Subordination and coefficient positivity for shifted multiple star polylogarithms

Prove that whenever a finite positive multi-index d400 strictly precedes another finite positive multi-index d400 in the paper's reverse lexicographic order, the shifted multiple star polylogarithm associated with d400 is subordinate to that associated with d400 on the unit disk, and prove that the Taylor coefficients of the corresponding subordinating map are all positive and have sum at most one.

Background

The paper defines subordination for holomorphic functions on the unit disk and introduces a reverse lexicographic order on finite positive multi-indices: a proper prefix is smaller, and at the first differing component the index with the smaller component is larger. The shifted multiple star polylogarithm is the normalized function H_d400(z)=Li\star_d400(z)/z.

The authors have established strong separation and monotonicity results for the representing densities and boundary curves of these functions. They then explicitly formulate a conjecture extending this ordering to analytic subordination and imposing coefficientwise positivity, together with the normalization that the coefficients sum to at most one. No proof of either asserted property is provided in the paper.

References

Now we propose the following conjecture. For ${\bf k}\succ {\bf l}$, we have (i) $\widetilde{\mathrm{Li}{\star}_{\bf k}(z) \succ \widetilde{\mathrm{Li}{\star}_{\bf l}(z)$ on $\mathcal{U}$. (ii) Furthermore, denote by $p_{{\bf k},{\bf l}}$ the holomorphic map $p_{{\bf k},{\bf l}}:\mathcal{U}\rightarrow\mathcal{U}$ which satisfies $\widetilde{\mathrm{Li}{\star}_{\bf k}\left( p_{{\bf k},{\bf l}}(z)\right) = \widetilde{\mathrm{Li}{\star}_{\bf l}(z)$ on $\mathcal{U}$. Let $p_{{\bf k},{\bf l}} (z) =\sum_{n\geq 1} c_n zn$, then $c_n>0,\quad \forall\, n\geq 1,\quad \sum_{n\geq 1} c_n\leq 1$.

— The derived set of multiple star polylogarithms  (2609.03653 - Li, 3 Sep 2026) in Conjecture immediately following Section 5, after Proposition 5.8