Research Problems in Function Theory — Problem 6.64

Definitions and mathematical statement

Let $\alpha\in{\mathbb{R}}$ and let \[ f(z)=z+\sum_{n=2}^{\infty}a_nz^n \] be analytic on $\mathbb D$. Assume $f(z)f'(z)/z\ne0$, with its removable value $1$ at $z=0$. Define $M_\alpha$, the class of $\alpha$-convex functions, by \[ \operatorname{Re}\left[(1-\alpha)\frac{zf'(z)}{f(z)} +\alpha\left(1+\frac{zf''(z)}{f'(z)}\right)\right]>0 \quad(z\in\mathbb D), \] where the expression has value $1$ at $0$. The source identifies $M_0=S^*$, the normalized starlike class, and $M_1=K$, the normalized convex class, and reports $M_\alpha\subset S^*$ for every real $\alpha$.

The coefficient criteria attributed to Clunie and Keogh are \[ \sum_{n=2}^{\infty}n|a_n|<1\ \Longrightarrow\ f\in M_0, \qquad \sum_{n=2}^{\infty}n^2|a_n|<1\ \Longrightarrow\ f\in M_1. \] Question (S. Miller). Find generalizations of these coefficient conditions sufficient for $f\in M_\alpha$, with the dependence on the real parameter $\alpha$ stated explicitly. The parameter is not restricted to $[0,1]$ or to nonnegative values in the supplied definition.

A natural coefficient-criterion format is to find explicit weights $w_n(\alpha)$ such that \[ \sum_{n=2}^{\infty}w_n(\alpha)|a_n|<1\ \Longrightarrow\ f\in M_\alpha, \] recovering weights $w_n(0)=n$ and $w_n(1)=n^2$ at the stated endpoints. This format explains the desired interpolation/generalization but is not an asserted formula or an additional requirement that every possible answer be a weighted-sum criterion. The source asks for sufficient conditions, without requiring an optimal necessary-and-sufficient characterization.

The supplied research summary records OPEN-TRIAGE, says the question was open in the 2018 edition, and reports no definitive resolution located. These are supplied status claims rather than independent verification.

Short English statement

Find coefficient-size conditions implying membership in the real-parameter class of $\alpha$-convex functions. They should generalize the known strict weighted-sum criteria with weights $n$ for starlikeness and $n^2$ for convexity, without silently restricting the parameter range.

Sources

• Local UnsolvedMath registry, permanent ID 2306064, source code catalogue entry, "Research Problems in Function Theory --- Problem 6.64"; attribution: S. Miller. Supplied status open, classification OPEN-TRIAGE, updated 2026-08-06.
• Hayman, Research Problems in Function Theory (2018), Problem 6.64, https://arxiv.org/abs/1809.07200. The record reports source-TeX extraction and access on 2026-07-29; the 2018 update was retained for status triage. Current status and rights require review; the record describes publicly available arXiv source TeX.
• The source attributes the two endpoint coefficient criteria to Clunie and Keogh but supplies no separate publication or theorem reference.