As the title states, I'm trying to construct a family of uniformly bounded holomorphic functions on the unit disk that are not uniformly Lipschitz on the unit disk.
More precisely, I want a family $\mathcal{F}$ of holomorphic functions such that there exists $M > 0$ such that $|f(z)| \leq M$ for all $f \in \mathcal{F}$ and $z \in \mathbb{D}$, the open unit disk, and these functions must NOT be uniformly Lipschitz. (By uniformly Lipschitz, I mean there exists $\lambda > 0$ such that $|f(z) - f(w)| \leq \lambda |z-w|$ for all $f \in \mathcal{F}$ and $z,w \in \mathbb{D}$.)
I already proved that such a family must actually be uniformly Lipschitz on compact sets, but I'm struggling to find a counterexample when we consider the entire unit disk. I've been playing around with functions like $z^n$ and seeing if I could modify it to get what I need but to no avail.
Let $f_n(z)=z^{n}$. Note that $|(1-\frac 1n)^{n}-(1-\frac 1{2n})^{n}|\leq\ \lambda \frac 1 {2n}$ cannot hold since LHS tends to $e^{-1}-e^{-1/2}$. So $(f_n)$ is an example of the type you want.