Notes on an inequality for sections of certain power series (Q1337808)

Let the power series \(f(x)= \sum^ \infty_{k=0} a_ k x^ k\) with positive coefficients be convergent in the interval \((-R,R)\), \(0< R\leq \infty\), and define \(J_ n(x)= f(x)-\sum^ n_{k=0} a_ k x^ k= \sum^ \infty_{k= n+1} a_ k x^ k\). In this note, we prove the following Theorem. Theorem. Let \(r,s\in \mathbb{N}\) and suppose there is a positive integer \(N\) such that \[ {a_{n+1}\over (a_{n+1-s})^{{r\over s+r}}(a_{n+1+r})^{{s\over s+ r}}}\geq {a_{n+2}\over (a_{n+2- s})^{{r\over s+r}}(a_{n+2+ r})^{{s\over s+r}}},\quad\text{for }n\geq N.\tag{1} \] Then, for all \(x\in (0,R)\) and \(n\geq N\), we have \[ J_ n(x)\leq {a_{n+1}\over (a_{n+1-s})^{{r\over s+r}}(a_{n+1+r})^{{s\over s+r}}} (J_{n-s}(x))^{{r\over s+r}} (J_{n+r}(x))^{{s\over s+r}}\tag{2} \] and the coefficient \(a_{n+1}/[(a_{n+1- s})^{r/(s+r)}(a_{n+1+r})^{s/(s+ r)}]\) cannot be replaced by a smaller number. The inequality in (2) is strict if at least one inequality in (1) is strict. As a consequence of this theorem, one can derive a result of \textit{H. Alzer} [Arch. Math. 55, No. 5, 462-464 (1990; Zbl 0723.26007)]. We also provide a different approach and prove an analogue of a result of \textit{K. Dilcher} [Arch. Math. 60, 339-344 (1993)].