# Equality holding in Cauchy's Inequality

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• September 21st 2012, 03:17 AM
Bingk
Equality holding in Cauchy's Inequality
Hello, this was a problem on an exam:

From Cauchy's Integral Formula, we get Cauchy's Inequality $|f^{(n)}(z_0)| \leq \frac{n!M}{R^n}$.
Prove that equality holds if and only if $f(z)= \frac{aMz^n}{R^n}$ for some $a \in \mathbb{C}$ with $|a| = 1$.

Proving $(\Leftarrow)$ is easy, just take the modulus of the nth derivative of f.
How do we prove the other direction?

Thanks!
• September 21st 2012, 03:45 AM
johnsomeone
Re: Equality holding in Cauchy's Inequality
What are M and R?
• September 21st 2012, 04:06 AM
Bingk
Re: Equality holding in Cauchy's Inequality
Ah, From Cauchy's Integral Formula and Inequality.
f is analytic on a ball of radius R
|f(z)| < M for all z in the ball
• September 21st 2012, 04:23 AM
johnsomeone
Re: Equality holding in Cauchy's Inequality
So the statement is:

Let $\Omega = \{z \in \mathbb{C} \ | \ \lVert z - z_0 \rVert < R \}$. Let $f \in H(\bar{\Omega})$ satisfy $\lVert f(z) \rVert \le M \ \forall z \in \Omega$.

Then $\lVert f^{(n)}(z_0) \rVert = \frac{n!M}{R^n} \Leftrightarrow \exists a \in \mathbb{C}, \lVert a \rVert = 1$, such that $f(z) = \frac{aMz^n}{R^n} \ \forall z \in \Omega$.

Correct?

Are you sure it isn't supposed to be $f(z) = \frac{aM(z-z_0)^n}{R^n}$ ?
• September 21st 2012, 04:53 AM
Bingk
Re: Equality holding in Cauchy's Inequality
What I wrote is what was on the exam, but I think that was justified by assuming that z0 = 0.

And yes, what you have is correct, as far as I understood the problem