Research Problems in Function Theory — Problem 6.104

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 $a_{-1}=0$. Define the covering constant and second-coefficient extremum by \[d_H=\sup\{d\geq0:\{w:|w|<d\}\subseteq f(\mathbb D)\text{ for all }f\in S_H^0\}, \qquad \alpha_H=\sup_{f\in S_H}|a_2|.\] The harmonic Koebe map is \[k_0(z)=\operatorname{Re}\frac{z+z^3/3}{(1-z)^3} +i\operatorname{Im}\frac{z}{(1-z)^2}.\] The source places $k_0$ in $S_H^0$ and states that its image omits the real ray $(-\infty,-1/6]$.

Statement recorded in the supplied source.

Prove or disprove $d_H=1/6$, and determine the exact value of $\alpha_H$. Both the covering problem for $S_H^0$ and the coefficient problem for the larger class $S_H$ are requested. [S2] [S3]

Short English statement

Determine the harmonic Koebe covering constant and, independently, the largest possible second analytic coefficient in the full normalized harmonic class.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306104 (AMR-022-6104). 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 6.104, 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.