Research Problems in Function Theory — Problem 6.8′

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 $p>0$ and a holomorphic $f$ on $\mathbb D$, let $n_f(w)$ count the preimages of $w$ with multiplicity. The function is areally mean $p$-valent when \[\int_0^{2\pi}\int_0^R n_f(\rho e^{i\theta})\rho\,d\rho\,d\theta\leq\pi pR^2\qquad(R>0).\] Writing $f(z)=\sum_{j=0}^{\infty}a_jz^j$, put $\mu_p(f)=\max\{|a_j|:j\in\mathbb Z,\ 0\leq j\leq p\}$, and let $S(p)$ be the mean $p$-valent functions with $\mu_p(f)=1$. For $0<r<1$, $\lambda>0$ and a holomorphic $u$, set \[I_\lambda(r,u)=\left(\frac1{2\pi}\int_0^{2\pi}|u(re^{i\theta})|^\lambda\,d\theta\right)^{1/\lambda}.\] For $M>0$ also consider $S(p;M)=\{f\in S(p):\sup_{\mathbb D}|f|\leq M\}$.

Statement recorded in the supplied source.

Determine the exact upper bounds, and hence the maximal orders as $r\uparrow1$, of both $I_\lambda(r,f)$ and $I_\lambda(r,f')$ as $f$ ranges over $S(p)$, for $p>0$ and $\lambda>0$, including the bounded subclasses $S(p;M)$. The source notes in particular that every bounded $f\in S(p)$ satisfies \[I_1(r,f')=o((1-r)^{-1/2}),\] and that the exponent $1/2$ cannot be improved for the full bounded mean-valent class. [S2]

Short English statement

Find sharp function and derivative integral-mean bounds for normalized mean-valent functions, including bounded functions.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306808 (AMR-022-6808). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Problem 6.8 prime, complete problem and update, Problem6.8, complete mean-valence notation, and introductory counterexample. https://arxiv.org/abs/1809.07200.