Alternate Proof of the Inner Analyticity of $w_{\epsilon }(z)$

Here we establish that $w_{\epsilon }(z)$ is analytic by showing that its real and imaginary parts satisfy the Cauchy-Riemann conditions. Consider an inner analytic function $w(z)$ and the corresponding filtered function within the open unit disk, with the real angular parameter $0<\epsilon\leq\pi$

$$w_{\epsilon}(z) = \frac{1}{2\epsilon} \int_{\theta-\epsilon}^{\theta+\epsilon}d\theta'\, w\!\left(z'\right).$$

Since $w(z)$ is analytic, we have $w(z)=f_{\rm c}(\rho,\theta)+\mbox{\boldmath$\imath$}f_{\rm s}(\rho,\theta)$ where $f_{\rm c}(\rho,\theta)$ and $f_{\rm s}(\rho,\theta)$ satisfy the Cauchy-Riemann conditions in polar coordinates,

$$\begin{eqnarray*} \frac{\partial f_{\rm c}}{\partial\rho}(\rho,\theta) & = & ... ... = & -\, \frac{\partial f_{\rm s}}{\partial\rho}(\rho,\theta). \end{eqnarray*}$$

It follows that the filtered function can be written as

$$\begin{eqnarray*} w_{\epsilon}(z) & = & f_{\epsilon,{\rm c}}(\rho,\theta)+\mb... ...theta+\epsilon}d\theta'\, f_{\rm s}\!\left(\rho,\theta'\right), \end{eqnarray*}$$

so that we have

$$\begin{eqnarray*} f_{\epsilon,{\rm c}}(\rho,\theta) & = & \frac{1}{2\epsilon}... ...theta+\epsilon}d\theta'\, f_{\rm s}\!\left(\rho,\theta'\right). \end{eqnarray*}$$

Since $f_{\rm c}(\rho,\theta)$ and $f_{\rm s}(\rho,\theta)$ are continuous and differentiable, it is clear that so are $f_{\epsilon,{\rm c}}(\rho,\theta)$ and $f_{\epsilon,{\rm s}}(\rho,\theta)$. If we calculate their partial derivatives with respect to $\rho$ we get

$$\begin{eqnarray*} \frac{\partial f_{\epsilon,{\rm c}}}{\partial\rho}(\rho,\thet... ...ac{\partial f_{\rm s}}{\partial\rho}\!\left(\rho,\theta'\right). \end{eqnarray*}$$

Using the Cauchy-Riemann relations for $w(z)$ we may write these as

$$\begin{eqnarray*} \frac{\partial f_{\epsilon,{\rm c}}}{\partial\rho}(\rho,\thet... ...,\theta+\epsilon)-f_{\rm c}(\rho,\theta-\epsilon)} {2\epsilon}. \end{eqnarray*}$$

If we now calculate the partial derivatives of $f_{\epsilon,{\rm c}}(\rho,\theta)$ and $f_{\epsilon,{\rm s}}(\rho,\theta)$ with respect to $\theta$ we get

$$\begin{eqnarray*} \frac{1}{\rho}\, \frac{\partial f_{\epsilon,{\rm c}}}{\parti... ...,\theta+\epsilon)-f_{\rm s}(\rho,\theta-\epsilon)} {2\epsilon}. \end{eqnarray*}$$

Comparing this pair of equations with the previous one we get

$$\begin{eqnarray*} \frac{\partial f_{\epsilon,{\rm c}}}{\partial\rho}(\rho,\thet... ...\frac{\partial f_{\epsilon,{\rm s}}}{\partial\rho}(\rho,\theta), \end{eqnarray*}$$

which establish the analyticity of $w_{\epsilon }(z)$, in the same domain as that of $w(z)$. Let us now examine the other properties defining an inner analytic function. For one thing we have $w(0)=0$, which means that

$$\begin{eqnarray*} \lim_{\rho\to 0} f_{\rm c}(\rho,\theta) & = & 0, \\ \lim_{\rho\to 0} f_{\rm s}(\rho,\theta) & = & 0. \end{eqnarray*}$$

If we calculate the corresponding limits for $w_{\epsilon }(z)$ we get

$$\begin{eqnarray*} \lim_{\rho\to 0} f_{\epsilon,{\rm c}}(\rho,\theta) & = & \... ...\rho\to 0} f_{\rm s}\!\left(\rho,\theta'\right) \\ & = & 0. \end{eqnarray*}$$

We have therefore that $w_{\epsilon}(0)=0$. Finally, $w(z)$ reduces to a real function over the interval $(-1,1)$ of the real axis, which means that its imaginary part is zero there, and therefore that $f_{\rm s}(\rho,0)=0$ and $f_{\rm s}(\rho,\pm\pi)=0$. If we write $f_{\epsilon,{\rm s}}(\rho,\theta)$ for the same values of $\theta$ we get

$$\begin{eqnarray*} f_{\epsilon,{\rm s}}(\rho,0) & = & \frac{1}{2\epsilon} \in... ...^{\pi+\epsilon}d\theta'\, f_{\rm s}\!\left(\rho,\theta'\right). \end{eqnarray*}$$

However, since $w(z)$ is an inner analytic function we have that $f_{\rm s}(\rho,\theta)$ is an odd function of $\theta$. In both cases above this implies that the integral is zero, and hence we conclude that $w_{\epsilon }(z)$ is an inner analytic function as well.