A conformal map can compress the plane so violently near the edge of a
domain that its derivative behaves like a large inverse power of the
distance to that edge. Brennan's conjecture bounds how strong this
compression can be. For a simply connected domain W in the plane and a
conformal bijection phi from W onto the open unit disk D, the conjecture
asserts that the derivative satisfies

    integral over W of |phi'(z)|^s  dA(z)  <  infinity   whenever  4/3 < s < 4 ,

with no hypothesis of any kind on the boundary of W and no hypothesis that
W be bounded. The endpoints 4/3 and 4 are genuinely lost: a single
example, the slit map, diverges there, and that divergence is part of the
subject rather than an accident of the proof. The problem was raised in
this form by James Brennan, whose paper improved the previously known
upper exponent beyond 3 and stated the full interval.

https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms/s2-18.2.261
https://www.mathnet.ru/eng/znsl3048

Passing to the inverse map turns every domain into the same source. Let
f = phi^{-1} be the univalent map of the disk onto W, and use the
conformal change of variables z = f(w), dA(z) = |f'(w)|^2 dA(w). The
integral over W becomes the integral over the disk of |f'|^(2-s), so with
the abbreviation t = 2 - s the statement becomes

    integral over D of |f'(w)|^t dA(w) < infinity whenever  -2 < t < 2/3 ,

for every injective holomorphic f on the disk. Positive t detects large
derivatives and is comparatively easy; negative t detects small
derivatives and is where the difficulty lives. The single hardest value
is t = -2, the image of the upper endpoint s = 4.

Normalizing removes the last freedom. Write S for the schlicht class of
injective holomorphic f on the disk with f(0) = 0 and f'(0) = 1; every
univalent f is an affine rescaling of an element of S, and the rescaling
multiplies the integral by a finite positive constant. For f in S, real t,
and 0 < r < 1, put

    M_t[f'](r) = (1 / 2 pi) integral from -pi to pi of |f'(r e^{i theta})|^t d theta ,

the circular integral mean of order t. The growth exponent at t is

    beta_f(t) = limsup as r -> 1 of  log M_t[f'](r) / log(1/(1-r)) ,

and the schlicht-class exponent is B_S(t) = sup over f in S of beta_f(t).
Qualitatively, M_t[f'](r) is comparable to (1-r)^(-beta_f(t)), and a
single uniform estimate controls the whole class at once. The central
claim is a uniform radial bound at t = -2 with an arbitrarily small
loss, together with the exact value of the exponent, and from these
the area statement follows. The historical development of these
formulations, including the equivalence between the area form and uniform
integral-mean forms, is due to Bertilsson and is recorded in his thesis.

https://www.diva-portal.org/smash/get/diva2:8593/FULLTEXT01.pdf
https://doi.org/10.1007/BF02384769

The main theorem has two parts. First, for every positive epsilon there is
a finite constant C_epsilon, depending on epsilon alone, such that

    M_{-2}[f'](r)  <=  C_epsilon (1-r)^(-1-epsilon)   for all f in S and 1/2 <= r < 1 ,

and moreover B_S(-2) = 1. Second, for every simply connected domain W whose
boundary in the Riemann sphere contains at least two points, every
conformal bijection phi : W -> D satisfies the area integrability above,
equivalently every univalent f on the disk satisfies the t-integrability
above for -2 < t < 2/3. The two parts are proved together: the uniform
estimate yields the area range, and the Koebe function shows both open
endpoints are sharp.

The proof is a contradiction argument organized around a positive
operator that measures how fast derivative information is lost under
rescaling. The distortion theorem already gives, for f in S,

    |f'(z)|^{-1}  <=  const (1-|z|)^{-1} ,

hence M_{-2}[f'](r) is at most const (1-r)^(-2). The target exponent is
1, not 2, and the entire gap is the statement that very small derivatives
cannot occupy too much of a circle. The strategy encodes a hypothetical
failure of that statement into a probability law on conformal maps, then
reads off two incompatible signs for a single average. The law is
algebraic; the opposing sign is geometric, coming from counting critical
points. The reduction to a growth rate for an operator, the construction
of the law, the compactness that makes the count meaningful, and the
final transport inequality are the four moving parts. Related operator
and weighted-Bergman methods appear in the work of Pommerenke, of
Hedenmalm and Shimorin, and of Sola, and the random-map viewpoint is
anticipated by the conformal snowflakes of Beliaev and Smirnov.

https://doi.org/10.1215/S0012-7094-04-12725-3
https://annals.math.princeton.edu/2010/172-1/p12
https://www.diva-portal.org/smash/get/diva2:11935/FULLTEXT01.pdf

Set up the half-plane model. Write H = { x + i y : x real, y > 0 } for the
upper half-plane and m(z) = (z - i)/(z + i) for the conformal map of H onto
D sending i to 0. Let K be the class of injective holomorphic maps F on H
satisfying F(i) = 0 and F'(i) = 1, topologized by locally uniform
convergence. The exact dictionary with the disk class is

    F(z) = 2 i f(m(z)) ,  f(w) = F(m^{-1}(w)) / (2 i) ,

because m maps the point i to 0 with derivative m'(i) = 1/(2 i). For F in
K write Q_F = 1/F' for the reciprocal derivative, a holomorphic never-zero
function on H.

Four classical facts are used repeatedly and are collected once. These are
the standard coefficient, distortion, and growth estimates and the Koebe
quarter theorem for the schlicht class, together with the normal-family
compactness of the class. The first gives, for f(w) = w + a_2 w^2 + ... in
S, the bound |a_2| <= 2 and, for |w| = r < 1,

    (1-r)/(1+r)^3  <=  |f'(w)|  <=  (1+r)/(1-r)^3 ,
    r/4  <=  |f(w)|  <=  r/(1-r)^2 ,

and, for any univalent holomorphic g on a disk B(z_0,R), the inclusion
B(g(z_0), R |g'(z_0)|/4) inside g(B(z_0,R)). These are the contents of the
first two chapters of Duren's monograph.

https://link.springer.com/book/9780387907956

The second classical fact is compactness. The upper growth bound through
the dictionary makes K locally uniformly bounded, so Montel's theorem
supplies a locally uniformly convergent subsequence from any sequence in
K. Cauchy's formula preserves the normalizations F(i) = 0 and F'(i) = 1.
Hurwitz's theorem, applied to F_n(z) - F_n(z_0) on the punctured plane
H minus a point z_0, shows the limit has no zero and cannot be constant on
a component, so the limit is injective and again lies in K. The compact-open
topology is metrizable on a countable compact exhaustion, so sequential
compactness gives compactness. Joint continuity of the evaluations
(F,z) -> F^{(j)}(z) and (F,z) -> Q_F^{(j)}(z), and the boundedness on K
cross a compact set of z, follow from Cauchy's formula and the nonvanishing
of F'. Two explicit bounds follow from these estimates. For real
C_0 >= 0,

    |Q_F(x + i y)|  <=  (C_0^2 + 4)^2 / y   for F in K, |x| <= C_0, 0 < y <= 1 ,

which is at most 64/y when |x| <= 2, and

    |F(z)|  >=  1/16   whenever F in K and |m(z)| >= 1/2 .

These are proved from the distortion bounds and the quarter theorem:
the strip bound uses the reciprocal-derivative formula and the lower
derivative bound, and the separation bound uses the Koebe quarter theorem
on the disk B(i,1/4) together with injectivity.

Next comes the rescaling that defines the operator. Identify z = x + i y in
H with the affine self-map w -> x + y w of the plane, sending i to z. These
affine maps form a group: the product is z * w = x + y w, the identity is i,
and the inverse of z is z# = (-x + i)/y. For F in K define the re-rooted map

    T_z F (w)  =  ( F(z * w) - F(z) ) / ( y F'(z) ) .

Differentiating shows T_z F again lies in K. The pieces satisfy clean
composition laws, proved by direct differentiation and matching of the
normalizations:

    Q_{T_z F}(w) = Q_F(z * w) / Q_F(z) ,   T_w (T_z F) = T_{z * w} F .

On the space C(K) of continuous complex-valued functions with the supremum
norm, define the positive operator

    A_z Phi (F) = |Q_F(z)|^2 Phi(T_z F) ,

and average the two base points z = -1/2 + i/2 and z = 1/2 + i/2:

    L = ( 1/2 ) ( A_{-1/2 + i/2}  +  A_{1/2 + i/2} ) .

Positivity means nonnegative functions go to nonnegative functions. In the
operator product the weights telescope, |Q_F(z)|^2 |Q_{T_z F}(w)|^2 =
|Q_F(z * w)|^2, which gives the operator identity A_z A_w = A_{z * w} and
makes iteration tractable. A word of length n in the two base points has
height 2^{-n} and real part a sum of the form -1 + (2 j + 1) 2^{-n}, each
value of j occurring once with coefficient 2^{-n}, and expansion gives the
grid formula

    L^n = v_n sum over j = 0 to 2^n - 1 of  A_{ u_{n,j} + i v_n } ,

with v_n = 2^{-n} and u_{n,j} = -1 + (2 j + 1) v_n. Because each L^n is a
positive sum, the operator norm equals the maximum over F in K of the value
at the constant function: || L^n || = max_F (L^n 1)(F). At the map
F(z) = z - i each summand equals one, giving || L^n || >= 1, and the strip
bound gives || L^n || <= 625 * 4^n. The logarithms of these positive norms
are subadditive, so the growth rate

    rho = lim as n -> infinity of || L^n ||^{1/n} = inf over n of || L^n ||^{1/n}

exists and satisfies 1 <= rho <= 4. Whenever lambda > rho there is a finite
C_lambda with || L^n || <= C_lambda lambda^n for all n.

The first half of the theorem is the following comparison. If rho <= 2 then
for every epsilon there is C_epsilon with

    M_{-2}[f'](r)  <=  C_epsilon (1-r)^(-1-epsilon)

for all f in S and 1/2 <= r < 1, with constant depending only on epsilon.
The argument compares a whole circle of the disk with the dyadic grid of
base points above a fixed interval. Fix f in S and 1/2 <= r < 1, let
delta = 1 - r, and choose n with v = 2^{-n} <= delta < 2 v. On the angular
interval I = [3 pi/4, 5 pi/4], write m^{-1}(r e^{i theta}) = x(theta) +
i y(theta) and D(theta) = 1 - 2 r cos(theta) + r^2. The inverse formula
gives x(theta) = -2 r sin(theta)/D(theta) and y(theta) =
(1 - r^2)/D(theta). On I the cosine is nonpositive, so D >= 1 + r^2 >= 2 r,
hence |x| < 1; further delta/2 <= y <= 2 delta, and d x / d theta >= 1/16.
The dyadic cells [u_{n,j - v, u_{n,j} + v}] cover [-1,1] with disjoint
interiors. If x(theta) lies in cell j then the relative point
w = (x(theta) - u_{n,j})/v + i y(theta)/v lies in the fixed compact triangle
E = { |u| <= 1, 1/2 <= eta <= 4 }. Let B_E be the maximum of |Q_G(w)| over
G in K and w in E, finite by the compactness bounds. The composition law
turns the derivative at the varying point into the derivative at the fixed
grid point,

    |Q_F(x(theta) + i y(theta))|  <=  B_E |Q_F(u_{n,j} + i v)| ,

which is the key comparison. The exact derivative conversion in the
dictionary is |f'(m(z))|^{-2} = 16 |Q_F(z)|^2 / |z + i|^4 <= 16 |Q_F(z)|^2,
and since x(theta) is increasing with d theta <= 16 d x, summation over the
cells of length 2 v gives

    integral over I of |f'(r e^{i theta})|^{-2} d theta
        <=  512 B_E^2 ( L^n 1 )(F) .

The four rotated maps f_alpha(z) = alpha^{-1} f(alpha z) for alpha in
{1, i, -1, -i} lie in S and cover the full circle by four rotated copies of
I, so M_{-2}[f'](r) <= C || L^n || for an absolute constant C. If rho <= 2,
apply the growth bound with lambda = 2^{1 + epsilon}: it yields
M_{-2}[f'](r) <= C C_lambda v^{-1-epsilon} <= const * (1-r)^{-1-epsilon},
and the operator and the compact set E were fixed before f and r, so the
constant depends only on epsilon. This completes the comparison.

The whole burden now rests on proving rho <= 2. Suppose, toward a
contradiction, that rho > 2, and set beta = log base 2 of rho, so beta > 1.
The excess growth is converted into a Borel probability law P on K whose
covariance under the affine source maps is exact. The statement is that
there is such a P with

    expectation over P of  |Q_F(z)|^2 Phi(T_z F)
        =  y^{-beta} expectation over P of  Phi(F)

for every z = x + i y in H and every nonnegative Borel Phi on K. The
construction has four steps and uses the Riesz representation theorem of
Folland to identify positive functionals with measures.

https://doi.org/10.1002/9781118625392

First, a probability eigenmeasure for L. For tau > rho the series
R_tau = sum over n >= 0 of tau^{-n} L^n converges in operator norm. Put
g_tau = R_tau 1 and M_tau = max over K of g_tau; positivity gives
1 <= g_tau <= M_tau, and the maximum is attained because K is compact and
g_tau continuous. One checks that M_tau -> infinity as tau decreases to
rho: if not, the resolvent identity L g_tau = tau (g_tau - 1) would force
|| L^n || to grow no faster than tau_j (1 - M_tau^{-1}) along a sequence
tau_j decreasing to rho, contradicting rho > 0. Choosing a maximizer F_tau
and setting mu_tau = delta_{F_tau} R_tau / M_tau gives a probability measure
with mu_tau L = tau mu_tau - (tau / M_tau) delta_{F_tau}. Taking a sequence
tau decreasing to rho and selecting a diagonal subsequence over a countable
dense subset of C(K), the bound on the total mass extends convergence to
all continuous functions, and the last term vanishes; the limit is a
probability measure mu with mu L = rho mu.

Second, one horizontal period. The grid formula reads L^n = v sum_j
A_{u_j + i v} with u_j = -1 + (2 j + 1) v. Right composition with A_{2 + i}
shifts u_j to u_j + 2 v, and the eigenmeasure identity cancels all interior
terms, leaving exactly

    mu A_{2 + i} - mu = rho^{-n} v mu ( A_{1 + v + i v} - A_{-1 + v + i v} ) .

Both remaining abscissae have modulus at most 2, so the strip bound gives
|Q_F(u + i v)| <= C / v and hence || mu A_{2 + i} - mu || <= C (2/rho)^n,
which tends to zero. The left side is independent of n, so mu A_{2 + i} =
mu, and composition with A_{-2 + i} gives the negative period too.

Third, horizontal and scale averages. Define the averaged functional
bar mu = (1/2) integral from 0 to 2 of mu A_{x + i} dx, a finite positive
functional of strictly positive mass whose mass is positive because the
weights |Q_F(x + i)|^2 are positive. The period shows bar mu A_{t + i} =
bar mu for every real t. To pass to half height, use the eigenmeasure
identity before expanding L:

    rho bar mu = (1/2) integral_0^2 mu L A_{x + i} dx
               = (1/2) integral_{-1/2}^{3/2} mu A_{s + i/2} ds ,

whose integrand has period two, so moving the interval to [0,2] gives
rho bar mu = bar mu A_{i/2} and therefore bar mu A_{2 i v} = rho^{-1}
bar mu A_{i v} for every v > 0. Since 2^beta = rho, the functional
H(v) = v^beta bar mu A_{i v} satisfies H(2 v) = H(v), and each bar mu A_{i v}
remains horizontally invariant because A_{i v} A_{t + i} = A_{v t + i} A_{i v}.
Averaging over one logarithmic period,

    nu = integral from 1 to 2 of H(v) dv / v ,

gives a finite positive functional of positive mass with nu A_{i y} =
y^{-beta} nu for every y > 0, by the substitution u = v y and the
periodicity of v -> H(v) with period log 2. Horizontal invariance persists,
and because A_{x + i y} = A_{x + i} A_{i y}, normalizing to the probability
P = nu / nu(1) gives P A_{x + i y} = y^{-beta} P on continuous functions.

Fourth, Borel tests. For fixed z = x + i y the map E -> integral over K of
|Q_F(z)|^2 1_E(T_z F) dP is a finite Borel measure on K, because its weight
and transformation are continuous in F and the weight is bounded. On
continuous functions this measure equals y^{-beta} P, and equality extends
to nonnegative simple functions and then by monotone convergence to all
nonnegative Borel functions, which is the stated covariance identity.
Equality also holds for integrable signed tests through the positive and
negative parts, and applies parameter by parameter when a test depends
jointly measurably on a parameter, followed by Tonelli's theorem.

The law immediately yields explicit moments of the logarithmic derivative.
Define, for z = x + i y in H,

    q_F(z) = i y Q'_F(z) / Q_F(z) = - i y F''(z) / F'(z) = a_F(z) + i b_F(z) ,

with a_F and b_F real. Taking Phi = 1 in the covariance identity gives
E |Q_F(x + i y)|^2 = y^{-beta}. The functions Q_F, Q'_F, Q''_F are uniformly
bounded on K over a compact neighborhood of any interior point, which
controls the first two real derivatives of |Q_F|^2 and justifies
differentiation under the expectation. Holomorphicity gives the pointwise
identities partial_y |Q_F|^2 = 2 a_F |Q_F|^2 / y, partial_x |Q_F|^2 =
2 b_F |Q_F|^2 / y, and Delta |Q_F|^2 = 4 |Q'_F|^2. At the point i one has
Q_F(i) = 1, so the y-derivative, the x-derivative, and the Laplacian of the
constant identity give 2 E a_F(i) = -beta, 2 E b_F(i) = 0, and
4 E |q_F(i)|^2 = beta (beta + 1), that is,

    E a_F(i) = -beta/2 ,   E b_F(i) = 0 ,   E |q_F(i)|^2 = beta (beta + 1)/4 .

Differentiating Q_{T_z F}(w) = Q_F(z * w)/Q_F(z) in w introduces the factor
y and gives the affine covariance q_{T_z F}(w) = q_F(z * w).

The target potential is fixed by

    k = (beta + 3)/4 > 1 ,   V_xi^F(x + i y) = y^k / |F(x + i y) - xi| ,

for xi in the plane, with the value plus infinity at the unique possible
point where F(z) = xi. The choice of k is dictated by the identity
4(k - 1) = beta - 1 and by 2k - 1 = (beta + 1)/2.

The next requirement is that points of a positive superlevel of V_xi^F
cannot escape to the real axis or to infinity, so that a critical-point
count on the boundary of a superlevel is legitimate. A weighted area
integral delivers this. Let S_0 = { |m(z)| >= 1/2 }, a region whose
complement has compact closure in H and contains the zero i of every F,
and put

    I(F) = integral over S_0 of  y^{beta + 1} |F(x + i y)|^{-4} dA .

Then I is Borel and E I(F) <= 256 pi, so I(F) is finite almost surely. The
soft step is an inverse-area identity. For z not equal to i, the
normalization gives (T_z F)(z#) = - F(z) Q_F(z)/y, and applying the
covariance identity with Phi(H) = |H(z#)|^{-4} gives

    E ( |F'(z)|^2 |F(z)|^{-4} ) = y^{-beta-4} E |F(z#)|^{-4} .

The involution (x,y) -> (u,Y) = (-x/y, 1/y) has determinant -y^{-3} on the
map and preserves S_0 because |m(z#)|^2 = |m(z)|^2; in the new variables
dx dy = Y^{-3} du dY. The separation bound |F| >= 1/16 on S_0 and injective
conformal change of variables give the deterministic estimate

    integral over S_0 of |F'|^2 |F|^{-4} dA
        = integral over F(S_0) of |w|^{-4} dA  <=  256 pi .

Integrating the inverse-area identity with Tonelli and the involution
converts E integral |F'|^2 |F|^{-4} into E I(F), proving the bound.

From I(F) < infinity one reads off escape at infinity and a bound on how
much of a horizontal line can map into a compact set. Write R(z) = |x| + y.
A disk B(z, y/2) lies in { R >= R(z)/3 } and eventually in S_0, where 1/F is
holomorphic, and the submean inequality for |1/F|^4 on that disk yields

    y^{beta + 3} |F(z)|^{-4}  <=  C_beta * integral over S_0 cap { R >= R(z)/3 }
                                     of  (Im z')^{beta + 1} |F(z')|^{-4} dA(z') ,

which tends to zero uniformly as R(z) grows, since the right side is the
tail of one integrable function. Because 4k = beta + 3, this says
y^k / |F(x + i y)| tends to zero as |x| + y tends to infinity, so at any
positive height |F| tends to infinity and the preimage of a compact set is
bounded and closed, hence compact in H. A horizontal second-derivative
integration by parts, applied to g = psi o F for a compactly supported psi
that dominates the indicator of the target set K, bounds the horizontal
preimage length uniformly in the height:

    integral over the real line of 1_K( F(x + i epsilon) ) dx  <=  C_{F,K}  for every epsilon > 0 .

The bound is one and the same for all heights and for every compact target.

Targets approached at small height still need control. For a fixed F with
I(F) < infinity one treats omitted targets first. For radius R, threshold
delta, and a dyadic height h <= 1, partition the real line into half-open
cells of length h and call a cell a witness cell if some x in it and some
height y in [h, 2h] and some omitted target xi with |xi| <= R satisfy
V_xi^F(x + i y) >= delta. At a witness z the quotient |F(z) - xi| is at
most (2h)^k / delta, and because the disk B(z,y) omits xi the quarter
theorem gives y |F'(z)| <= 4 |F(z) - xi| <= 4 (2h)^k / delta. For any other
point x' of the cell the relative point lies in a fixed compact set, so by
the local bounds the whole cell maps into a target disk of radius
C_delta h^k at height h. The trace bound therefore limits the number of
witness cells to C_{F,R,delta}/h, and covering each witness with a disk of
radius C_delta h^k puts all omitted targets witnessed at scale h into a set
E_h of area at most C_{F,R,delta} h^{2k-1}. The sum over dyadic h is finite,
so the upper limit of E_{2^{-n}} has area zero, being bounded by every tail
of a convergent sum. A countable union over integer R and delta = 1/n
leaves, for every omitted target outside the union,

    sup over x of V_xi^F(x + i y) -> 0  as y -> 0 .

For an interior target xi choose a disk of radius a about xi inside the
image of F; the inverse image of its closure is compact with positive
minimum height, below which |F - xi| >= a and V_xi^F <= y^k / a, so the same
conclusion holds. Consequently, for such a target and any delta > 0, small
heights are excluded by the last display and large |x| + y are excluded by
the escape estimate, while the superlevel { V_xi^F >= delta } is closed and
contains its possible pole; it lies in a bounded box of positive minimum
height and is compact in H. This proves compact positive superlevels for P-
almost every F and area-almost every xi.

The condition that these superlevels be compact is Borel in (F,xi). Using
the exhaustion C_m = { |x| <= m, 1/m <= y <= m }, compactness of every closed
positive superlevel is equivalent to: for every n there is m with
{ V_xi^F > 1/n } contained in C_m. For fixed n and m this containment holds
exactly when |F(q) - xi| >= n (Im q)^k for every rational point q of H
outside C_m, and each such inequality is closed in the parameters by
continuity of evaluation; countably many closed tests give a Borel set.

Now comes the critical map. Define the smooth real map

    G_F(z) = F(z) - (i y / k) F'(z) ,  z = x + i y ,

which satisfies G_F(i) = -i/k for every F by the normalization. Its
normalized signed Jacobian is J_F(z) = det D G_F(z) / |F'(z)|^2, the
determinant taken for G_F as a map of two real planes; the sign of J_F
distinguishes saddles from peaks. Since F''/F' = i q_F / y, the two real
derivative columns are (G_F)_x = (F'/k)(k + q_F) and
(G_F)_y = (i F'/k)(k - 1 + q_F), and because the real determinant of two
complex vectors u, v is Im(conjugate u times v), applying this to the
columns gives

    k^2 J_F(z) = k(k - 1) + (2k - 1) a_F(z) + |q_F(z)|^2 .

The map F -> J_F(i) is bounded and continuous on K, and combining the last
formula with the moments of q_F at i, and the cancellation 2k - 1 =
(beta + 1)/2 of the cross terms, yields the strictly positive mean

    k^2 E J_F(i) = k(k - 1) > 0 .

Away from the unique possible pole put l = log V_xi^F. Since log|F - xi| is
harmonic off the pole, a computation gives Delta l = -k / y^2, and the
critical points of l are exactly the solutions of G_F(z) = xi. At such a
point,

    V_xi^F(z) = k y^{k-1} |Q_F(z)| ,
    Hess l (z) = (k / y^2) times the matrix with entries
        -(k + a_F),  b_F ;  b_F,  k - 1 + a_F ,
    det Hess l (z) = -(k^4 / y^4) J_F(z) .

The Hessian of l has trace -k / y^2 < 0, so a negative Hessian
determinant gives a saddle and a positive one gives two negative
eigenvalues, hence a strict maximum. Because Hess V_xi^F = V_xi^F Hess l at
a critical point, the classification transfers: a solution with J_F(z) > 0
is a nondegenerate saddle and one with J_F(z) < 0 is a nondegenerate peak.

Call a pair (F,xi) good when every closed positive superlevel of V_xi^F is
compact in H and xi is a regular value of G_F, so every solution of
G_F(z) = xi has J_F(z) not equal to zero, an empty fiber counted as regular.
The set of good pairs is Borel and, for P-almost every F, its section over
xi has full planar area. Borelness of the superlevel condition follows
from the countable rational-point test, and regularity is tested on
compact boxes C_m, where
failure means some w in C_m has G_F(w) = xi and J_F(w) = 0, whose solution
set is closed because C_m is compact. For each fixed F the critical values
of G_F have area zero, which is the plane case of Sard's theorem; the
short covering argument covers a compact source box by O(h^{-2}) squares of
side h, and on a square meeting the critical set the linear part of G_F at
a chosen critical point has rank at most one, so the image of the square
lies in an O(h^2)-neighborhood of a segment of length O(h) and has area
O(h^3), whence the critical image has outer area O(h) and area zero. The
full-area section statement combines this with the almost-sure compactness
of superlevels.

https://doi.org/10.1090/S0002-9904-1942-07811-6

The good-pair property is affine natural. For F, xi, and t = x_0 + i y_0 in
H, set xi-tilde = (xi - F(t)) / (y_0 F'(t)), with w -> t * w the
affine product. The derivative identity (T_t F)'(w) = F'(t * w)/F'(t) gives

    G_{T_t F}(w) = ( G_F(t * w) - F(t) ) / ( y_0 F'(t) ) ,
    q_{T_t F}(w) = q_F(t * w) ,   J_{T_t F}(w) = J_F(t * w) ,
    V^{T_t F}_{xi-tilde}(w) = y_0^{1 - k} |F'(t)| V_xi^F(t * w) .

The source affine map is a homeomorphism of H and the target change is a
nonsingular similarity, so (F,xi) is good exactly when (T_t F, xi-tilde) is
good.

The counting statement is the engine. Fix a good pair (F,xi). There are only
finitely many critical points of V_xi^F with value at least h, for any h > 0,
and for every such h,

    number of peaks with V_xi^F >= h  >=  number of saddles with V_xi^F >= h ,

the possible pole excluded from the peak count. The solutions of G_F = xi
are isolated by regularity and closed in H, isolation following from the
real inverse-function theorem, so those above a level lie in a compact set
and are finite; a pole cannot accumulate because at a preimage z_0 of xi
the difference G_F(z_0) - xi equals -i (Im z_0) F'(z_0)/k, which is nonzero.
Choose a positive level lambda that is not a critical value and let U be the
open superlevel { V_xi^F > lambda }, with compact closure in H and boundary
exactly the level. That boundary is a compact smooth one-manifold whose
components are finitely many regular simple closed curves; compactness makes
the normalized-gradient flow complete, its orbits are open in each
component, and a component is a single periodic orbit giving such a curve.
For a bounded Jordan interior of one curve, the Jordan curve theorem of
Munkres puts its closure in H. The strict subharmonicity relation
Delta l = -k/y^2 < 0 forces V_xi^F > lambda throughout the interior, so
nested curves are impossible and the interiors are precisely the
components of U.

https://www.ams.org/books/surv/155/
https://link.springer.com/book/10.1007/978-1-4419-9982-5

Inside each component, remove small disjoint disks about the finitely many
critical points and the possible pole. On the remaining compact smooth
domain the gradient of l is nonzero and its angle form is closed, and
Stokes' theorem makes the winding on the outer boundary equal the sum of
windings on the positively oriented small circles. On the positive
orientation of each boundary curve the gradient is an inward normal with
winding number plus one: this is the tangent-turning theorem, for which
Hopf's secant proof suffices, parametrizing the curve periodically, lifting
the angle of the unit secant continuously on the closed triangle, and reading
the total turn of 2 pi.

https://www.numdam.org/item/CM_1935__2__50_0/

At a nondegenerate critical point the gradient on a small circle is homotopic
through nonzero vectors to its linear part, so its index is the sign of the
determinant of the Hessian, plus one at a peak and minus one at a saddle.
At the pole, the simple zero of F - xi gives l(z) = -log|z - z_0| plus a
smooth function, and the negative radial leading field of the gradient has
winding number plus one, so the pole contributes plus one without being a
finite critical point. Writing C for the number of components of U and B for
the number of poles in {0,1}, the sum of index identities is

    C = (peaks with V_xi^F > lambda) - (saddles with V_xi^F > lambda) + B .

When B = 1 the pole lies in U so C >= B; the inequality C >= B is automatic
when B = 0. Hence peaks are at least as numerous as saddles above every
regular positive level. For a prescribed h, finiteness of the critical
points of value at least h/2 gives a regular lambda in (h/2, h) with no
critical value in [lambda, h), and the counts above lambda coincide with
those at or above h, including ties.

The cumulative count gives an injective pairing. For a fixed good pair
(F,xi), order the saddles by decreasing value of V_xi^F, breaking ties by
increasing lexicographic order of (x,y), and order the peaks independently by
the same rule. Each point has finitely many predecessors, its zero-based
rank being the number of its predecessors; ranks are distinct, and no rank is
missing below an attained one. If a saddle has rank j and value v, there are
at least j + 1 saddles of value at least v, hence by the count at least j + 1
peaks of value at least v, which form an initial segment of the peak order;
the peak of rank
j therefore exists and has value at least v. Assigning each saddle to the
peak of the same rank is injective within the fiber, satisfies
V_xi^F(peak) >= V_xi^F(saddle), and places a peak of the opposite sign at the
same target. Distinct fibers cannot share a point, so the assignment is
injective for fixed F across its whole domain.

This assignment is realized by a Borel map that is affine equivariant. Let
the pairing domain be the set of pairs (F,z) in K cross H with J_F(z) > 0
and (F, G_F(z))
good, and let R be its map, R(F,z) = R_F(z), carrying each good-target
saddle to its paired peak, so that with z' = R_F(z) and xi = G_F(z),

    G_F(z') = xi ,  J_F(z') < 0 ,  V_xi^F(z') >= V_xi^F(z) ,

the assignment is injective for each fixed F, and when (F, t * w) belongs
to the pairing domain,

    (T_t F, w) belongs to the pairing domain ,
    R_{T_t F}(w) = t# * R_F(t * w) .

The construction is a countable-Borel enumeration. The nonsingular solution
graph Gamma = { (F,xi,w) : G_F(w) = xi, J_F(w) not equal to zero } is covered
by countably many continuous branches over open parameter sets: at a point
(F_0, xi_0, w_0) of Gamma with invertible matrix M = D G_{F_0}(w_0), joint
continuity of derivatives on source compacts gives a ball and a parameter
neighborhood on which w -> w - M^{-1}(G_F(w) - xi) is a contraction of the
ball into its interior, whose unique fixed point solves G_F(w) = xi, is
nonsingular, and depends continuously on (F,xi) by the contraction principle
and uniqueness. Because Gamma is second countable, a countable family of
such branch graphs covers it. Enumerate the branches, activate branch j where
its parameter lies in its open set and its value differs from every earlier
branch, and the activation sets are Borel, since equality of two
continuous branches is closed on the overlap and only finitely many
earlier indices matter; each nonsingular root occurs in exactly one active
branch. On a branch the critical value can be evaluated without dividing by
F(w) - xi, through the positive continuous function H(F,w) =
k (Im w)^{k-1} |Q_F(w)|, which equals V_xi^F(w) at every root. Signs of
J_F, comparisons and equalities of H, and lexicographic comparisons of
coordinates are all Borel, so the rank of an active branch among those of its
sign is a countable sum of Borel indicators and is Borel and finite on good
fibers. Requiring a positive-sign and a negative-sign active branch to share
a finite rank is a countable Borel test, and selecting the unique such branch
that passes the test at the target G_F(z) defines a Borel map on the
pairing domain. Under the affine source change w -> t * w the critical
fibers and signs correspond by
the covariance identities, all critical heights in a fiber are multiplied by
one positive constant, and the real source scale preserves lexicographic
order including ties, so the two orders, the ranks, and the equal-rank
assignment correspond exactly, giving the displayed equivariance. The
enumeration enters only through measurability and not through the geometric
definition of the assignment.

https://www.ams.org/books/surv/155/

The pairing is used through one weight. At each critical point the identity
4(k - 1) = beta - 1 and the expression for the critical height give

    y^{beta - 1} |Q_F(z)|^4  =  ( V_xi^F(z) / k )^4 ,

so the pairing raises the weight on the left, because the paired peak has at
least the saddle's value.

Now the transport argument, which converts the law and the pairing into
incompatibility. Keep rho > 2, beta = log base 2 of rho, k = (beta + 3)/4,
and the probability P, and write J_{F,+} = max(J_F,0), J_{F,-} =
max(-J_F,0), and omega_F(x + i y) = y^{beta - 1} |Q_F(x + i y)|^2. For fixed
F the open set { J_F > 0 } has a countable cover by sets on which G_F is a
diffeomorphism; disjointifying and applying absolute-Jacobian change of
variables followed by Tonelli's theorem, and using |det D G_F| =
|F'|^2 |J_F|, gives, for every nonnegative Borel h on H,

    integral over H of omega_F J_{F,+} h dA
      = integral over the plane of
          ( sum over G_F(z) = xi with J_F(z) > 0 )  y^{beta - 1} |Q_F(z)|^4 h(z)  dA(xi) ,

and the same with both signs reversed; no global injectivity of G_F and no
finiteness of the number of preimages is needed, since all terms are
nonnegative.

https://doi.org/10.1002/9781118625392

The one exceptional target that normalization forces must be handled. The
root target is G_F(i) = -i/k for every F, but the almost-everywhere target
statement gives full area only section by section. Applying the change of
variables on the positive sign with h supported on the box B_0 = { |x| <= 1,
1 <= y <= 2 } times the indicator that (F, G_F(z)) is not good, the target
integral is carried by a null set for P-almost every F, so the source
integral vanishes. With Phi(F) = J_{F,+}(i) times the indicator that
(F, -i/k) is not good, the affine equivariance of J and of goodness gives
Phi(T_z F) equal to J_{F,+}(z) times the indicator that (F, G_F(z)) is not
good, and the covariance identity gives E of omega_F(z) times Phi(T_z F)
equal to y^{-1} E Phi. Taking expectations and using Tonelli yields
0 = 2 (log 2) E Phi, so

    E ( J_{F,+}(i)  times the indicator that (F, -i/k) is not good ) = 0 .

For the box comparison, for each positive integer N let D_N be the Borel set
of pairs (F,z) with J_F(z) > 0, (F, G_F(z)) good, and the paired point
z' = R_F(z) = x' + i y' satisfying 1/N <= y'/y <= N and |x' - x| <= N y. The
Borel property comes from the pairing map, its equivariance gives
1_{D_N}(F,z) = 1_{D_N}(T_z F, i), the sets D_N increase with N, and every
good-target saddle belongs to some D_N because its partner has finite
coordinates and positive height. For R > 0 and Y > 1 put B_{R,Y} = { |x| <=
R, 1 <= y <= Y } and B'_{R,Y,N} = { |x| <= R + N Y, 1/N <= y <= N Y }. If z
lies in B_{R,Y} and (F,z) lies in D_N, the partner lies in B'_{R,Y,N}, and at
a preimage the critical-point identity gives y^{beta - 1} |Q_F(z)|^4 =
(V_xi^F(z)/k)^4 with 4(k - 1) = beta - 1. Because the pairing is injective on
each target fiber and the partner's value is at least the saddle's, applying
the change of variables to both signs and comparing target sums produces,
for every F,

    integral over B_{R,Y} of omega_F J_{F,+} 1_{D_N} dA
        <= integral over B'_{R,Y,N} of omega_F J_{F,-} dA .

Both source integrands are jointly Borel and nonnegative in (F,z), so
Tonelli's theorem allows averaging this fixed-F inequality. By covariance and
the equivariance of D_N,

    E ( omega_F(z) J_{F,+} 1_{D_N}(F,z) ) = y^{-1} E ( J_{F,+}(i) 1_{D_N}(F,i) ) ,

and likewise E ( omega_F(z) J_{F,-} ) = y^{-1} E J_{F,-}(i). Averaging the box
inequality over B_{R,Y} gives 2 R (log Y) times E(J_{F,+}(i) 1_{D_N}(F,i))
at most 2 (R + N Y) (log(N^2 Y)) times E J_{F,-}(i); the root quantities are
integrable because q_F(i) is uniformly bounded on K and J_F(i) is a fixed
quadratic expression in that jet. Holding N fixed, setting R = Y^2, and
letting Y grow, the ratio of the right box factor to the left one,
(1 + N/Y)(1 + 2 log N / log Y), tends to one, so

    E ( J_{F,+}(i) 1_{D_N}(F,i) ) <= E J_{F,-}(i) .

Letting the integer N grow, monotone convergence exhausts the good-target
saddles, and the root exception accounts for the remaining positive-J weight,
so

    E J_{F,+}(i) <= E J_{F,-}(i) ,   hence   E J_F(i) <= 0 .

For the same fixed J_F and the same law P, the moment computation gives

    k^2 E J_F(i) = k(k - 1) - (2k - 1) beta/2 + beta (beta + 1)/4
                 = k(k - 1) > 0 ,

because 2k - 1 = (beta + 1)/2 and k > 1. The two inequalities for the same
number are incompatible, so the assumption rho > 2 is false. Therefore
rho <= 2, and the disk comparison already proved gives, for every epsilon,
the uniform bound M_{-2}[f'](r) <= C_epsilon (1-r)^{-1-epsilon}, with
constant depending only on epsilon.

The sharpness of the exponent at t = -2 is a separate one-line computation
on the Koebe map f_K(z) = z/(1 - z)^2, whose derivative is (1 + z)/(1 - z)^3
and which is injective on the disk because equality of its values forces
(z - u)(1 - z u) = 0. Writing delta = 1 - r and taking z = r e^{i(pi + theta)}
with |theta| <= delta, one has |1 + z| <= 2 delta and |1 - z| >= 1, so

    M_{-2}[f_K'](r) >= (1/2 pi) integral from -delta to delta of d theta/(4 delta^2)
                    = 1/(4 pi delta) .

No exponent smaller than 1 bounds these means near the boundary, so
beta_{f_K}(-2) >= 1, and together with the uniform estimate this gives
B_S(-2) = 1.

The full negative tail is immediate from the distortion bound. For any real
t <= -2 put a = |t| - 2 >= 0; the lower derivative bound and the estimate at
t = -2 give

    M_t[f'](r) <= ((1 + r)^3/(1 - r))^a M_{-2}[f'](r) <= 8^a C_epsilon (1-r)^{-(a+1)-epsilon} ,

and since a + 1 = |t| - 1 this bounds M_t[f'](r) by (1-r)^{-(|t|-1)-epsilon},
so B_S(t) = |t| - 1 for every t <= -2. On the Koebe arc the same computation
reverses, with |f_K'| <= 2 delta, giving M_t[f_K'](r) at least
2^{-|t|} delta^{1-|t|}/pi and beta_{f_K}(t) >= |t| - 1, hence equality.

For the interior negative range -2 < t < 0, put q = -t/2 in (0,1) and apply
Holder's inequality on the circle of total measure one:

    M_t[f'](r) = (1/2 pi) integral of ( |f'|^{-2} )^q
               <= M_{-2}[f'](r)^q .

Choosing 0 < epsilon < 1/q - 1 makes q(1 + epsilon) < 1, so the uniform
bound gives M_t[f'](r) <= C (1 - r)^{-q(1+epsilon)} with q(1 + epsilon) < 1,
and radial integration, together with smoothness of f' on compact subdisks,
gives integral over D of |f'|^t dA = 2 pi integral from 0 to 1 of
r M_t[f'](r) dr finite, for every f in S and -2 < t < 0. The value t = 0
gives the area pi.

The positive exponents 0 < t < 2/3 assemble from two elementary estimates for
the function itself. First, for every 0 < p < 1/2 the circular means of
|f|^p are uniformly bounded: away from the simple zero at 0, Delta |f|^p =
p^2 |f|^{p-2} |f'|^2 is locally integrable, the Green mean identity of Lee
expresses the mean of |f|^p as a disk integral of |f|^{p-2}|f'|^2 log(r/|z|)
over |z| < r, the inner-circle boundary terms vanish as the radius shrinks
because f(z) = z + O(z^2), and splitting at |z| = 1/2, one uses the classical
bounds on the inner piece and the conformal change of variables w = f(z) with
the growth bound on the outer piece to get a finite constant C_p. Second, for
every 0 <= a < 1 the full-disk integral of (1 + |f|)^a is finite: the case
a = 0 is immediate, and for 0 < a < 1 choose p with max{0, a - 1/2} < p < 1/2
so that the bounded p-means control the a-means when a <= p, while when a > p
the growth bound bounds the a-means by C (1 - r)^{-2(a - p)} with
2(a - p) < 1, and radial integration finishes. From these, for 0 < t < 2/3
choose 0 < eta < 2/t - 3 and set a = (2 + eta) t/(2 - t) in (0,1) and
u_eta(z) = |f'(z)|^2 (1 + |f(z)|)^{-2-eta}; injective conformal change of
variables gives integral of u_eta equal to the polar integral over the plane
of (1 + |w|)^{-2-eta}, which is finite. The exact factorization |f'|^t =
u_eta^{t/2} ( (1 + |f|)^a )^{(2-t)/2}, together with Holder's inequality with
exponents 2/t and 2/(2-t), now bounds integral over D of |f'|^t dA by the
product of a finite constant and the finite integral of (1 + |f|)^a, so
integral of |f'|^t is finite for every f in S and 0 < t < 2/3.

The disk statement transfers to domains. For an arbitrary univalent f on the
disk set g(z) = (f(z) - f(0))/f'(0), an element of S, so
integral over D of |f'|^t equals |f'(0)|^t integral over D of |g'|^t, a
finite positive multiple. For a conformal bijection phi : W -> D with inverse
f, the identities phi'(f(w)) = 1/f'(w) and dA(f(w)) = |f'(w)|^2 dA(w) give

    integral over W of |phi'|^s dA  =  integral over D of |f'|^{2-s} dA ,

valid for nonnegative integrands whether or not the integrals are finite, and
since 4/3 < s < 4 is the same as -2 < 2 - s < 2/3, the desired domain
integral is finite. The disk and domain formulations have exactly the same
scope: a univalent disk image is simply connected and cannot be the whole
plane, because its inverse would be a bounded nonconstant entire function,
and if its boundary in the sphere had at most one point it would be a
nonempty clopen piece of the connected sphere with that one point removed,
that is, the whole sphere or a punctured sphere, contrary to being a proper
plane domain; so the boundary condition holds and the change of variables
applies, and conversely every conformal bijection in the domain formulation
has a univalent inverse on the disk.

One reformulation is the Sobolev pullback theorem of Gol'dshtein and Ukhlov.
For a plane domain U and 1 <= p <= infinity, let L^1_p(U) denote the
homogeneous Sobolev space of locally integrable weakly differentiable real
functions with seminorm ||grad u||_{L^p(U)} finite. For a simply connected
domain Omega whose boundary in the sphere has at least two points, a
conformal bijection phi : Omega -> D, exponents 2 < p < infinity and
4/3 < s < 4, and q = p s/(p + s - 2), the pullback u -> u o phi is bounded
from L^1_p(D) to L^1_q(Omega), and also from L^1_infinity(D) to L^1_s(Omega)
for every 4/3 < s < 4, with finite bounds that may depend on the map and the
exponents. This is the known Sobolev equivalence, since 1 < q < p and
(p - 2) q/(p - q) = s, and the derivative integrability supplied above is
exactly what the equivalence requires.

https://arxiv.org/abs/1212.3999

The area statement also implies the individual endpoint integral-mean bound
that closes the equivalence with B_S(-2) = 1. Fix f in S and 0 < eta < 2.
The function |f'|^{-2+eta} is subharmonic because f' is nonvanishing, so its
circular means are nondecreasing; if its disk area integral is finite then,
for r >= 1/2, the area integral dominates 2 pi M_{-2+eta}[f'](r) times the
integral from r to 1 of u du, giving M_{-2+eta}[f'](r) <= C_{f,eta}
(1 - r)^{-1}. The distortion lower bound gives |f'(r e^{i theta})|^{-eta} <=
C_eta (1 - r)^{-eta}, and multiplying the two bounds yields M_{-2}[f'](r) <=
C'_{f,eta} (1 - r)^{-1-eta}; letting eta shrink to zero gives beta_f(-2) <= 1
for every f. In the other direction that individual exponent bound, together
with Holder's inequality, gives every negative interior area exponent, and
the positive exponents were proved independently, so the equivalence with
B_S(-2) = 1 holds. The uniform constant in the endpoint bound arises
directly from the transfer-operator argument rather than from the
individual area statement.

Finally the open area range is sharp at both ends. For the same Koebe map and
sufficiently small |w|, one has |f_K'(-1 + w)| = |w|/|2 - w|^3 of size |w|
and |f_K'(1 - w)| = |2 - w|/|w|^3 of size |w|^{-3}. The fixed sector
0 < |w| < 1/2 with |arg w| < pi/3 lies inside the disk under both
substitutions z = -1 + w and z = 1 - w, since in both cases
|z|^2 = 1 - 2 Re w + |w|^2 < 1. The area integral at t = -2 near -1 and the
area integral at t = 2/3 near 1 each dominate a positive multiple of the
integral from 0 to 1/2 of u^{-1} du, which diverges; more generally these
sectors diverge for t <= -2 and for t >= 2/3. Because the image of the Koebe
map satisfies the domain hypotheses, its inverse gives, through the change
of variables, divergence of the domain integral at s = 4 and at s = 4/3.
Thus both open endpoints are sharp, and the interval 4/3 < s < 4 cannot be
enlarged for the full class of simply connected plane domains.

The result settles Brennan's original question in the affirmative and
identifies the sharp inverse-square endpoint B_S(-2) = 1, in agreement with
the partial results and numerical estimates of earlier work: the linear
spectral tail for sufficiently negative parameters obtained by Carleson and
Makarov, the bounds B_S(-2) <= 1.218 of Hedenmalm and Shimorin and the
improved B_S(-2) <= 1.206 of Sola, and the conditional integrability for
semigroups of Aleman and Kouroupis, for polynomial basins of Zheng, and for
normalized rational maps of Jin. Recent computations of the real
integral-means spectrum for special classes and Schwarzian-multiplier
criteria for individual maps are consistent with, and strictly weaker than,
the uniform statement for the full schlicht class.

https://doi.org/10.1007/BF02559522
https://arxiv.org/abs/2409.15074
https://arxiv.org/abs/2604.12240
https://arxiv.org/abs/2512.09330