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

Let $R>0$ and let $f$ be a holomorphic function on an open neighbourhood of the closed disk $\{z:|z|\le R\}$. Assume $f(0)\ne 0$ and that $f$ has no zeros of a function on the boundary circle $|z|=R$. List its interior zeros of a function as $a_1,\dots ,a_N$, repeating each according to its multiplicity of a zero. With the natural logarithm, Jensen's formula statesThe empty sum is zero. This boundary-zero-free version is the holomorphic special case of the general identity in the source.^[ref]

Every interior zero of a function makes a positive contribution; a zero nearer the centre contributes more. Consequently boundary growth relative to $|f(0)|$ limits how many zeros of a function can lie well inside the disk. The Jensen zero-count bound makes this quantitative, while the integrated zero count retains the individual radial weights. The boundary statistic here is the circle log-mean of a holomorphic function; exponentiating gives a circle geometric mean of a holomorphic function.

Under the hypotheses of Jensen's formula, the circle log-mean is the angular average of the natural logarithm of the complex modulus. The factor $1/(2\pi )$ normalizes the integral to a mean. It averages values at equally weighted angles on the circle, and Jensen's formula computes it from $|f(0)|$ and the interior zeros of a function.

The continuous angular geometric mean of the positive boundary complex modulus isHere the empty product is one. This differs from the angular arithmetic mean $A_f(R)=(2\pi {)}^{-1}{\int }_0^{2\pi }|f(R{e}^{i\theta })|d\theta$. In general $G_f(R)\le A_f(R)$ by the concavity of the logarithm; equality requires constant boundary complex modulus. For a holomorphic function with no zeros of a function in the closed disk, Jensen's formula gives $G_f(R)=|f(0)|$, even when the boundary complex modulus varies with angle. Thus replacing the mean of a logarithm by the logarithm of an arithmetic mean loses the exact zero identity.

For a polynomial $p(z)=c{\prod }_j(z-a_j)$ with $c\ne 0$, no polynomial root at zero, and $R>0$ different from every $|a_j|$, Jensen's formula gives the displayed expression. The polynomial roots are repeated according to their multiplicity of a zero: an exterior root contributes $\log |a_j|$, whereas an interior root contributes $\log R$.

For an explicit example, take the polynomialIts two simple zeros are $1/4$ and $-1/2$. The circle log-mean of a holomorphic function is thereforeAt $R=0.2$ the open disc contains no zeros of a function, at $R=0.4$ it contains $1/4$ only, and at $R=0.75$ it contains both. These three circles avoid the polynomial roots; the radii $1/4$ and $1/2$ are excluded from the displayed version.

The corresponding values of the circle log-mean of a holomorphic function and the circle geometric mean of a holomorphic function areFor example, at $R=0.4$ the contribution of the sole interior zero of a function is $\log (0.4/0.25)=\log (8/5)$; adding it to $\log (3/8)$ gives $\log (3/5)$. The second polynomial root still affects $f(0)$, but contributes no additional interior-zero term at this radius.

Fix a holomorphic function on a neighbourhood of a closed disk, with $f(0)\ne 0$. On an interval of positive radii whose circles contain no zeros of a function, differentiating Jensen's formula with respect to $x=\log R$ givesEach interior zero of a function contributes slope one for each unit of its multiplicity of a zero. Between consecutive zero moduli, $L_f(e^x)$ is an affine function; passing the modulus of a zero of multiplicity of a zero $m$ increases the adjacent slopes by $m$. The finite sum $\log |f(0)|+{\sum }_j\max (0,x-\log |a_j|)$ defines a continuous convex function of $x$ whose values agree with the circle log-mean of a holomorphic function at every boundary-zero-free radius. This describes the adjacent line segments without asserting a boundary-singularity extension of the integral.

For a nonzero holomorphic function, $n_f(r)$ counts its zeros of a function in the centred open disc of radius $r$, with multiplicity of a zero. Thus a double zero counts twice, even though it is a single point. On every smaller closed disk in the domain the count is finite. The strict inequality $|a|<r$ specifies the convention at a zero modulus; elsewhere it agrees with counting in the closed disk. In the polynomial circle log-mean example the count is zero, one and two on the three displayed intervals.

Under the hypotheses of Jensen's formula, $f(0)\ne 0$ ensures that the radial zero-counting function vanishes for sufficiently small positive $t$. Integrating each zero's contribution separately givesThe values of $n_f$ at its finitely many jump points do not change the integral. For $0<r<R$, every zero in the smaller open disc supplies at least $\log (R/r)$ per unit of multiplicity of a zero, soThe second inequality bounds the circle log-mean of a holomorphic function by the logarithm of the largest boundary complex modulus, recovering the Jensen zero-count bound. An upper bound on boundary growth and a fixed nonzero centre value therefore constrain interior zeros. The factor $\log (R/r)$ also explains why the same estimate becomes weaker as the smaller radius approaches the boundary.

Divide $f$ by its finitely many interior zero factors, with each factor repeated according to its multiplicity of a zero. The resulting holomorphic function $g$ has no zeros of a function in the closed disk, so $\log |g|$ is a harmonic function and its circle average is $\log |g(0)|$ by the mean value property for harmonic functions. For each removed factor $z-a$, with $0<|a|<R$, the circle average of $\log |R{e}^{i\theta }-a|$ is $\log R$: factor out $R{e}^{i\theta }$ and apply the mean value property for harmonic functions to $\log |1-(a/R)w|$ on a neighbourhood of the closed unit disk. Reassembling the factors gives $\log (R/|a|)$ for each zero, proving Jensen's formula.

For a meromorphic function with no zero or pole on $|w|=R$, set $G_R(z,a)=\log |(R^2-\overline{a}z)/(R(z-a))|$. At source points that are neither zeros nor poles, the displayed identity expresses its logarithmic modulus as the Poisson integral of its boundary values, minus its zero Green potentials plus its pole Green potentials. The Poisson kernel is $(R^2-|z{|}^2)/|R{e}^{i\theta }-z{|}^2$ with normalized angular measure. Subtracting the corresponding logarithmic singularities leaves a harmonic function, so the harmonic Poisson formula proves the identity. At zero it reduces to Jensen's formula.

For a meromorphic function near $0<|z|\le 1$, and a target $a$ off its image of the unit circle, let $u_a=-\log k(f,a)$ for the chordal metric, $M_a(s)$ its normalized angular mean, and $I(a)=M_a^{\prime }(1)$. The weighted spherical area $T(r)=(4\pi {)}^{-1}{\int }_{r<|z|<1}\log (|z|/r)(f_{\mathrm{round}}^{\#}{)}^{2}dA$ satisfies the displayed formula, where $N$ counts $a$-points with multiplicity and weight $\log (|z|/r)$ and $\overline{m}=M_a(r)-M_a(1)$. Indeed $\Delta u_a=(f_{\mathrm{round}}^{\#}{)}^2/2-2\pi \sum {\mathrm{deg}}_z(f){\delta }_z$, and Green's second identity with the radial logarithmic weight proves the formula. The angular mean must include $1/(2\pi )$.

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.

A nonzero entire function vanishing at every nonzero point of the integer lattice has at least a constant times $R^2$ zeros in the disc of radius $R/2$. Each contributes at least $\log 2$ to Jensen's formula at radius $R$. Therefore the circle mean of $\log |F|$, and hence the logarithm of its maximum modulus, is bounded below by $c{R}^2$ for all sufficiently large boundary-zero-free radii. Choosing maximizing points yields $|F(z_j)|>e^{c^{\prime }|z_j{|}^2}$ for some $c^{\prime }>0$ and $|z_j|\to \infty$. If zero is also a zero, first factor out its finite multiplicity; the additional $m\log R$ term does not weaken the quadratic lower bound.