Disproof of a conjecture of Ruscheweyh et al. on universal convexity of Ramanujan generating function

@misc{he2026disproof,
  title = {{Disproof of a conjecture of Ruscheweyh et al. on universal convexity of Ramanujan generating function}},
  author = {Bing He},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.07479},
  eprint = {2611.07479},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.07479},
}

TY  - UNPB
TI  - Disproof of a conjecture of Ruscheweyh et al. on universal convexity of Ramanujan generating function
AU  - Bing He
PY  - 2026
DA  - 2026/11/05
PB  - arXiv
DO  - 10.48550/arxiv.2611.07479
UR  - https://arxiv.org/abs/2611.07479
AB  - We disprove a conjecture of Ruscheweyh, Salinas and Sugawa concerning the universal convexity of the normalized generating function associated with the Ramanujan sequence θ(n). More precisely, we prove that the function σ(z)=1/(θ(1)−1/3)∑_n≥1(θ(n)−1/3)z^n is not universally convex on the slit domain ℂbackslash[1,∞). The proof uses Watson\textquoteright s integral representation of θ(n), asymptotic estimates for an auxiliary function h, and the representation of derivatives of universally convex functions via logarithmic integrals. As a general extension of this result, we also show that the generalized Ramanujan generating function σ_α(z)=1/(θ(1)−1/3)∑_n≥1(θ(n)−1/3)z^n/n^α is universally convex if and only if α≥1.
ER  -