An inequality for section of certain power series
The following result is proved: Suppose that the power series \(f(x)=\sum^ \infty_{j=0}a_ jx^ j\) with positive coefficients is convergent in the interval \((-R,R)\), \(0<R \leq \infty\), and define \[ I_ n(x)=f(x)-\sum^ n_{j=0}a_ jx^ j= \sum^ \infty_{j=n+1} a_ jx^ j. \] Further, suppose there is a positive integer \(N\) such that for all \(n \geq N\) we have \[ {a_{n+1} \over a_ n}-{a_{n+2} \over a_{n+1}} \geq 0 \quad \text{ and } \quad {a_{n+1} \over a_ n}-2{a_{n+2} \over a_{n+1}}+{a_{n+3} \over a_{n+2}} \geq 0. \tag{*} \] Then for all \(n \geq N\) and \(x \in (0,R)\) we have \[ I_{n-1}(x)I_{n+1}(x) \geq {a_ na_{n+2} \over (a_{n+1})^ 2} \bigl( I_ n(x) \bigr)^ 2, \tag{**} \] and the coefficient \(a_ na_{n+2}/a^ 2_{n+1}\) is best possible in the sense that it cannot be replaced by a larger number. The inequality \((**)\) is strict if the inequalities \((*)\) are all strict.
scientific article; zbMATH DE number 446308