Research Problems in Function Theory — Problem 5.3

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.

Let $\mathbb D=\{z\in\mathbb C:|z|<1\}$ and let $f(z)=\sum_{n=0}^\infty a_nz^n$ be holomorphic in $\mathbb D$. For $\lambda>0$ and $0<r<1$ define the integral mean \[I_\lambda(r,f)=\left(\frac1{2\pi}\int_0^{2\pi}|f(re^{i\theta})|^\lambda\,d\theta\right)^{1/\lambda}. \] The coefficient estimates refer to $n\to\infty$, and mean estimates to $r\uparrow1$. Constants in $O(\cdot)$ may depend on $f$ and the indicated exponent, but not on $n$ or $r$. A function is weakly univalent here if, for every radius $R>0$, one of the following alternatives holds: every value $w$ on $|w|=R$ is assumed exactly once in $\mathbb D$, or at least one value $w_R$ on that circle is omitted throughout $\mathbb D$. Preimages in the first alternative are counted with their holomorphic multiplicities. This condition is imposed directly, without requiring an independently chosen omitted-value sequence. The earlier estimates recorded in the problem are $I_\lambda(r,f)=O((1-r)^{-2})$ for $\lambda>0$ and $|a_n|=O(n^2)$.

Statement recorded in the supplied source.

For weakly univalent $f$, obtain sharper general bounds for the integral means $I_\lambda(r,f)$ and for $|a_n|$. In particular, decide whether the stronger bounds \[I_1(r,f)=O\left((1-r)^{-1}\right),\qquad |a_n|=O(n) \] hold, and seek improvements of the quadratic estimates even independently of these full-strength targets. [S2]

Short English statement

Sharpen the integral-mean and coefficient bounds for weakly univalent functions, with the first-mean boundary-distance bound and linear coefficients as the stronger targets.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2305003 (AMR-022-5003). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory, arXiv:1809.07200, 2018. Chapter 5 introduction, equations (5.1)--(5.6), Problem 5.3, and complete Updates 5.1--5.3. https://arxiv.org/pdf/1809.07200.