Research Problems in Function Theory — Problem 6.24

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 $\mathbb T=\partial\mathbb D$.For a nonconstant holomorphic function $f$, let $n_f(w)$ be the number of solutions of $f(z)=w$ in $\mathbb D$, counted with multiplicity and allowed to be infinite. It is areally mean $p$-valent, for $p>0$, if \[\int_0^{2\pi}\int_0^R n_f(\rho e^{it})\rho\,d\rho\,dt\le\pi pR^2\qquad(R>0).\] Constants in coefficient asymptotics may depend on the particular function. Coefficients are not normalized unless stated.Let $S(1)$ be the class of mean-univalent holomorphic functions with $\max(|a_0|,|a_1|)=1$. Here further require $f(z)=z+\sum_{n\ge2}a_nz^n$. For $0<r<1$, derivative and logarithmic-derivative bounds are pointwise bounds on $|z|=r$; a zero at $z\ne0$ makes $|f'/f|$ infinite there.

Statement recorded in the supplied source.

Prove or disprove $|f(z)|\le r/(1-r)^2$ for every such $f$ and every $|z|=r<1$. Also determine the sharp bounds for $|f'(z)|$ and $|f'(z)|/|f(z)|$ in this class. The source records the covering and lower-distortion conclusion $|f(z)|\ge r/(1+r)^2$ as background. [S2]

Short English statement

Establish the proposed upper distortion bound for normalized mean-univalent functions and determine their sharp derivative and logarithmic-derivative bounds.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306024 (AMR-022-6024). 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 6.24, complete update and preceding notation. https://arxiv.org/abs/1809.07200.