JOVANA
Explore Library Glossary Getting Started Three Levels Fields How it works Mission
Join the mission
All guides

Jensen's Formula: Growth and Zeros

An entire function's zeros and its growth are not two separate facts — they are two faces of one accounting identity. Jensen's formula is the ledger that balances them: the more zeros a function packs into a disk, the faster |f| must grow to pay for them.

The question Jensen answers

In the previous guide you learned to measure how fast an entire function grows: the order rho captures the exponential scale of the maximum modulus M(r) = max over |z| = r of |f(z)|, and the type refines it. But growth is only half the life of an entire function. The other half is its zeros — where does f vanish, and how thickly? A polynomial of degree n grows like r^n and has exactly n zeros; the two numbers are the same n. That is no coincidence, and Jensen's formula is the law behind it: for any holomorphic function, growth and zeros are locked together by a single exact identity.

Here is the puzzle stated sharply. Suppose f is entire and you only get to watch |f| on a circle |z| = r — the boundary data. Can the boundary tell you anything about the zeros hiding inside that circle? Intuitively a zero is a place where |f| dives to 0, a deep valley; to climb back out to the circle's boundary, |f| has to rise, and the more valleys you dig the higher the rim must be. Jensen turns this hand-waving into an equation that holds to the last decimal. It is, in spirit, the mean value theorem for harmonic functions corrected for the zeros that would otherwise break it.

From the mean value property to the correction

Start with the clean case: a function with no zeros. If f is holomorphic and never zero on the closed disk |z| <= r, then log|f(z)| is a harmonic function there — it is the real part of the (locally defined) holomorphic log f, and real parts of holomorphic functions are harmonic. Harmonic functions obey the mean value property: their value at the center equals their average over any surrounding circle. So with no zeros around, log|f(0)| is exactly the average of log|f| over the circle of radius r. The center value is the boundary average, nothing more.

The repair is a beautiful trick: divide the zeros out. Suppose f has zeros at a_1, ..., a_n inside |z| < r (listed with multiplicity, so a double zero appears twice), and for simplicity f(0) is not 0. For each zero a_k, the Blaschke-type factor (r^2 - a_k-bar z)/(r(z - a_k)) is a gadget that cancels the zero at a_k yet — crucially — has modulus exactly 1 on the circle |z| = r. Multiply f by all of them and you build a function F with no zeros but the same boundary modulus as f. Apply the clean mean value property to F, then unwind the factors, and out drops the correction.

The formula itself

When the dust settles, here is what the bookkeeping says. The boundary average of log|f| exceeds the center value log|f(0)| by exactly a sum of one term per zero — and each term is the log of how much room that zero has, the ratio r/|a_k| of the circle's radius to the zero's distance from the center.

(1 / 2 pi) * integral from 0 to 2 pi of  log|f(r e^(i theta))| d theta

   =  log|f(0)|  +  sum over zeros a_k inside |z| < r  of  log( r / |a_k| )

Each factor r/|a_k| > 1, so every interior zero ADDS to the boundary average.
(If f(0) = 0, replace log|f(0)| by the leading Taylor-coefficient term.)
Jensen's formula. The left side is the average of log|f| around the circle of radius r. The right side splits it into the center value plus one positive contribution log(r/|a_k|) for each zero inside the disk, counted with multiplicity.

Read it as a budget. The boundary average of log|f| is the income; log|f(0)| is the baseline; every interior zero charges a positive fee log(r/|a_k|), large for zeros near the center (small |a_k|) and shrinking to 0 for zeros right on the rim. The equation says the income must cover the baseline plus all the fees, exactly. This is Jensen's formula, and it is an equality, not an estimate — astonishing for a statement that ties together two things, growth and zeros, that look so different. Notice the same minus signs we saw before: more zeros means a bigger right-hand sum, which forces a bigger left-hand average, which means |f| must be larger on the circle. Zeros literally cost growth.

Counting zeros, and the bridge to order

Now squeeze the formula for what it is worth. Let n(r) be the zero-counting function: the number of zeros of f in the disk |z| <= r, counted with multiplicity. A standard manipulation rewrites Jensen's sum over individual zeros as an integral of n(t)/t — this packages the whole formula in terms of how the zero count grows with radius. The upshot is a clean inequality: the average of log|f| on the circle, which is bounded above by log M(r), dominates the accumulated zero count. Growth controls zeros from above.

  1. Apply Jensen on the circle of radius 2r instead of r. The boundary average is at most log M(2r), since log|f| <= log M(2r) everywhere on that circle.
  2. On the right, throw away the zeros in the annulus r <= |a_k| <= 2r and keep only those with |a_k| <= r. Each of the n(r) kept zeros contributes at least log(2r / r) = log 2.
  3. Chain the two: n(r) log 2 <= log M(2r) - log|f(0)|. Hence n(r) <= (log M(2r) - log|f(0)|) / log 2 — the zero count is bounded by the growth of log M.
  4. Read off the consequence for order. If f has order rho, then log M(r) grows like r^rho, so n(r) grows no faster than r^rho. An entire function of finite order cannot have too many zeros: their density is capped by the growth rate.

This is the payoff that powers the rest of the rung. The zeros of a finite-order entire function are not allowed to pile up arbitrarily fast — their counting function is throttled by r^rho. That single bound is exactly the input the next guide needs: it guarantees that the infinite product over the zeros actually converges, which is what makes the Hadamard factorization theorem possible. Jensen is the quantitative heart; Hadamard is the structural conclusion that grows out of it.

A tiny worked check, and honest fine print

Test it on something you can see. Take f(z) = z, with its single zero at the origin — but Jensen as stated wants f(0) not 0, so use f(z) = z - a for a fixed a with |a| < r, a simple zero at z = a. The right side is log|f(0)| + log(r/|a|) = log|a| + log(r/|a|) = log r. The left side is the average of log|r e^(i theta) - a| over the circle. That average is a classic integral, and it equals exactly log r whenever |a| < r — a fact you can check by symmetry or by the mean value property applied to log|z - a| away from a. Both sides equal log r. The ledger balances to the last symbol.

Three honest caveats. First, Jensen's formula needs f to have no zeros on the circle |z| = r itself; a zero exactly on the boundary makes log|f| spike to minus infinity right on the path of integration. The fix is to nudge r slightly, and almost every r works since zeros are isolated, by the isolation of zeros. Second, the case f(0) = 0 is handled, not excluded: you factor out the leading z^m and the term log|f(0)| is replaced by log of the first nonzero Taylor coefficient plus m log r. Third — and this is the deep one — Jensen counts only zeros, where f hits the single value 0. The grand generalization that counts how often f hits every value w, weighting growth against value-attainment for all w at once, is Nevanlinna's characteristic, the climax of this rung. Jensen is the seed of the entire theory of value distribution.