Research Problems in Function Theory — Problem 6.36

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 $S$ consist of holomorphic injective functions $f(z)=z+\sum_{n\ge2}a_nz^n$ on $\mathbb D$; set $a_0=0$, $a_1=1$. A function in $S$ is starlike if its image is starlike about $0$, and convex if its image is a convex domain.For real $p>0$, define $B_{n,p}(f)$ by the branch normalized at $0$: \[(f(z)/z)^p=\sum_{n\ge0}B_{n,p}(f)z^n,\qquad B_{0,p}=1.\] For integral $p$, the coefficient of $z^{n+p}$ in $f^p$ is $B_{n,p}$. Put $A_{n,p}=\sup_{f\in S}|B_{n,p}(f)|$ and $b_{n,p}=\Gamma(2p+n)/(\Gamma(n+1)\Gamma(2p))$, the corresponding coefficient for the Koebe function; $\Gamma$ is Euler's gamma function. This reindexing distinguishes coefficients of $f^p$ from those of $(f/z)^p$.

Statement recorded in the supplied source.

Determine sharp bounds for the coefficients of $f^p$, equivalently $A_{n,p}$. In the range $1/4\le p<1$, where the source update records a limit $K_p=\lim_{n\to\infty}A_{n,p}/b_{n,p}$, determine $K_p$ and prove or disprove $K_p=1$ if and only if $p\ge1/2$. [S2]

Short English statement

Find sharp coefficient bounds for powers of univalent functions and settle the update’s limiting extremal ratio conjecture for fractional powers.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306036 (AMR-022-6036). 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 6.36, complete update and preceding notation. https://arxiv.org/abs/1809.07200.