Before I state my actual question, I think it is important to clearly outline the definitions I am working with.
The below definitions can be found in the book Logarithmic Potentials with External Fields by Edward B. Saff, Vilmos Totik, on pages 89-91:
With these definitions, my question is the following:
In the below pictures, the curve $\gamma$ is of class $C^{1 + \delta}$ for some fixed $\delta > 0$, and is simple and regular, i.e. it is injective and its derivative is nowhere vanishing. The part that confuses me in the argument of the picture shown below is how the integral over $A_{h} z_0$ is computed to be bounded below by the quantities to the right, I can work out for myself how the bound is achieved assuming $|t - z_0| \geq h$ but am totally lost on the bound obtained for when $|t - z_0| < h$ (here $t$ is always assumed to lie on $\gamma$).
Moreover, I am not sure how to prove that one can draw the segment $A_{h} z_0$ so that up to $z_0$, it is wholly contained in a left neighborhood of $z_0 \in \gamma$ contained in $D_{+}$. The biggest hurdle here for me is that I do not know how to use the definition of a 'sector' to show that :
For any positive $\theta > 0$, such a sector is nonempty and contains points arbitrarily close to $z_0$
For points in the sector sufficiently close to $z_0$, one can always draw a line segment like $A_{h} z_0$ contained wholly in the left neighborhood of $z_0$ that the sector is contained in (except for $z_0 \in \gamma$).
Thank you in advance for any help.