An inequality for residual of Maclaurin expansion (Q1924898)

Let \(I_n(x)\) be the residual after the \(n\)th term in Maclaurin expansion of a function \(f\). In this paper we show that the inequality \[ {I_{n-1} (x) I_{n + 1} (x) \over I^2_n(x)} \geq {n + 1\over n + 2} \] holds for a class of functions with log-convex derivatives. Further, we give a sufficient condition (in terms of logarithmic convexity) for the inequality \[ {I_{n - 1} (x) I_{n + 1} (x) \over I^2_n (x)} \;\geq \;{f^{(n)} (0) f^{(n + 2)} (0) \over (f^{(n + 1} (0))^2} \cdot\;{n + 1 \over n + 2} . \]