Research Problems in Function Theory — Problem 6.107

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 $a_{-1}=0$.

Statement recorded in the supplied source.

For every $f\in S_H^0$ and integer $n\geq2$, does \[\left||a_n|-|a_{-n}|\right|\leq n\] hold? The outer absolute value is part of the assertion. [S2] [S3]

Short English statement

Is the absolute difference between the magnitudes of the analytic and coanalytic nth coefficients at most n?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306107 (AMR-022-6107). 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 6.107, 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.