Solution to the monotonicity problem of an interesting class of sequences of real numbers

Recently we have presented some sufficient conditions so that the real sequence $\big (1+\frac{\alpha }{n}\big )^{n+p}$, $n\in {\mathbb{N}}$, strictly increasingly converges to $e^{\alpha }$, when $0\le 2p<\alpha $, and shown that the sequence is strictly decreasing if and only if $0<\alpha \le 2p$. Here, we give a complete characterization of the monotonicity character of the sequence in the case $0\le 2p<\alpha $, solving the most difficult case and the problem on the monotonicity character completely. To do this, among other things, we use several interesting new analytic inequalities.