Jensen's formula
/ Jensen -> YEN-sen /
Here is the bridge that makes all of value-distribution theory possible. On one side stands the size of a holomorphic function, measured by the average of log|f| around a circle. On the other side stand the zeros of f inside that circle — where f vanishes. Naively these seem unrelated: how big f is on the boundary, versus where it hits zero in the interior. Jensen's formula reveals they are rigidly linked by a single equation. The growth of |f| pays an exact, accountable price for every zero the function has.
The statement: let f be holomorphic on the closed disk |z| <= r with f(0) not zero, and let a_1, ..., a_n be its zeros inside the disk listed with multiplicity. Then the average of log|f(r e^(i theta))| over theta from 0 to 2 pi equals log|f(0)| plus the sum over the zeros of log(r / |a_k|). Equivalently, (1 / 2 pi) times the integral from 0 to 2 pi of log|f(r e^(i theta))| d theta = log|f(0)| + sum log(r / |a_k|). Each zero a_k contributes a positive term log(r / |a_k|) — positive because |a_k| < r — so more zeros, or zeros closer to the center, force the boundary average of log|f| to be larger. Rewriting the sum as an integral against the zero-counting function n(t) gives the form log M is controlled by the integral of n(t)/t dt, the workhorse inequality.
Jensen's formula is the engine behind the central theorem of this field: that an entire function of finite order rho cannot have too many zeros — the number of zeros out to radius r grows no faster than r^(rho + epsilon). Every Hadamard factorization, every Nevanlinna characteristic, every counting estimate ultimately traces back to this one identity. The hypothesis f(0) not zero is removable: if f has a zero of order m at the origin, factor out z^m first and apply the formula to what remains. The deep content is that growth and zeros are not independent — you cannot have a slowly growing function with wildly many zeros.
Take f(z) = z - a with 0 < |a| < r, a single zero at a. Jensen's formula predicts the boundary average of log|f| equals log|f(0)| + log(r/|a|) = log|a| + log(r/|a|) = log r. And indeed the average of log|r e^(i theta) - a| over the circle is exactly log r whenever |a| < r — a classical computation. The single zero is fully paid for by the term log(r/|a|).
One interior zero raises the boundary average of log|f| by exactly log(r/|a|) — growth pays for zeros.
The formula needs f holomorphic with no zeros ON the circle |z| = r (or the boundary integral becomes singular) and f(0) not zero; both are routine to arrange. It is an exact equality, not an inequality — every zero is accounted for to the penny.