Research Problems in Function Theory — Problem 6.13

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 $\mathbb D=\{z\in\mathbb C:|z|<1\}$, $\mathbb T=\partial\mathbb D$, and $\mathbb D^*=\{z\in\mathbb C:|z|>1\}$. Let $S$ be the class of analytic injective functions $f:\mathbb D\to\mathbb C$ normalized by $f(0)=0$, $f^{\prime}(0)=1$, with expansion $f(z)=z+\sum_{n=2}^{\infty}a_nz^n$ and $a_1=1$. Fix positive integers $k,m,n$. Coefficients $c_0,\ldots,c_m\in\mathbb C$ may depend on $f,k,m,n$, and must satisfy $c_0=1$ and $|c_m|\ge4^{-m}$. For an exponent $\beta_m$, the desired estimates are \[\left|\sum_{i=0}^mc_i a_{n+j+i}\right|\le K n^{\beta_m},\qquad 0\le j\le k,\] with a constant $K$ allowed to depend on $f,k,m$ but independent of $n$. A function in $S$ is starlike if its image is starlike about $0$: $tw\in f(\mathbb D)$ for every $w\in f(\mathbb D)$ and $0\le t\le1$.

Statement recorded in the supplied source.

For each $m$, determine the best possible exponents $\beta_m$ for which these estimates hold for every $f\in S$ and every $k$, with the stated quantifiers on $K$ and the $c_i$. Preserve the proposed refinements $\beta_2=-1/3$, estimates with $\beta_3=-1/2+\varepsilon$ for every $\varepsilon>0$, and an admissible $\beta_m<-1/2$ for $m>4$. For the starlike subclass the source gives the sharp exponent $-1+2/(m+1)$; the update provides an improved general bound approaching $-1/2$. [S2]

Short English statement

Determine the optimal decay exponents of short coefficient combinations of univalent functions, retaining the coefficient-normalization constraints and the proposed small-order refinements.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306013 (AMR-022-6013). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory (2018), arXiv:1809.07200. Problem 6.13, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.