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

Abstract

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 \[ σ(z)=\frac{1}{θ(1)-1/3}\sum_{n\geq1}(θ(n)-1/3)z^{n} \] is not universally convex on the slit domain $\mathbb{C}\backslash[1,\infty).$ 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 \[ σ_α(z)=\frac{1}{θ(1)-1/3}\sum_{n\geq1}(θ(n)-1/3)\frac{z^{n}}{n^α} \] is universally convex if and only if $α\geq1.$