Modular graph (modulus surface) of an analytic function

Prove the identity:

The function $w = f(z)$ is regular in a simply connected domain bounded by the curve $\left|w(z)\right| = c$. Show that $\arg w$ varies monotonically as $z$ traverses the curve $\left|w(z)\right| = c$ in the positive direction.

We take the cylinder over the disk $\left|z\right|\leq r$ and intersect it with the modular graph of the function

The volume of that part of the cylinder that is contained between the $z$-plane and the modular graph

Let $\gamma (z)$ denote the angle between the $x,y$-plane and the tangential plane of the modular graph at $z$, $\left|f(z)\right|^{2}$. Then

A point of the modular graph with a horizontal tangent plane belongs to one of two types, it is a pit'' or a saddle point'': If the tangent plane is the $x,y$-plane, then there are only isolated points of the graph on it, pits. If the tangent plane is not the $x,y$-plane it intersects the modular graph along a curve (level line) $2n$ branches of which meet under equal angles, $\frac{2\pi }{2n}$, at the point of contact which is a saddle point. (For a mountain pass, a saddle point in an actual landscape, we expect $n=2$.) The $2n$ regions of the modular graph determined locally in this way lie alternately above and below the tangent plane. (All the pits are in the $x,y$-plane, all the saddle points above, at diverse heights.)

The intersection of the modular graph of a polynomial with real zeros only and of a plane perpendicular to the $x$-axis is a convex curve the lowest point of which lies in the $x,\zeta$-plane.

What does the maximum principle say about the modular graph?

Consider the domain bounded by a closed level line ($\left|f(z)\right|$ is constant along this curve) without selfintersections and lying inside the region where $f(z)$ is regular. It contains at least one zero of $f(z)$ unless $f(z)$ is a constant.

Let the function $f(z)$ be regular on the disk $\left|z\right|\leq r$ and $z_{0}$, $\left|z_{0}\right|=r$, be a point at which $\left|f(z)\right|$ assumes its maximum. Then $z_{0}\dfrac {f’(z_{0})}{f(z_{0})}$ is real and positive. [Problems 103, 132.]

Here $L$ denotes a closed continuous curve without double points and $\mathfrak {D}$ the closed interior of $L$.

The function $f(z)$ has one zero more than $f'(z)$ inside $L$ under the hypothesis of problem 191. (More informative than problem 142.) This means geometrically: Inside a closed level line without double points the modular graph has more pits than saddle points, namely precisely one more.