Threshold Phenomena and Bounds in Normalized Remainders of Degenerate Exponential Functions

In this work, we study a normalized remainder $T_{n,\lambda}[\e_\lambda]$ for the degenerate exponential $\e_\lambda(u)=(1+\lambda u)^{1/\lambda}$ ($\lambda>0$). We establish an integral representation, an exact monotonicity threshold at $\lambda=1/(n+1)$, and rigorous conditions for the local failure of logarithmic convexity at the origin. We then prove a sharp asymptotic result: for every $\lambda$ in the increasing regime $(0,1/(n+1))$, the second logarithmic derivative satisfies $u^2L(u)\to -\alpha<0$ as $u\to\infty$, showing that global logarithmic convexity on $(0,\infty)$ fails throughout this regime. We further give a necessary and sufficient condition for absolute monotonicity, showing it holds only on a countable, measure-zero set of parameters, and we derive explicit two-sided truncation-error bounds that are pointwise sharp at the origin.