Summary: Let \[ I_n(x)= e^{-x}- \sum^n_{k=0} (-1)^k {x^k\over k!}= \sum^\infty_{k=n+1}(-1)^k {x^k\over k!}. \] We prove: if \(\alpha,\beta> 0\) are real numbers and \(n\geq 1\) is an integer, then the inequalities \[ {n+1\over n+2} {(1+{x\over n+\alpha})^2\over (1+{x\over n- 1+\alpha})(1+ {x\over n+1+\alpha})}< {I_{n- 1}(x)I_{n+1}(x)\over (I_n(x))^2}< {n+1\over n+2} {(1+{ x\over n+\beta})^2\over (1+{x\over n-1+\beta}) (1+{x\over n+ 1+\beta})} \] hold for all real numbers \(x> 0\) if and only if \(\alpha\leq 1\) and \(\beta\geq 2\). Our result improves inequalities published by M. Merkle [J. Math. Anal. Appl. 212, No. 1, 126-134 (1997; Zbl 0886.26014)].