in $\R_{>0}$.

Let $\Gamma$ be a circle in $\C$ with center at $z_0$ and radius $r$.

Let $f$ be analytic on $\Gamma$ and its interior.

Let $t \in \C$ be such that $\cmod {t - z_0} < r$.


Then a real number $M$ exists such that, for every $n \in \N$:

$\displaystyle \cmod {\map {f^{\paren n} } t} \le \frac {M r \, n!} {\paren {r - \cmod {t - z_0} }^\paren {n + 1} }$


Proof

Lemma (Analytic Function Bounded on Circle)

Let $f$ be a complex function.

Let $z_0$ be a point in $\C$.

Let $\Gamma$ be a circle in $\C$ with center at $z_0$ and radius in $\R_{>0}$.

Let $f$ be analytic on $\Gamma$.


Then $f$ is bounded on $\Gamma$.


We have:

$f$ is analytic on $\Gamma$ and its interior
$t$ is in the interior of $\Gamma$

Therefore:

$\displaystyle \map {f^{\paren n}} t = \frac {n!} {2 \pi i} \int_\Gamma \frac {\map f z} {\paren {z - t}^{\paren {n + 1} } } \rd z$ by Cauchy's Integral Formula for Derivatives

where $\Gamma$ is traversed counterclockwise.


We have that $f$ is bounded on $\Gamma$ by Lemma (Analytic Function Bounded on Circle).

Therefore, there is a positive real number $M$ that satisfies:

$M \ge \cmod {\map f z}$ for every $z$ on $\Gamma$


We have $\cmod {t - z_0} < r$.

Therefore:

$0 < r - \cmod {t - z_0}$

We observe that $r - \cmod {t - z_0}$ is the minimum distance between $t$ and $\Gamma$.

Therefore:

$\paren {r - \cmod {t - z_0}} \le \cmod {z - t}$ for every $z$ on $\Gamma$

We get:

\(\displaystyle \cmod {\map {f^{\paren n} } t}\) \(=\) \(\displaystyle \cmod {\frac {n!} {2 \pi i} \int_\Gamma \frac {\map f z} {\paren {z - t}^{\paren {n + 1} } } \rd z}\)
\(\displaystyle \) \(\le\) \(\displaystyle \frac {n!} {2 \pi} \int_\Gamma \frac {\cmod {\map f z} } {\cmod {z - t}^{\paren {n + 1} } } \cmod {\d z}\)
\(\displaystyle \) \(\le\) \(\displaystyle \frac {n!} {2 \pi} \int_\Gamma \frac M {\cmod {z - t}^{\paren {n + 1} } } \cmod {\d z}\) as $M \ge \cmod {\map f z}$ for every $z$ on $\Gamma$
\(\displaystyle \) \(\le\) \(\displaystyle \frac {n!} {2 \pi} \int_\Gamma \frac M {\paren {r - \cmod {t - z_0} }^{\paren {n + 1} } } \cmod {\d z}\) as $0 < \paren {r - \cmod {t - z_0} } \le \cmod {z - t}$ for every $z$ on $\Gamma$
\(\displaystyle \) \(=\) \(\displaystyle \frac {n!} {2 \pi} \frac M {\paren {r - \cmod {t - z_0} }^{\paren {n + 1} } } \int_\Gamma \cmod {\d z}\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {n!} {2 \pi} \frac M {\paren {r - \cmod {t - z_0} }^{\paren {n + 1} } } 2 \pi r\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {M r \, n!} {\paren {r - \cmod {t - z_0} }^{\paren {n + 1} } }\)

$\blacksquare$