Research Problems in Function Theory — Problem 6.45

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 Taylor series $f(z)=\sum a_nz^n$ and $g(z)=\sum b_nz^n$, their Hadamard convolution is $(f*g)(z)=\sum a_nb_nz^n$, and their integrated convolution is $(f\otimes g)(z)=\sum_{n\ge1}(a_nb_n/n)z^n$. These series are holomorphic on $\mathbb D$ when both original functions are. For $0<\alpha<1$, let $S^*_{\mathrm{str}}(\alpha)=\{f\in S:|\arg(zf'(z)/f(z))|<\alpha\pi/2\text{ on }\mathbb D\}$; extend the quotient by $1$ at $0$. The argument is the continuous branch equal to $0$ at $0$.

Statement recorded in the supplied source.

Is $S^*_{\mathrm{str}}(\alpha)$ closed under $\otimes$ for every $0<\alpha<1$? More generally, for every $0<\alpha,\beta<1$, does there exist $0<\gamma(\alpha,\beta)<1$ such that $f\in S^*_{\mathrm{str}}(\alpha)$ and $g\in S^*_{\mathrm{str}}(\beta)$ imply $f\otimes g\in S^*_{\mathrm{str}}(\gamma)$? Investigate analogous closure and mixed-order questions for other convolutions, including the Hadamard convolution $*$. [S2]

Short English statement

Test closure and mixed-order bounds for strongly starlike functions under integrated convolution and explore other convolutions, including the Hadamard product.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306045 (AMR-022-6045). 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 6.45, complete update and preceding notation. https://arxiv.org/abs/1809.07200.