Let $h$ be a function from $\mathbb{R}$ to $\mathbb{R}$, continuous and $2\pi$-periodic on $\mathbb{R}$. For any complex number $z$ such that $|z| < 1$, set $$g(z) = \frac{1}{2\pi} \int_0^{2\pi} h(t) \mathcal{P}(t,z)\, \mathrm{d}t \quad \text{where} \quad \mathcal{P}(t,z) = \operatorname{Re}\left(\frac{\mathrm{e}^{\mathrm{i}t} + z}{\mathrm{e}^{\mathrm{i}t} - z}\right)$$ Using Heine's theorem, show that, for all $\varepsilon > 0$, there exists $\delta > 0$ such that, for all real number $\varphi$ and all complex number $z$ satisfying $|z| < 1$, $$|g(z) - h(\varphi)| \leqslant \frac{\sup_{t \in \mathbb{R}} |h(t)|}{\pi} \int_{\varphi+\delta}^{\varphi+2\pi-\delta} \mathcal{P}(t,z)\, \mathrm{d}t + \varepsilon$$
Let $h$ be a function from $\mathbb{R}$ to $\mathbb{R}$, continuous and $2\pi$-periodic on $\mathbb{R}$. For any complex number $z$ such that $|z| < 1$, set
$$g(z) = \frac{1}{2\pi} \int_0^{2\pi} h(t) \mathcal{P}(t,z)\, \mathrm{d}t \quad \text{where} \quad \mathcal{P}(t,z) = \operatorname{Re}\left(\frac{\mathrm{e}^{\mathrm{i}t} + z}{\mathrm{e}^{\mathrm{i}t} - z}\right)$$
Using Heine's theorem, show that, for all $\varepsilon > 0$, there exists $\delta > 0$ such that, for all real number $\varphi$ and all complex number $z$ satisfying $|z| < 1$,
$$|g(z) - h(\varphi)| \leqslant \frac{\sup_{t \in \mathbb{R}} |h(t)|}{\pi} \int_{\varphi+\delta}^{\varphi+2\pi-\delta} \mathcal{P}(t,z)\, \mathrm{d}t + \varepsilon$$