\right)^\frac12\xi\right\|_2^2$ exist and equal $\langle c_j\xi,\xi\rangle$, finite for any $\xi\in\mathcal H$. Thus, the family $\left\{\left\|\frac{\left(\Im f(\alpha+iyv_j)\right)^{1/2}}{\sqrt{y}}
\xi\right\|_2\colon y\in(0,1)\right\}$ is bounded for any $\xi\in\mathcal H$. By the Banach-Steinhaus Theorem and the positivity of the operators $\frac{\Im f(\alpha+iyv_j)}{{y}}$, it follows that $\left\{\left\|\frac{\Im f(\alpha+iyv_j)}{{y}}\right\|\colon 
y\in(0,1)\right\}$ is a bounded set. Moreover, as it will be seen in the proof of Theorem \[JC\], if $z$ tends to zero nontangentially and $\lim_{y\downarrow 0}\frac{\langle\Im f(\alpha+iyv_j)\xi,\xi\rangle}{y}$ is finite, then $\left\{\frac{\langle\Im f(\alpha+\Im zv_j)\xi,\xi\rangle}{\Im z}\colon|z|<1,z\in\Gamma\right\}$ stays bounded for any closed cone $\Gamma\subset\mathbb C^+\cup\{0\}$. A bound for $c_j$ is $\|c_j\|\leq\limsup_{y\to0}\left\|\frac{\Im f
(\alpha+iyv_j)}{{y}}\right\|$. Thus, $\{\|W(z,\zeta,\varepsilon)\|\colon
z,\zeta\in\Gamma,|z|,|\zeta|<1\}$ is bounded for any closed cone $\Gamma\subset\mathbb C^+$ with vertex at zero. The lemma follows by letting $\varepsilon\downarrow0$.

We note that the bounds depend exclusively on $c_j,v_j (j=1,2),w$. Moreover, the dependence can be bounded (at most) linearly in terms of $\|w\|,\|v_1\|,\|v_2\|,\|v_1^{-1}\|$ and $\|v_2^{-1}\|$.

For the sake of completeness, let us use the results of Proposition \[prop:3.1\] to give a short, elementary proof of Theorem \[JC\].

Assume equation holds. By Proposition \[prop:3.1\], $$\left|\frac{f(z)-f(z')}{\sqrt{\Im f(z)\Im f(z')}}\right|\leq
\left|\frac{z-z'}{\sqrt{\Im z\Im z'}}\right|,\quad z,z'\in
\mathbb C^+.$$ This is equivalent to $$\label{9}
\left|\frac{f(z)-f(z')}{z-z'}\right|^2\leq
\left|\frac{\Im f(z)\Im f(z')}{\Im z\Im z'}\right|,\quad z,z'\in
\mathbb C^+, z\neq z'.$$ Consider a sequence $\{z_n'\}_{n\in\mathbb N}\subset\mathbb C^+$ converging to $\alpha$ such 