Research Problems in Function Theory — Problem 6.23

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.For $f(z)=\sum_{n\ge0}a_nz^n$, define $\Delta_n=\big||a_{n+1}|-|a_n|\big|$. For symmetric series use $f(z)=\sum_{n\ge0}b_nz^{An+B}$, with integers $A\ge1$, $B\ge0$, and measure differences of successive $b_n$.

Statement recorded in the supplied source.

Find the sharp power exponents in bounds $\Delta_n=O(n^s)$ for mean $p$-valent functions, especially for $1/4<p<1$, where the source gives $s=2p-2\sqrt p$. Determine the corresponding sharp exponents for the symmetric series with parameters $A,B$. In particular, sharpen the bound $\big||b_{n+1}|-|b_n|\big|=O(n^{1-\sqrt2})$ for odd mean-univalent functions $f(z)=z+\sum_{n\ge1}b_nz^{2n+1}$. [S2]

Short English statement

Find optimal decay or growth powers for successive coefficient-modulus differences of mean-valent functions, including symmetric series and the odd mean-univalent case.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306023 (AMR-022-6023). 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 6.23, complete update and preceding notation. https://arxiv.org/abs/1809.07200.