# Jensen's Inequality (Complex Analysis)

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*This proof is about Jensen's Inequality in the context of Complex Analysis. For other uses, see Jensen's Inequality.*

## Theorem

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

Let $R > 0$ be such that $\map B {0, R} \subset D$.

Let $f: D \to \C$ be analytic with $\map f 0 \ne 0$.

Let $\cmod {\map f z} \le M$ for $\cmod z \le R$.

Let $0 < r <R$.

Then the number of zeroes of $f$, counted with multiplicity, for $\cmod z \le r$, is at most:

- $\dfrac {\map \log {M / \cmod {\map f 0} } } {\map \log {R / r} }$

## Proof

## Source of Name

This entry was named for Johan Jensen.

## Sources

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