Proof. The coefficient and derivative bounds are the classical estimates just cited. The classical lower growth bound \(r/(1+r)^2\) implies the weaker bound \(r/4\) recorded here. A univalent holomorphic map has nonzero derivative, so for the last assertion the map \[\zeta\longmapsto \frac{g(z_0+R\zeta)-g(z_0)}{R g'(z_0)}\] belongs to \(\mathcal S\). Applying the Koebe quarter theorem to this normalized map gives (4). ◻

Proof. The upper growth bound and (5) give local uniform boundedness. Montel’s theorem therefore gives a locally uniformly convergent subsequence from every sequence in \(\mathcal K\); see (Duren 1983, sec. 1.1 and 1.3) for this normal-family argument and the univalent-limit form of Hurwitz’s theorem. Cauchy’s formula preserves the derivative normalization, so a limit \(F\) has \(F'(i)=1\) and is nonconstant. For any fixed \(z_0\in\mathbb H\), apply Hurwitz’s theorem to \(F_n(z)-F_n(z_0)\) on the connected domain \(\mathbb H\setminus\{z_0\}\). The limit is not identically zero and has no zero there. Thus \(F\) is injective and belongs to \(\mathcal K\). The compact-open topology is metrizable using a countable compact exhaustion of \(\mathbb H\), so this sequential compactness proves compactness.

Cauchy’s formula gives locally uniform convergence of all derivatives. The limiting derivative is nonzero, so reciprocals also converge locally uniformly, as do their derivatives. These observations give the asserted joint continuity. Compactness of \(\mathcal K\times E\) then gives the uniform bounds.

For the explicit estimate, put \(D=|x+i(y+1)|^2\) and \(r=|m(x+iy)|\). Then \[1-r^2=\frac{4y}{D},\qquad |Q_F(x+iy)|=\frac{D}{4|f'(m(x+iy))|}.\] The lower derivative bound in (3) gives \[|Q_F(x+iy)| \le\frac{D(1+r)^3}{4(1-r)} =\frac{D^2(1+r)^4}{16y} \le\frac{D^2}{y}.\] When \(|x|\le C_0\) and \(y\le1\), this proves (6).

Finally \(|m(z)|<1/7\) on \(B(i,1/4)\), while (4) gives \(B(0,1/16)\subset F(B(i,1/4))\). Injectivity prevents any point outside \(B(i,1/4)\) from having an image of modulus less than \(1/16\). This proves (7). ◻

Proof. Differentiating (9) proves the first identity in (11). Applying the normalization twice proves the second. In the operator product the weights telescope: \[|Q_F(z)|^2|Q_{T_zF}(w)|^2=|Q_F(z\cdot w)|^2.\] This gives the third identity, with its displayed order of multiplication. Joint continuity of \((z,F)\mapsto T_zF\) follows from locally uniform convergence on affine images of compact source sets and the nonzero normalizing denominator. Together with Lemma 4, this proves boundedness of \(A_z\). For fixed \(\Phi\), joint continuity on \(\mathcal K\) times a compact neighborhood of \(z\) gives the asserted supremum-norm continuity.

A word of length \(n\) in the affine maps \(-1/2+i/2\) and \(1/2+i/2\) has height \(2^{-n}\) and real part \(\sum_{l=1}^n\sigma_l2^{-l}\), where \(\sigma_l\in\{-1,1\}\). Writing \(\sigma_l=2\epsilon_l-1\) and \(j=\sum_{l=1}^n\epsilon_l2^{n-l}\) gives real part \(-1+(2j+1)2^{-n}\). Each \(j\) occurs once and each word has coefficient \(2^{-n}\). Expansion of \(L^n\) proves the grid formula; at \(n=0\) it reduces to \(A_i=I\).

For the positive sums defining \(L^n\), \[|L^n\Phi|\le L^n|\Phi|\le\|\Phi\|_\infty L^n\mathbf1.\] Testing with \(\Phi=\mathbf1\) proves the norm identity. At the map \(F(z)=z-i\) every summand in \(L^n\mathbf1(F)\) equals one, so \(\|L^n\|\ge1\). Since \(|u_{n,j}|\le1\), (6) gives \[\|L^n\|\le625\,4^n.\] The logarithms of these positive norms are subadditive. More explicitly, for fixed \(N\ge1\) write \(n=qN+r\), \(0\le r<N\), and use \(\|L^n\|\le\|L^N\|^q\|L^r\|\). Taking the upper limit of the \(n\)th roots and then the infimum over \(N\) proves (13); the displayed lower and upper bounds give \(1\le\rho\le4\). If \(\lambda>\rho\), the root limit bounds \(\|L^n\|\) by \(\lambda^n\) for all sufficiently large \(n\). Increasing a constant to cover the finitely many remaining \(n\) proves the last claim. ◻

Proof. Fix \(f\in\mathcal S\) and \(1/2\le r<1\). Put \(\delta=1-r\) and choose \(n\ge1\) with \(v=2^{-n}\le\delta<2v\). On the fixed angular interval \(I=[3\pi/4,5\pi/4]\), write \[m^{-1}(re^{i\theta})=x(\theta)+iy(\theta),\qquad D(\theta)=1-2r\cos\theta+r^2.\] The inverse formula gives \[ x(\theta)=\frac{-2r\sin\theta}{D(\theta)},\qquad y(\theta)=\frac{1-r^2}{D(\theta)}. \tag{15}\] On \(I\) we have \(\cos\theta\le0\) and hence \(D\ge1+r^2\ge2r\); therefore \(|x|\le|\sin\theta|<1\). Also \[\frac\delta2\le y\le2\delta,\qquad \frac{dx}{d\theta} =\frac{4r^2-2r(1+r^2)\cos\theta}{D(\theta)^2} \ge\frac1{16}.\] The height bounds use \(D\le(1+r)^2\), \(D\ge1\), and \(1+r\le2\). For the derivative bound use \(4r^2\ge1\) and \(D^2\le16\).

The cells \([u_{n,j}-v,u_{n,j}+v]\) cover \([-1,1]\) with disjoint interiors. If \(x(\theta)\) belongs to cell \(j\), then \[w=\frac{x(\theta)-u_{n,j}}v+i\frac{y(\theta)}v \in E:=\{u+i\eta:|u|\le1,\ 1/2\le\eta\le4\}.\] Let \(B_E=\max_{G\in\mathcal K,\,w\in E}|Q_G(w)|\), which is finite by Lemma 4. The first identity in (11) gives \[ |Q_F(x(\theta)+iy(\theta))| \le B_E|Q_F(u_{n,j}+iv)|, \qquad F=2i f\circ m. \tag{16}\] Thus the comparison covers every point of the varying-height arc in each cell.

The exact derivative conversion in (5) is \[|f'(m(z))|^{-2} =\left|\frac{m'(z)}{m'(i)}\right|^2|Q_F(z)|^2 =\frac{16}{|z+i|^4}|Q_F(z)|^2 \le16|Q_F(z)|^2.\] Since \(x(\theta)\) is increasing and \(d\theta\le16\,dx\), (16) and the cell lengths \(2v\) give \[\begin{align*} \int_I|f'(re^{i\theta})|^{-2}\,d\theta &\le256B_E^2\sum_j|Q_F(u_{n,j}+iv)|^2 \bigl|[u_{n,j}-v,u_{n,j}+v]\cap x(I)\bigr|\\ &\le512B_E^2 v\sum_j|Q_F(u_{n,j}+iv)|^2 =512B_E^2 L^n\mathbf1(F). \end{align*}\] Here the bars around the intersection denote interval length.

For \(\alpha\in\{1,i,-1,-i\}\), the map \(f_\alpha(z)=\alpha^{-1}f(\alpha z)\) belongs to \(\mathcal S\) and has derivative \(f'(\alpha z)\). The four rotated copies of \(I\) cover the full circle, with overlap only at endpoints. Applying the preceding estimate to these four maps therefore proves \[M_{-2}[f'](r)\le C\|L^n\|\] with an absolute constant \(C\). If \(\rho\le2\), apply Lemma 5 with \(\lambda=2^{1+\varepsilon}\). It gives \[M_{-2}[f'](r)\le C C_\lambda v^{-1-\varepsilon} \le C C_\lambda 2^{1+\varepsilon}(1-r)^{-1-\varepsilon}.\] The operator \(L\) and the compact set \(E\) were fixed before \(f\) and \(r\) were chosen, so this constant depends only on \(\varepsilon\). ◻

Proof. For \(z\ne i\), normalization gives \[(T_zF)(z^\#)=-\frac{F(z)Q_F(z)}y.\] Apply (27) with \(\Phi(H)=|H(z^\#)|^{-4}\). Both evaluations avoid the unique zero, and we obtain \[ \mathbb E\!\left[|F'(z)|^2|F(z)|^{-4}\right] =y^{-\beta-4}\mathbb E|F(z^\#)|^{-4}. \tag{30}\] The transformation \((x,y)\mapsto(u,Y)=(-x/y,1/y)\) is an involution, with \[\det\frac{\partial(u,Y)}{\partial(x,y)} =\det\begin{pmatrix}-1/y&x/y^2\\0&-1/y^2\end{pmatrix} =y^{-3}.\] Consequently \(dx\,dy=Y^{-3}du\,dY\). Moreover, \[|m(z^\#)|^2 =\frac{x^2+(y-1)^2}{x^2+(y+1)^2}=|m(z)|^2,\] so the involution preserves \(S_0\).

Lemma 4 gives \(|F|\ge1/16\) on \(S_0\). Injectivity and conformal change of variables therefore imply \[ \int_{S_0}|F'(z)|^2|F(z)|^{-4}\,dA(z) =\int_{F(S_0)}|w|^{-4}\,dA(w) \le\int_{|w|\ge1/16}|w|^{-4}\,dA(w)=256\pi. \tag{31}\] All the integrands are nonnegative and jointly Borel, by continuity of interior evaluations. Integrating (30), using Tonelli and the involution, gives \[\begin{split} \mathbb E\int_{S_0}|F'|^2|F|^{-4}\,dA &=\int_{S_0}y^{-\beta-4}\mathbb E|F(z^\#)|^{-4}\,dx\,dy\\ &=\int_{S_0}Y^{\beta+1}\mathbb E|F(u+iY)|^{-4}\,du\,dY =\mathbb EI(F). \end{split}\] The same joint measurability makes \(I\) Borel. The conclusion follows from (31). ◻

Proof. Write \(R(z)=|x|+y\). If \(z'=x'+iy'\in B(z,y/2)\), then \[y/2<y'<3y/2,\qquad R(z)\le |x'|+3y'\le3R(z').\] Thus these disks lie in \(\{R(z')\ge R(z)/3\}\) and eventually in \(S_0\). On them \(1/F\) is holomorphic. Submean for \(|1/F|^4\), followed by comparison of \(y\) with \(y'\), gives \[ y^{\beta+3}|F(z)|^{-4} \le C_\beta\!\int_{S_0\cap\{R(z')\ge R(z)/3\}} (\Im z')^{\beta+1}|F(z')|^{-4}\,dA(z') \longrightarrow0. \tag{34}\] The convergence follows from the tail of the single integrable function in (29), and is uniform in \(z\) as \(R(z)\to\infty\). Since \(4k=\beta+3\), this proves (32).

At heights \(y\ge\epsilon>0\), (32) also gives \(|F(z)|\to\infty\) as \(R(z)\to\infty\). Hence the preimage of a compact \(K\), restricted to these heights, is bounded. It is closed in the plane, because a limit stays at height at least \(\epsilon\) and \(F\) is continuous there. It is therefore compact in \(\mathbb H\). Taking \(\epsilon=1\) and then including the points below height one proves the bound on \(H_{F,K}\).

To prove (33), choose a nonnegative \(\psi\in C_c^\infty(\mathbb C)\) with \(\psi\ge1\) on \(K\), and fix \(\epsilon>0\). The function \(g=\psi\circ F\) has compact support when restricted to \(y\ge\epsilon/2\), by the compactness just proved. Thus \[h(y)=\int_{\mathbb R}g(x,y)\,dx\] is smooth for \(y\ge\epsilon\), differentiation passes under the integral, and \(h,h'\) vanish at sufficiently large height. The horizontal second derivative of \(g\) integrates to zero. Therefore \[\begin{split} \int_{y>\epsilon}(y-\epsilon)\Delta g(x,y)\,dx\,dy &=\int_\epsilon^\infty(y-\epsilon)h''(y)\,dy\\ &=[(y-\epsilon)h'(y)]_\epsilon^\infty -[h(y)]_\epsilon^\infty=h(\epsilon). \end{split}\] Write \(\Omega=F(\mathbb H)\) and \(Y(w)=\Im F^{-1}(w)\) for \(w\in\Omega\). Since \(\Delta(\psi\circ F)=|F'|^2(\Delta\psi)\circ F\), conformal change of variables gives \[h(\epsilon)= \int_{\Omega\cap\{Y>\epsilon\}}(Y(w)-\epsilon) \Delta\psi(w)\,dA(w).\] Taking the absolute value of this signed integral yields \[\int_{\mathbb R}\mathbf1_K(F(x+i\epsilon))\,dx \le h(\epsilon) \le H_{F,\operatorname{supp}\psi} \|\Delta\psi\|_{L^1(\mathbb C)}.\] This finite bound is independent of \(\epsilon\), as required. ◻

Proof of Proposition 9. Fix \(F\) with \(I(F)<\infty\), and put \(\Omega=F(\mathbb H)\). For each fixed \(\xi\in\mathbb C\) and \(\epsilon>0\), Lemma 11 implies \[ V_\xi^F(x+iy)\longrightarrow0 \quad\text{as }|x|+y\longrightarrow\infty,\quad y\ge\epsilon. \tag{35}\] Indeed \(|F|\to\infty\) in this region, so eventually \(|F-\xi|\ge|F|/2\).

For small heights first consider omitted targets. Fix \(R,\delta>0\) and a dyadic \(h\le1\). Partition \(\mathbb R\) into the half-open cells \(I_j=[jh,(j+1)h)\), \(j\in\mathbb Z\). Call a cell a witness cell if there exist \(x\in I_j\), \(h\le y\le2h\), and \(\xi\notin\Omega\) with \(|\xi|\le R\) such that \(V_\xi^F(x+iy)\ge\delta\). At any such witness \(z=x+iy\), \[|F(z)-\xi|\le(2h)^k/\delta.\] The disk \(B(z,y)\) lies in \(\mathbb H\), and its image omits \(\xi\). Lemma 3 therefore gives \[ y|F'(z)|\le4|F(z)-\xi|\le4(2h)^k/\delta. \tag{36}\] For every \(x'\in I_j\), the relative point \[(x'-x)/y+i h/y \in[-1,1]+i[1/2,1]\] belongs to a fixed compact subset of \(\mathbb H\). The values of \(T_zF\in\mathcal K\) there are bounded by a common constant \(M\), by Lemma 4. The definition of \(T_zF\) and (36) imply \[ |F(x'+ih)-\xi| \le (1+4M)(2h)^k/\delta =:C_\delta h^k. \tag{37}\]

In particular, each witness cell at height \(h\) maps entirely into the fixed target disk \(K_{R,\delta}=\{|w|\le R+C_\delta\}\). By (33), the disjoint cells, each of length \(h\), have number at most \(C_{F,R,\delta}/h\). This count permits all the omitted targets with \(|\xi|\le R\) at once. For every witness cell place a closed target disk of radius \(C_\delta h^k\) at the image of its midpoint at height \(h\). Estimate (37), applied to any witness in that cell, shows that these disks cover every target witnessed at scale \(h\). Their union \(E_h\) has area at most \[ |E_h|\le C_{F,R,\delta}h^{2k-1}. \tag{38}\] The sum over dyadic \(h\to0\) is finite. Hence \(\limsup_{n\to\infty}E_{2^{-n}}\) has area zero: its area is bounded by each tail of that convergent sum.

Take the countable union of these null sets for positive integer \(R\) and \(\delta=1/n\), \(n\ge1\). Every omitted target outside the union satisfies \[ \sup_{x\in\mathbb R}V_\xi^F(x+iy)\longrightarrow0 \qquad(y\downarrow0). \tag{39}\] Indeed each positive threshold eventually has no witness in any smaller dyadic slab, and these slabs cover all sufficiently small heights. This argument only uses finite disk covers for the fixed map \(F\).

For an interior target \(\xi\in\Omega\), choose \(a>0\) so that \(\overline B(\xi,a)\subset\Omega\). Its image under the continuous inverse \(F^{-1}\) is compact in \(\mathbb H\), and has positive minimum height. Below that height, \(|F(z)-\xi|\ge a\), so \(V_\xi^F(z)\le y^k/a\). Thus (39) holds for every interior target as well.

For each target retained above and each \(\delta>0\), (39) first excludes all sufficiently small heights from \(\{V_\xi^F\ge\delta\}\). Equation (35) then excludes all sufficiently large \(|x|+y\). Finally \[\{V_\xi^F\ge\delta\} =\{z\in\mathbb H:|F(z)-\xi|\le(\Im z)^k/\delta\}\] is closed, includes the possible pole, and lies in a bounded box with positive minimum height. It is compact in \(\mathbb H\). The assertion follows from Lemma 10. ◻

Proof. Use the compact exhaustion \(C_m=\{x+iy:|x|\le m,\ 1/m\le y\le m\}\), \(m\ge1\). The stated property is equivalent to \[\text{for every }n\ge1\text{ there is }m\ge1 \text{ such that }\{V_\xi^F>1/n\}\subset C_m.\] Necessity follows by containing the closed \(1/n\)-superlevel in a box. For sufficiency, given \(\delta>0\) choose \(1/n<\delta\); the corresponding box contains the closed \(\delta\)-superlevel. For fixed \(n,m\), the containment is equivalent to \[|F(q)-\xi|\ge n(\Im q)^k \quad\text{for every }q\in(\mathbb Q+i\mathbb Q)\cap(\mathbb H\setminus C_m).\] Any violation occurs in a nonempty open subset of the open complement of \(C_m\) and is therefore detected at a rational point. Each displayed inequality is closed in the parameters by continuity of evaluation. The countable tests prove the claim. ◻

Proof. Since \(F''/F'=iq_F/y\), the two real derivative columns are \[(G_F)_x=\frac{F'}{k}(k+q_F),\qquad (G_F)_y=\frac{iF'}{k}(k-1+q_F).\] For complex vectors \(u,v\), their real determinant is \(\operatorname{Im}(\overline u v)\). Applying this identity to the displayed columns gives (42). Local derivative continuity and compactness of \(\mathcal K\) give the asserted continuity and boundedness at \(i\). Lemma 8 gives \[\mathbb Ea_F(i)=-\frac\beta2,\qquad \mathbb E|q_F(i)|^2=\frac{\beta(\beta+1)}4.\] Because \(2k-1=(\beta+1)/2\), the last two terms in the expectation of (42) cancel. This proves (43). ◻

Proof. Write \(r=F'/(F-\xi)\). Since \(\log|F-\xi|\) is harmonic off the pole, \[\ell_x=-\operatorname{Re}r,\qquad \ell_y=\frac{k}{y}+\operatorname{Im}r,\] and (44) follows. Thus the critical equation is \(r=-ik/y\), or \(F-\xi=iyF'/k\), which is precisely \(G_F(z)=\xi\) and gives (45). A pole cannot solve this equation because \(y>0\) and \(F'\ne0\).

At a critical point the holomorphic derivative of \(r\) is \[r'=\frac{F''}{F'}r-r^2 =\frac{k(k+q_F)}{y^2}.\] Differentiating the two real gradient entries now gives (46), whose determinant is (47) by (42). Its trace is \(-k/y^2<0\). A negative determinant therefore gives a saddle; a positive determinant gives two negative eigenvalues and hence a strict maximum. At a critical point, \(\operatorname{Hess}V^F_\xi=V^F_\xi\operatorname{Hess}\ell\), so these classifications apply to \(V^F_\xi\) as well. ◻

Proof. The compact-superlevel condition is Borel by Lemma 12. For completeness, regularity can be tested on the compact source boxes \(C_m=\{|x|\le m,\ 1/m\le y\le m\}\): failure means that for some \(m\) there is \(w\in C_m\) with \[G_F(w)=\xi,\qquad J_F(w)=0.\] The set defined by these equalities is closed in \(\mathcal K\times\mathbb C\times C_m\); its projection onto \(\mathcal K\times\mathbb C\) is closed because \(C_m\) is compact. A countable union gives a Borel exceptional set.

For each fixed \(F\), the critical values of \(G_F\) have area zero. This is the plane case of Sard’s theorem (Sard 1942, Theorem 4.1); we give the short covering argument for the smooth maps at hand. Fix a compact source box and a slightly larger compact box in \(\mathbb H\). For small \(h>0\), cover the first box by \(O(h^{-2})\) squares of side \(h\). If a square meets the critical set, choose a critical point there. Taylor expansion at that point, using the bounded second derivative on the larger box, puts the image of the square in an \(O(h^2)\)-neighborhood of a line segment of length \(O(h)\): the linear part has rank at most one. This neighborhood has area \(O(h^3)\). The image of all critical points in the first box consequently has outer area \(O(h)\), and therefore area zero. Exhausting \(\mathbb H\) by boxes proves the claim. Combine it with Proposition 9 to obtain the asserted full-area sections for almost every \(F\).

The derivative identity \((T_tF)'(w)=F'(t\cdot w)/F'(t)\) gives (48) by substitution and gives the \(q\) identity in (49). Formula (42) then gives the \(J\) identity. The definition of \(V\) gives (50). Its multiplying factor is positive and independent of \(w\). The source affine map is a homeomorphism of \(\mathbb H\), while the target change is a nonsingular similarity. They preserve the two conditions defining \(\mathcal G\) in both directions. ◻

Proof. Put \(\ell=\log V^F_\xi\) away from the pole. The solutions of \(G_F(z)=\xi\) are isolated by regularity, and form a closed subset of \(\mathbb H\); isolation follows from the real inverse-function theorem (Lee 2013, Theorem C.34). Those with \(V^F_\xi\ge h\) lie in a compact set and are therefore finite. This includes the possibility of a pole in that compact set without allowing accumulation there: if \(F(z_0)=\xi\), then \(G_F(z_0)-\xi=-i\operatorname{Im}(z_0)F'(z_0)/k\ne0\).

Choose \(\lambda>0\) which is not a critical value of \(V^F_\xi\) and put \[U_\lambda=\{z\in\mathbb H:V^F_\xi(z)>\lambda\},\] including the pole if it exists. This is open, with compact closure in \(\mathbb H\). Its boundary is exactly the level \(\{V^F_\xi=\lambda\}\): the pole is an interior point of \(U_\lambda\), and at every level point the nonzero gradient gives values on both sides of \(\lambda\). The level is a compact smooth one-manifold without boundary. Its components are open in that manifold, and thus there are finitely many. Each component is a smooth regular simple closed curve. One way to see the last assertion is to flow a unit tangent field, obtained by rotating \(\nabla\ell/|\nabla\ell|\) along the level. The flow is complete by compactness. Its orbits are open in each connected component, so a component is a single orbit. If this orbit were not periodic, its parametrization would be an injective surjective local homeomorphism from \(\mathbb R\) to that compact component, which is impossible. The return times form a closed subgroup with a gap around zero by local injectivity; its least positive element gives a regular simple closed parametrization.

Consider the bounded Jordan interior \(D\) of one level curve \(\gamma\). The Jordan curve theorem is used here in its plane separation form (Munkres 2014, Theorem 63.4). The closure of \(D\) lies in \(\mathbb H\): points below the positive minimum height of \(\gamma\) connect to infinity without crossing it and hence lie in the unbounded complementary component. In fact \[ V^F_\xi>\lambda\quad\hbox{throughout }D. \tag{52}\] If a point inside had smaller value, \(\ell\) would attain a minimum below its boundary value \(\log\lambda\) at an interior point. A neighborhood of any pole can be excluded when taking this minimum, since \(\ell\) tends to \(+\infty\) there. Such a minimum contradicts \(\Delta\ell=-k/y^2<0\). Once smaller values are excluded, equality at an interior point is likewise a local minimum and is impossible.

The interiors of the level curves are disjoint and are exactly the components of \(U_\lambda\). Indeed, nested curves are excluded by (52). Conversely, a nonempty component of \(U_\lambda\) has a boundary point because its closure is compact. This point is on the level: a boundary point cannot belong to a component of the open set \(U_\lambda\). Near it, regularity and (52) identify the superlevel side with the interior side of its level curve. The component therefore meets and contains that connected Jordan interior, and cannot leave it without crossing the curve. This proves the assertion, including that no component has a hole.

The superlevel components are therefore bounded Jordan domains with smooth boundaries. It remains to turn this topological information into a count of their critical points, keeping the possible pole visible in the index sum. On the positive orientation of each boundary curve, \(\nabla\ell\) is an inward normal and has winding number \(+1\). We recall the elementary tangent-turning argument of Hopf (Hopf 1935, sec. 2, pp. 53–55) to fix this sign. Parametrize a regular simple closed curve periodically by \(\gamma:[0,1]\to\mathbb R^2\), positively oriented and starting at a leftmost point. Its initial tangent points down. The unit secant \[\frac{\gamma(t)-\gamma(s)}{|\gamma(t)-\gamma(s)|}, \qquad 0\le s<t\le1,\quad(s,t)\ne(0,1),\] extends continuously to the closed triangle: on the diagonal its value is the unit tangent, and at \((0,1)\) it is the negative initial tangent. The latter extension follows by the endpoint expansion \[\gamma(t)-\gamma(s) =-\bigl((1-t)+s\bigr)\gamma'(0) +o\bigl((1-t)+s\bigr).\] The extended secant has a continuous angle lift \(A(s,t)\) on the triangle. For example, uniform continuity permits one common number of subdivisions of every segment from \((0,0)\) to \((s,t)\) so that successive unit-vector ratios have a unique small continuous argument; adding those arguments constructs the lift. Along \((0,t)\), secants lie in the closed right semicircle, from the downward tangent to its upward negative. Thus \(A(0,1)-A(0,0)=\pi\). The secant along \((t,1)\) is the negative of that along \((0,t)\), including their extensions, so continuity gives \(A(t,1)-A(0,t)=\pi\) throughout. Therefore \(A(1,1)-A(0,0)=2\pi\). This is the turn of the tangent along the diagonal; rotation to the inward normal preserves its winding number.

Inside one component of \(U_\lambda\), remove small disjoint disks about its finitely many critical points and its pole, if present. On the remaining compact smooth domain, \(X=\nabla\ell\) is nonzero and its angle form \[\frac{X_1\,dX_2-X_2\,dX_1}{X_1^2+X_2^2}\] is closed. Stokes’ theorem (Lee 2013, Theorem 16.11) shows that the winding on the outer boundary equals the sum of the windings on the positively oriented small circles. At a nondegenerate critical point, the gradient on a sufficiently small circle is homotopic through nonzero vectors to its invertible linear part. Its index is consequently the sign of \(\det\operatorname{Hess}\ell\), namely \(+1\) at a maximum and \(-1\) at a saddle.

If \(z_0\) is the pole, the simple zero of \(F-\xi\) gives \[\ell(z)=-\log|z-z_0|+\text{a smooth function},\qquad \nabla\ell(z)=-\frac{z-z_0}{|z-z_0|^2}+O(1),\] where vectors are identified with complex numbers in the second formula. The negative radial leading field has winding number \(+1\). Thus the pole also contributes \(+1\), although it is not a finite critical point.

Let \(C_\lambda\) be the number of components of \(U_\lambda\), and let \(B\in\{0,1\}\) be the number of poles. Summing the index identity gives \[ C_\lambda =\#\{\text{maxima with }V^F_\xi>\lambda\} -\#\{\text{saddles with }V^F_\xi>\lambda\}+B. \tag{53}\] If \(B=1\), its pole belongs to \(U_\lambda\), so \(C_\lambda\ge B\). The same inequality is automatic when \(B=0\). Hence maxima are at least as numerous as saddles above every regular positive level. An empty superlevel gives the same identity with all counts zero.

Finally, for a given \(h>0\), finiteness of the critical points of value at least \(h/2\) permits a regular \(\lambda\in(h/2,h)\) with no critical value in \([\lambda,h)\). The counts above \(\lambda\) are exactly those at or above \(h\), including all ties. This proves (51). ◻

Proof. Fix a good pair \((F,\xi)\). Order its saddles by decreasing value of \(V^F_\xi\), breaking ties by increasing lexicographic order of \((x,y)\). Order its maxima separately by the same rule. Every point has finitely many predecessors, since all predecessors have value at least its own positive value. Give it the zero-based rank equal to the number of its predecessors. Distinct points have distinct ranks: if one precedes another, its predecessors and the point itself precede the latter. Moreover, if rank \(j\) occurs, the finite ordered set of its predecessors has ranks \(0,\ldots,j-1\). Thus there are no missing ranks below an attained one, whether the entire fiber is finite or infinite.

Suppose a saddle has rank \(j\) and value \(v\). There are at least \(j+1\) saddles of value at least \(v\). Lemma 17 gives at least \(j+1\) maxima of value at least \(v\). These maxima form an initial segment of the maximum order, so the maximum of rank \(j\) exists and has value at least \(v\). Assign the saddle to this maximum. This is injective within the fiber, and gives (54) by Lemma 14. Distinct fibers cannot share a point, so the assignment is injective for fixed \(F\) across its entire domain.

Figure 1 illustrates the two ordered lists.

We give the Borel construction explicitly. Countable Borel enumeration is the subject of the Lusin–Novikov theorem; see (Moschovakis 2009, 4F.6 and 4F.17). Here nonsingularity supplies continuous local branches, and the rank definition will also supply affine equivariance. Consider the nonsingular solution graph \[\Gamma=\{(F,\xi,w)\in\mathcal K\times\mathbb C\times\mathbb H: G_F(w)=\xi,\ J_F(w)\ne0\}.\] It can be covered by countably many continuous branches over open parameter sets in \(\mathcal K\times\mathbb C\). To prove this without differentiating in the parameter \(F\), fix \((F_0,\xi_0,w_0)\in\Gamma\) and the invertible real matrix \(D=DG_{F_0}(w_0)\). Joint continuity of derivatives on source compacts gives a closed ball \(\overline B(w_0,r)\subset\mathbb H\) and an open parameter neighborhood \(U\) such that \[\|I-D^{-1}DG_F(w)\|\le\frac12 \quad(w\in\overline B(w_0,r)),\qquad |D^{-1}(G_F(w_0)-\xi)|<\frac r2\] for every \((F,\xi)\in U\). The map \[w\longmapsto w-D^{-1}(G_F(w)-\xi)\] is a contraction of the closed ball into its interior. Its unique fixed point exists by the contraction principle (Lee 2013, Lemma C.35). This point solves \(G_F(w)=\xi\), is nonsingular, and depends continuously on \((F,\xi)\). Continuity follows, for instance, by taking any convergent parameter sequence and using compactness of the ball and uniqueness to identify every subsequential limit of its fixed points.

Consequently \(\Gamma\cap(U\times B(w_0,r))\) is the graph of one continuous branch over \(U\), and is an open neighborhood in \(\Gamma\). The space \(\Gamma\) is second countable, being a subspace of \(\mathcal K\times\mathbb C\times\mathbb H\). A countable collection of these branch neighborhoods therefore covers it. Enumerate the resulting continuous partial branches as \(b_j:U_j\to\mathbb H\), \(j\ge1\). Activate branch \(j\) at a parameter if the parameter is in \(U_j\) and its value differs from every earlier branch defined there. These activation sets are Borel: there are finitely many earlier indices for a given \(j\), and equality of two continuous branches is closed relative to their open overlap. Each nonsingular root occurs in exactly one active branch.

On a branch, its critical value can be evaluated without division by \(F(w)-\xi\), using the positive continuous function \[H(F,w)=k(\operatorname{Im}w)^{k-1}|Q_F(w)|.\] At every root this equals \(V^F_\xi(w)\) by (45). The signs of \(J_F\), comparisons of \(H\), equalities of values, and lexicographic comparisons of coordinates are Borel. The rank of an active branch among branches of its sign is thus the countable sum of the Borel indicators of its active predecessors. This is an extended-integer-valued Borel function, and is finite on good fibers. Requiring a positive-sign and a negative-sign active branch to have the same finite rank gives a countable Borel test for the assigned pair.

The set \(\mathcal D\) is Borel by Lemma 16 and continuity of \((F,z)\mapsto(F,G_F(z))\). At this target, require additionally that the positive-sign branch equal \(z\). Exactly one active negative-sign branch passes the equal-rank test, by the existence and uniqueness proved above. Selecting its value defines a Borel map on \(\mathcal D\): the preimage of an open source set is the countable union of the Borel tests just described with the selected branch constrained to that set.

Finally, under the affine source change \(w\mapsto t\cdot w\), the critical fibers and signs correspond by (48)–(49). All critical heights in a fiber are multiplied by the same positive constant in (50). The positive real source scale preserves lexicographic order, including ties of the first coordinate. Hence the two orders, their ranks, and the equal-rank assignment correspond exactly, giving (55). The enumeration was used only to prove measurability and does not enter the geometric definition of the assignment. ◻

Proof. For \(P\)-almost every \(F\), the complement of \(\{\xi:(F,\xi)\in\mathcal G\}\) has planar area zero by Lemma 16. Apply (57) on the positive sign with \[h(z)=\mathbf 1_{B_0}(z) \mathbf 1_{\{(F,G_F(z))\notin\mathcal G\}}, \qquad B_0=\{x+iy:|x|\leq1,\ 1\leq y\leq2\}.\] Its target integral is supported on that null set, so \[ \int_{B_0}\omega_F(z)J_{F,+}(z) \mathbf 1_{\{(F,G_F(z))\notin\mathcal G\}}\,dA(z)=0 \tag{59}\] for almost every \(F\).

Define the nonnegative Borel function \[\Phi(F)=J_{F,+}(i) \mathbf 1_{\{(F,-i/k)\notin\mathcal G\}}.\] The normalized Jacobian and goodness are affine-equivariant, so \[\Phi(T_zF)=J_{F,+}(z) \mathbf 1_{\{(F,G_F(z))\notin\mathcal G\}}.\] The weighted covariance identity consequently gives \[\mathbb E[\omega_F(z)\Phi(T_zF)] =y^{\beta-1}y^{-\beta}\mathbb E\Phi =y^{-1}\mathbb E\Phi.\] Taking expectations in (59) and using Tonelli yields \(0=2\log2\,\mathbb E\Phi\), which proves the claim. ◻

Proof. Continue under the contrary assumption \(\rho>2\). By weighted covariance and (61), \[\mathbb E[\omega_F(z)J_{F,+}(z)\mathbf 1_{D_N}(F,z)] =y^{-1}\mathbb E[J_{F,+}(i)\mathbf 1_{D_N}(F,i)].\] Likewise \(\mathbb E[\omega_F(z)J_{F,-}(z)]=y^{-1}\mathbb EJ_{F,-}(i)\). Thus expectation in (63) yields \[ \begin{split} 2R\log Y\, \mathbb E[J_{F,+}(i)\mathbf 1_{D_N}(F,i)] \leq{}& 2(R+NY)\log(N^2Y)\,\mathbb EJ_{F,-}(i). \end{split} \tag{64}\] The root quantities are integrable: \(q_F(i)\) is uniformly bounded on the compact normalized class, and \(J_F(i)\) is a fixed quadratic expression in this jet.

Hold \(N\) fixed, set \(R=Y^2\), and let \(Y\to\infty\). The ratio of the right box factor to the left one is \[\left(1+\frac NY\right) \left(1+\frac{2\log N}{\log Y}\right)\longrightarrow1.\] Dividing (64) by \(2R\log Y\) gives \[\mathbb E[J_{F,+}(i)\mathbf 1_{D_N}(F,i)]\leq\mathbb EJ_{F,-}(i).\] Now let integer \(N\to\infty\). Monotone convergence exhausts the good-target saddles, and Lemma 19 accounts for the remaining positive-J weight. Hence \[\mathbb EJ_{F,+}(i)\leq\mathbb EJ_{F,-}(i),\qquad \mathbb EJ_F(i)\leq0.\] For the same \(J_F\) and the same law \(P\), Lemma 13 gives \[k^2\mathbb EJ_F(i) =k(k-1)-(2k-1)\frac{\beta}{2} +\frac{\beta(\beta+1)}4 =k(k-1)>0,\] since \(2k-1=(\beta+1)/2\) and \(k>1\). This contradiction proves \(\rho\leq2\). ◻

Proof of the integral-means assertion in Theorem 1. Proposition 20 gives \(\rho\leq2\). Proposition 6 therefore implies that, for every \(\epsilon>0\), there is \(C_\epsilon<\infty\), independent of \(f\in\mathcal S\), such that \[ M_{-2}[f'](r)\leq C_\epsilon(1-r)^{-1-\epsilon}, \qquad \frac12\leq r<1. \tag{65}\] In particular \(\beta_f(-2)\leq1\) for every \(f\in\mathcal S\).

For the reverse inequality consider the Koebe map \[f_{\mathrm K}(z)=\frac{z}{(1-z)^2},\qquad f_{\mathrm K}'(z)=\frac{1+z}{(1-z)^3}.\] It belongs to \(\mathcal S\): equality of its values at \(z,u\in\mathbb D\) gives \((z-u)(1-zu)=0\), hence \(z=u\). Write \(\delta=1-r\) and take \(r\) sufficiently close to \(1\). On the arc \(z=re^{i(\pi+\theta)}\) with \(|\theta|\leq\delta\), \[|1+z|\leq\delta+r|\theta|\leq2\delta,\qquad |1-z|\geq1.\] Consequently \[M_{-2}[f_{\mathrm K}'](r) \geq \frac1{2\pi}\int_{-\delta}^{\delta} \frac{d\theta}{4\delta^2} =\frac1{4\pi\delta}.\] No exponent smaller than \(1\) can bound these means near the boundary. Hence \(\beta_{f_{\mathrm K}}(-2)\geq1\), and \(B_{\mathcal S}(-2)=1\). ◻

Proof. Put \(a=|t|-2\geq0\). The distortion lower bound (3) and (65) give \[\begin{aligned} M_t[f'](r) &\leq\left(\frac{(1+r)^3}{1-r}\right)^a M_{-2}[f'](r)\\ &\leq 8^a C_\epsilon(1-r)^{-(a+1)-\epsilon}. \end{aligned}\] Since \(a+1=|t|-1\), this proves (66) with \(C_{t,\epsilon}=8^{|t|-2}C_\epsilon\). At \(t=-2\) it is exactly (65). For each fixed \(f\) and \(t\), letting \(\epsilon\downarrow0\) in the definition of \(\beta_f(t)\) gives \(\beta_f(t)\leq|t|-1\); taking the supremum over \(f\) gives the same upper bound for \(B_{\mathcal S}(t)\).

For the Koebe map and the arc used above, \(|f_{\mathrm K}'(z)|\leq2\delta\). Thus, for \(\delta=1-r\) sufficiently small, \[M_t[f_{\mathrm K}'](r) \geq\frac1{2\pi}\int_{-\delta}^{\delta}(2\delta)^{-|t|}\,d\theta =\frac{2^{-|t|}}{\pi}\delta^{1-|t|}.\] Hence \(\beta_{f_{\mathrm K}}(t)\geq|t|-1\), proving the equality. ◻

Proof. Away from the simple zero of \(f\) at \(0\), \[\Delta |f|^p=p^2|f|^{p-2}|f'|^2.\] The right side is locally integrable also at \(0\), where it is \(O(|z|^{p-2})\). Green’s radial mean identity gives \[ \frac1{2\pi}\int_{-\pi}^{\pi}|f(re^{i\theta})|^p\,d\theta =\frac{p^2}{2\pi}\int_{|z|<r} |f(z)|^{p-2}|f'(z)|^2\log\frac r{|z|}\,dA(z). \tag{71}\] To justify it at the zero, apply Green’s theorem on an annulus (Lee 2013, Theorem 16.17) with inner radius \(\delta\). The circular mean at that radius is \(O(\delta^p)\) and its derivative is \(O(\delta^{p-1})\), because \(f(z)=z+O(z^2)\). The inner terms are consequently \(O(\delta^p(1+|\log\delta|))\) and vanish as \(\delta\downarrow0\).

Bound the logarithm in (71) by \(\log(1/|z|)\). On \(|z|\leq1/2\), the classical bounds give \(|f(z)|\geq c|z|\) and \(|f'(z)|\leq C\), uniformly in \(\mathcal S\). Since \(p-2<0\), this part is bounded by a constant times \[\int_0^{1/2}u^{p-1}\log(1/u)\,du<\infty.\] On \(|z|>1/2\), the growth bound \(|f(z)|\leq |z|/(1-|z|)^2\) implies \[\log\frac1{|z|} \leq C(1-|z|) \leq C\min\{1,|f(z)|^{-1/2}\}.\] The conformal change of variables \(w=f(z)\) therefore bounds the remaining integral by a constant times \[\int_{\mathbb C}|w|^{p-2}\min\{1,|w|^{-1/2}\}\,dA(w) =2\pi\left(\int_0^1u^{p-1}\,du +\int_1^\infty u^{p-3/2}\,du\right)<\infty.\] This proves (69).

For (70), the case \(a=0\) is immediate. For \(0<a<1\), choose \[\max\{0,a-\tfrac12\}<p<\tfrac12.\] If \(a\leq p\), the power-mean inequality and (69) bound the means of \(|f|^a\). If \(a>p\), the growth bound gives instead \[\frac1{2\pi}\int_{-\pi}^{\pi}|f(re^{i\theta})|^a\,d\theta \leq \left(\max_{|z|=r}|f(z)|\right)^{a-p}C_p \leq C_p(1-r)^{-2(a-p)}.\] Thus the means of \((1+|f|)^a\) are bounded by \(C(1-r)^{-2(a-p)_+}\), where \((b)_+=\max\{b,0\}\). Since \(2(a-p)_+<1\), radial integration proves the claim. ◻

Proof. Choose \(0<\eta<2/t-3\) and set \[a=\frac{(2+\eta)t}{2-t}\in(0,1),\qquad u_\eta(z)=|f'(z)|^2(1+|f(z)|)^{-2-\eta}.\] Injective conformal area change gives \[ \int_{\mathbb D}u_\eta\,dA =\int_{f(\mathbb D)}(1+|w|)^{-2-\eta}\,dA(w) \leq\frac{2\pi}{\eta(1+\eta)}. \tag{72}\] Here the last constant is the polar integral over \(\mathbb C\). The exact factorization \[|f'|^t=u_\eta^{t/2}\bigl((1+|f|)^a\bigr)^{(2-t)/2}\] and Hölder’s inequality with exponents \(2/t\) and \(2/(2-t)\) now yield \[\int_{\mathbb D}|f'|^t\,dA \leq \left(\int_{\mathbb D}u_\eta\,dA\right)^{t/2} \left(\int_{\mathbb D}(1+|f|)^a\,dA\right)^{(2-t)/2}<\infty\] by Lemma 22. ◻

Proof of the area assertion in Theorem 1. Equations (68) and Lemma 23, together with \(t=0\), prove the disk integral for \(f\in\mathcal S\) throughout \(-2<t<2/3\). For an arbitrary univalent \(f:\mathbb D\to\mathbb C\), \[g(z)=\frac{f(z)-f(0)}{f'(0)}\in\mathcal S,\qquad \int_{\mathbb D}|f'|^t\,dA =|f'(0)|^t\int_{\mathbb D}|g'|^t\,dA.\] The multiplier is finite and positive for every real \(t\).

If \(\varphi:W\to\mathbb D\) is a conformal bijection and \(f=\varphi^{-1}\), the identities \(\varphi'(f(w))=1/f'(w)\) and \(\,dA(f(w))=|f'(w)|^2\,dA(w)\) give \[ \int_W|\varphi'(z)|^s\,dA(z) =\int_{\mathbb D}|f'(w)|^{2-s}\,dA(w). \tag{73}\] This change of variables is valid for nonnegative integrands, whether or not the integrals are finite. Since \(4/3<s<4\) is equivalent to \(-2<2-s<2/3\), the desired domain integral is finite. ◻

Proof. This is the known Sobolev formulation of Brennan’s conjecture due to Gol’dshtein and Ukhlov (Gol’dshtein and Ukhlov 2013, Equivalence Theorem and its following Remark 2). Indeed, \(1<q<p\) and \((p-2)q/(p-q)=s\), so Theorem 1 supplies the derivative integrability required by their equivalence. The cited remark gives the case \(p=\infty\), with \(q=s\). ◻

Proof of Corollary 2. Use the same Koebe map \(f_{\mathrm K}\). For sufficiently small \(|w|\), \[|f_{\mathrm K}'(-1+w)|=\frac{|w|}{|2-w|^3}\asymp |w|, \qquad |f_{\mathrm K}'(1-w)|=\frac{|2-w|}{|w|^3}\asymp |w|^{-3}.\] The fixed sector \[0<|w|<\tfrac12,\qquad |\arg w|<\tfrac{\pi}{3}\] lies inside \(\mathbb D\) under either substitution \(z=-1+w\) or \(z=1-w\): in both cases \(|z|^2=1-2\Re w+|w|^2<1\). The area integral at \(t=-2\) near \(-1\) and the one at \(t=2/3\) near \(1\) therefore each dominate a positive multiple of \[\int_0^{1/2}u^{-2}u\,du=\infty.\] More generally these sectors diverge for \(t\leq-2\) and \(t\geq2/3\), respectively. The image of \(f_{\mathrm K}\) meets the domain hypotheses just verified, so its inverse, by (73), also gives divergence at \(s=4\) and \(s=4/3\). Thus both open endpoints are sharp. ◻