Research Problems in Function Theory — Problem 6.56

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Write $\mathbb D=\{z\in\mathbb C:|z|<1\}$ and $\mathbb T=\partial\mathbb D$. Let $S$ consist of holomorphic injective functions $f(z)=z+\sum_{n\ge2}a_nz^n$ on $\mathbb D$; set $a_0=0$, $a_1=1$. A function in $S$ is starlike if its image is starlike about $0$, and convex if its image is a convex domain. For Taylor series $f(z)=\sum a_nz^n$ and $g(z)=\sum b_nz^n$, their Hadamard convolution is $(f*g)(z)=\sum a_nb_nz^n$, and their integrated convolution is $(f\otimes g)(z)=\sum_{n\ge1}(a_nb_n/n)z^n$. These series are holomorphic on $\mathbb D$ when both original functions are. A $q$-quasiconformal extension is an orientation-preserving homeomorphism of the Riemann sphere fixing $\infty$ (equivalently a homeomorphism of the whole plane extended at infinity), locally in $W^{1,2}$ in charts, agreeing with the function on its domain and satisfying $|\partial_{\bar z}F|\le q|\partial_zF|$ almost everywhere for $0\le q<1$.For $0\le q<1$, let $S_R(q)$ comprise the real-coefficient functions in $S$ admitting a $q$-quasiconformal extension.

Statement recorded in the supplied source.

Determine the values $0\le r<1$ for which the implication $f\in S_R(p),\ g\in S_R(q)\Longrightarrow f\otimes g\in S_R(r)$ holds for every pair $f,g$, as a function of $0\le p,q<1$. In particular, does a uniform choice $r=r(p,q)<1$ always exist? [S2]

Short English statement

Does integrated convolution preserve quasiconformal extendibility for real-coefficient univalent maps, and what extension bound follows from the two initial bounds?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306056 (AMR-022-6056). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), arXiv:1809.07200, Chapter 6, Problem 6.56, complete update and preceding notation. https://arxiv.org/abs/1809.07200.