Jensen's formula ($\log |f(0)|+{\sum }_{|a_j|<R}\log (R/|a_j|)=\frac{1}{2\pi }{\int }_0^{2\pi }\log |f(R{e}^{i\theta })|d\theta$)

For a holomorphic function nonzero at zero and without boundary zeros, the formula relates its interior zeros, counted by multiplicity, to the circle mean of its logarithmic modulus. Divide out the finite interior zero factors; the remaining logarithmic modulus is harmonic, and each zero contributes $\log (R/|a_j|)$ by the harmonic mean-value formula. Boundary radii follow by limits where appropriate.

Every zero in the smaller disk contributes at least $\log (R/r)$ in Jensen's formula. Bounding the outer circle average by its maximum proves the estimate. With $R=2r$, an exponential $O(r\log r)$ maximum bound gives an $O(r\log r)$ zero count.