A note on a new exponential bound for M-acceptable random variables

We present a new exponential inequality as a generalization of that of Sung $\mathit{\text{et}}\mathit{\text{al.}}$ \cite{sun2011} for $M$-acceptable random variables, and hence for extended negative ones. Our result is based on the simple real inequality $e^x\le 1+x+(\mathrm{\mid }x\mathrm{\mid }\mathrm{/}2)e^{\mathrm{\mid }x\mathrm{\mid }},x\in \mathbb{R}$, in place of the following one: $e^x\le 1+x+(x^2\mathrm{/}2)e^{\mathrm{\mid }x\mathrm{\mid }},x\in \mathbb{R}$, used by many authors. We compare the given bound with former ones.