Let $\mathbb{D}$ be the open unit disc in $\mathbb{C}.$
Let $w_1, w_2 \in \overline{\mathbb{D}}$ with $w_1\neq w_2.$
Let $\tilde{c}(w_1, w_2)= \sup_f\left\{\left|\frac{f(w_1)-f(w_2)}{1-\overline{f(w_2)}f(w_1)}\right| : f\in \tilde{A}(\mathbb{D})\right\}$, where
$\tilde{A}(\mathbb{D})= \{ f\in$ Disc algebra$ : f(\overline{\mathbb{D}})\subset \mathbb{D}\}$.
Where the disc Algebra is the space of all complex valued functions that are analytic on the unit disc $\mathbb{D}$ and are continuous on $\overline{\mathbb{D}}.$ Is it true that $\tilde{c}(w_1, w_2)$ is always strictly less than 1?
This question is similar to a previous question. The difference is that in the previous question the supremum was taken over bounded continuous functions, but here the supremum is taken over bounded analytic functions.
The Schwarz-Pick theorem states that $$ \left |\frac{f(w_1) - f(w_2)}{1 - \overline{f(w_2)} f(w_1)} \right| \le \left |\frac{w_1 - w_2}{1 - \overline{w_2} w_1} \right| $$ for all holomorphic functions $f: \Bbb D \to \Bbb D$. It follows that $$ \tilde{c}(w_1, w_2) \le \left |\frac{w_1 - w_2}{1 - \overline{w_2} w_1} \right| < 1 \, . $$