Convexity properties of power series with logarithmically \(s\)-concave coefficients (Q2367054)

Let \(f(t)=\sum^ \infty_ 0 a_ n t^ n\) \((a_ n>0,\;n=0,1,\dots)\) be analytic in a neighbourhood of the origin and let \(r_ n(s):=\Bigl(1+ {s-1\over n}\Bigr){a_{n-1}\over a_ n}\), \(1\leq s<\infty\), \(r_ n(\infty):= {1\over n}\cdot{a_{n-1}\over a_ n}\) \((a_ n>0,\;n=1,2,\dots)\). Fixing \(s\), \(1\leq s\leq \infty\), the sequence \(\{a_ n\}^ \infty_ 0\) of positive real numbers is said to be logarithmically \(s\)-concave \((1\leq s\leq \infty)\) if the sequence \(\{r_ n(s)\}\) is monotone non-decreasing. The authors consider the power series: \[ F_ s(t):= \sum^ \infty_{n=0} {a_ n\over s^ n}{s-1+n\choose n} t^ n,\quad F_ \infty(t)= \sum^ \infty_{n=0} {a_ n\over n!} t^ n, \] further the classes \(A_ s(a_ 0,a_ 1):=\Bigl\{f(t)=\sum^ \infty_ 0 a_ n t^ n\in{\mathbf H}\mid r_ n(s)\leq r_{n+1}(s)\), \(n=1,2,\dots\Bigr\}\), where \({\mathbf H}\) denotes the set of all functions analytic in a neighbourhood of the origin having positive coefficients. Noting that the sequence of coefficients of \(F_ s(t)\) for each fixed \(s\), \(1\leq s\leq \infty\), is logarithmically \(s\)-concave if and only if \(a^ 2_ n\geq a_{n-1} a_{n+1}\), \(n=1,2,\dots\), a main result asserts that under this condition the function \([F_ s(t)]^{-1/s}\) is convex, or equivalently \[ \Bigl(1+\textstyle{{1\over s}}\Bigr)\bigl(F_ s'(t)\bigr)^ 2- F_ s(t) F_ s^{\prime\prime}(t)\geq 0,\quad (1\leq s<\infty,\;t>0). \] Properties of the set \(A_ s(a_ 0,a_ 1)\) are given; e.g., that if \(f(t)\) is element of the normalized class \(A_ s(1,1)\), then \(f(t)\leq f_ s(t;1,1)= 1/\Bigl(1-{t\over s}\Bigr)^ s\) \((1\leq s<\infty\); \(| t|<s)\) and for \(s=\infty\;f_ \infty(t;1,1)=e^ t\); this means that \(f_ s(t;1,1)\) is the extremal function of the normalized class.