Research Problems in Function Theory — Problem 6.103

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 $S_H$ consist of injective, sense-preserving harmonic functions on $\mathbb D$ with the expansion \[f(z)=\sum_{n=1}^{\infty}a_nz^n+\sum_{n=1}^{\infty}a_{-n}\overline z^{\,n},\qquad a_1=1.\] Harmonicity means that the real and imaginary parts satisfy the planar Laplace equation. Writing $f=h+\overline g$ with $h,g$ holomorphic, sense preservation is $|h'(z)|^2-|g'(z)|^2>0$ throughout $\mathbb D$. The subclass $S_H^0$ is defined by $a_{-1}=0$. Let \[k(z)=2\operatorname{Re}\frac{z+z^3/3}{(1-z)^3} =\sum_{n=1}^{\infty}\frac{2n^2+1}{3}(z^n+\overline z^{\,n}).\] The closure $\overline{S_H}$ means locally uniform limits on $\mathbb D$ of maps in $S_H$; the source places $k$ in this closure.

Statement recorded in the supplied source.

Is $k$ extremal for the coefficient bounds in $S_H$? Precisely, for every $n\geq2$, are the sharp upper bounds \[\sup_{f\in S_H}|a_n|=\sup_{f\in S_H}|a_{-n}|=\frac{2n^2+1}{3},\] with the displayed limit map furnishing the extremal coefficients? [S2] [S3]

Short English statement

Are the quadratic-in-degree coefficients of the displayed limiting harmonic map the sharp coefficient bounds for all normalized univalent harmonic maps?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306103 (AMR-022-6103). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Problem 6.103, complete problem and update, harmonic-class notation preceding Problem6.103. https://arxiv.org/abs/1809.07200.
• S3 S. Ponnusamy and A. Sairam Kaliraj, On the Coefficient Conjecture of Clunie and Sheil-Small on Univalent Harmonic Mappings (2014), Section1, pp.1--2, definitions of $S_H$, $S_H^0$, and harmonic Jacobian. https://arxiv.org/abs/1403.5619.