# Math Help - cauchy inequalities [complex analysis]

1. ## cauchy inequalities [complex analysis]

hello all,..

i need help with this problem

If $f$ is a holomorphic function on the strip $-1 < y < 1, x \in \mathbb{R}$ with $|f(z)| \leq A (1 + |z|)^{\lambda}$, where $\lambda$ is a fixed real number, for all z in that strip.

show that for each integer $n \geq 0$ there exist $A_n \geq 0$ s.t

$|f^{(n)}(x)| \geq A_n(1 + |x|)^{\lambda}$ for all $x \in \mathbb{R}$.

my uncomplete solution :

fixed n. for any $x_0 \in \mathbb{R}$, we can make a circle $C$centered at $x_0$ with radius $R < 1$.

by cauchy inequalities, we have

$|f^{(n)}(x_0)| \leq \frac{n!||f||_C}{R^n}$

with $||f||_C = \sup_{z \in C}|f(z)|$.

let $\sup_{z \in C}|f(z)| = |f(z_0)|$

because $|f(z_0)| \leq A (1 + |z_0|)^{\lambda}$, then

$|f^{(n)}(x_0)| \leq \frac{n!A (1 + |z_0|)^{\lambda}}{R^n}$

what should I do next?

thx for any comment

2. Originally Posted by dedust
hello all,..

i need help with this problem

If $f$ is a holomorphic function on the strip $-1 < y < 1, x \in \mathbb{R}$ with $|f(z)| \leq A (1 + |z|)^{\lambda}$, where $\lambda$ is a fixed real number, for all z in that strip.

show that for each integer $n \geq 0$ there exist $A_n \geq 0$ s.t

$|f^{(n)}(x)| \geq A_n(1 + |x|)^{\lambda}$ for all $x \in \mathbb{R}$.

my uncomplete solution :

fixed n. for any $x_0 \in \mathbb{R}$, we can make a circle $C$centered at $x_0$ with radius $R < 1$.

by cauchy inequalities, we have

$|f^{(n)}(x_0)| \leq \frac{n!||f||_C}{R^n}$

with $||f||_C = \sup_{z \in C}|f(z)|$.

let $\sup_{z \in C}|f(z)| = |f(z_0)|$

because $|f(z_0)| \leq A (1 + |z_0|)^{\lambda}$, then

$|f^{(n)}(x_0)| \leq \frac{n!A (1 + |z_0|)^{\lambda}}{R^n}$

what should I do next?

thx for any comment

Cauchy inequalities

If $f$ is holomorphic in an open set that contains the closure of a disc $D$ centered at $z_0$ and of radius $R$, then

$|f(z_0)| \leq \frac{n!||f||_C}{R^n}$,

where $||f||_C = \sup_{z \in C} |f(z)|$ denotes the supremum of $|f|$ on the boundary circle $C$