Threshold Phenomena and Bounds in Normalized Remainders of Degenerate Exponential Functions

In this work, we study a normalized remainder $T_{n,\lambda }[{\text{e}}_{\lambda }]$ for the degenerate exponential ${\text{e}}_{\lambda }(u)=(1+\lambda u{)}^{1\mathrm{/}\lambda }$ ($\lambda >0$). We establish an integral representation, an exact monotonicity threshold at $\lambda =1\mathrm{/}(n+1)$, 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 $(0,1\mathrm{/}(n+1))$, the second logarithmic derivative satisfies $u^{2}L(u)\to -\alpha <0$ as $u\to \mathrm{\infty }$, showing that global logarithmic convexity on $(0,\mathrm{\infty })$ 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.