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For $g^{\prime }(0)\ne 0$, apply Jensen's formula with the smooth curvature term to the hyperbolic derivative density. Its distributional Laplacian is $H_g^2+2\pi \sum m_a{\delta }_a$, giving the displayed identity with ramification points defined by $g^{\prime }(a)=0$. If $g^{\prime }(z)=c{z}^m+\cdots$ at zero, use $m\log R$ as the origin counting term and replace the base constant by $\log {\lim }_{z\to 0}{H}_g(z)/|z{|}^m$. Without this regularization the basepoint-critical formula is undefined.

There are two necessary qualifications to the printed formula. Critical points satisfy $g^{\prime }=0$, not $g^{\prime }=1$. At a point with $g^{\prime }=1$, the local degree is one and the printed summand is zero. In addition, $\log \Vert g^{\prime }(0)\Vert$ is finite only when $g^{\prime }(0)\ne 0$; constant maps and origin-critical maps need separate treatment.

These are genuine issues even for otherwise ordinary disc maps. The map $g(z)=(z-1/2{)}^2/4$ maps the unit disc into itself and has $g^{\prime }(0)\ne 0$. Its derivative never equals one in the disc, so the printed count is zero, but it has a ramification point at $1/2$ of multiplicity one. For $R>1/2$ the correct identity has the additional term $\log (2R)$, proving that the displayed identity with the literal printed count is false. Also $g(z)=z^2$ shows the undefined basepoint logarithm in an origin-critical example.

Prove the intended identity first for a nonconstant map with $g^{\prime }(0)\ne 0$. Let $H(z)=2|g^{\prime }(z)|/(1-|g(z){|}^2)$ be its hyperbolic derivative density and $u=\log H$. Away from derivative zeros, $\log |g^{\prime }|$ is harmonic, and direct differentiation givesAt a zero $a$ of $g^{\prime }$ of multiplicity $m_a$, factor $g^{\prime }=(z-a{)}^{m_a}h$ with $h(a)\ne 0$. Its logarithm contains $m_a\log |z-a|$, whose distributional Laplacian is $2\pi m_a{\delta }_a$. ThusThe local map degree is $m_a+1$, so this is the required excess-degree weighting.

For a smooth function, differentiating its circular mean and applying the divergence theorem gives ${\overline{u}}^{\prime }(r)=(2\pi r{)}^{-1}{\int }_{|z|<r}\Delta udA$. Integrating in $r$ gives the logarithmic Green-Jensen formula. Factoring the isolated logarithmic singularities extends it to the present $u$:Substitution yields the critical-point Jensen identity for the hyperbolic derivativewhere $N_{\mathrm{crit}}={\sum }_{|a|<R:g^{\prime }(a)=0}{m}_a\log (R/|a|)$. This is the requested formula with the critical-point typo corrected. Initially choose a circle containing no ramification point on its boundary; continuity and integrability of logarithmic boundary singularities extend the result to the remaining radii.

By Schwarz-Pick theorem, $H(z)\le 2/(1-|z{|}^2)$. ThereforeThe normalized hyperbolic distance is $\rho (R)=\log ((1+R)/(1-R))$, andConsequentlyOne can take $C_1=1$ and $C_2=\log (2/H(0))\ge 0$. Since the corrected count is nonnegative in the noncritical-basepoint case, this also proves the growth bound for $T_{\mathbb{D}}$ with the literal printed, identically zero count, though not its false identity.

For completeness, if $g^{\prime }$ vanishes to order $m\ge 1$ at zero, put $c={\lim }_{z\to 0}H(z)/|z{|}^m>0$ andApply the same formula to $\log H-m\log |z|$, which is regular at zero. It givesThe origin contribution is a regularized Jensen term, not the undefined $\log (R/0)$. If one instead omits the origin term, the upper bound gains $-m\log R$; this is bounded for $R\ge 1/2$, while the characteristic and off-origin count are bounded on any smaller compact disc. Thus a bound $C_1\rho +C_2$ still holds for that nonnegative-count convention after enlarging $C_2$. For a constant map $H=0$ and $T_{\mathbb{D}}=0$, so its characteristic growth bound is trivial; the logarithmic identity and a finite critical-divisor count are not defined. These qualifications resolve all maps allowed by the original opening sentence without asserting an undefined equality.