Let's assume that $f$ is a quasiconformal homeomorphism of $\mathbb{C}$ with Beltrami coefficient $\mu = \frac{\bar{\partial} f}{\partial f}$. Notice that by definition $\Vert \mu \Vert _{L^{\infty}} < 1$. It is well known that if $\mu_n \rightarrow \mu$ in $L^{\infty}$ then $f_n \rightarrow f$ in the compact open topology (hence pointwise at least). I believe (but am not sure) that the rate of convergence is also known.

My question is - In the above setting if $\mu_n \rightarrow \mu$ in $L^p$ for some $\infty >p>1$ then can we say (preferably with effective convergence estimates) that $f_n$ still goes to $f$ in the compact-open topology (or atleast pointwise)?

Let's assume that $f$ is a quasiconformal homeomorphism of $\mathbb{C}$ with Beltrami coefficient $\mu = \frac{\bar{\partial} f}{\partial f}$. Notice that by definition $\Vert \mu \Vert _{L^{\infty}} < 1$. It is well known that if $\mu_n \rightarrow \mu$ in $L^{\infty}$ then $f_n \rightarrow f$ (the f's are all normalised to fix three points) in the compact open topology (hence pointwise at least). I believe (but am not sure) that the rate of convergence is also known.

My question is - In the above setting if $\mu_n \rightarrow \mu$ in $L^p$ for some $\infty >p>1$ then can we say (preferably with effective convergence estimates) that $f_n$ still goes to $f$ in the compact-open topology (or atleast pointwise)? It is also assumed that $\Vert \mu _n \Vert _{L^{\infty}} \rightarrow \Vert \mu \Vert _{L^{\infty}}$.