Given a subGaussian variable $\xi$ with $E[\exp(\lambda \xi)]\leq \frac{\lambda^2 A^2}{2}$, where $\lambda\in R$ and $A>0$.
Now I am interested in the property of $E[\exp(\lambda |\xi|)]$, especially for the upper bound of $E[\exp(\lambda |\xi|)]$.
By noticing the $\varphi_2$ condition in subGaussian variable, i.e., $\exists a>0$ such that $E[\exp(a \xi^2)]\leq 2$, we can derive the following property as \begin{equation} P[|\xi|\geq \frac{\sqrt{t}}{\sqrt{a}}]\leq 2\exp(-t),~\text{with}~t\in R \end{equation}
But this equation seems helpless for the derivation of upper bound of $E[\exp(\lambda |\xi|)]$.
Could someone give me some hints on this issue? Thanks a lot.