# Jensen's Inequality (Complex Analysis)

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*This proof is about Jensen's Inequality in complex analysis. For other uses, see Jensen's Inequality.*

## Contents

## Theorem

Let $D \subset \C$ be an open set with $0 \in D$.

Let $R > 0$ be such that $B \left({0, R}\right) \subset D$.

Let $f: D \to \C$ be analytic with $f \left({0}\right) \ne 0$.

Let $\left\lvert{f \left({z}\right)}\right\rvert \le M$ for $\left\lvert{z}\right\rvert \le R$.

Let $0 < r <R$.

Then the number of zeroes of $f$, counted with multiplicity, for $\left\lvert{z}\right\rvert \le r$, is at most:

- $\dfrac {\log \left({M / \left\lvert{f \left({0}\right)}\right\rvert}\right)} {\log \left({R / r}\right)}$

## Proof

## Source of Name

This entry was named for Johan Jensen.

## Sources

- 2006: H. L. Montgomery and R. C. Vaughan:
*Multiplicative Number Theory: I. Classical Theory*... (next) $6$: The Prime Number Theorem: $\S6.1$: A zero-free region: Lemma $6.1$