Let be $X$ an arbitrary random variable with characteristic function $\varphi(t)$. Prove that $$\psi(t)=\frac{1}{2-\varphi(t)},$$ and $$\chi(t)=\int_{0}^{\infty}\varphi(st)e^{-s}ds$$ are also characteristic functions.
The later is easy I think. It can be seen that it is continous at zero, $\chi(0)=1$ and positive definiteness can be derived. For any $n\in\mathbb N, \{u_i\}\in\mathbb R$ and $\{c_j\}\in\mathbb C$ where $i,j=0...n$ $$\sum_{i,j=1}^{n}c_i\bar{c_j}\int_{0}^{\infty}\varphi(su_i-su_j)e^{-s}ds=\int_{0}^{\infty}\sum_{i,j=1}^{n}c_i\bar{c_j}\varphi(su_i-su_j)e^{-s}ds\ge0$$ since the later is sum of non-negative numbers for any fixed $s$, given that $\varphi(t)$ is a charactheristic function, thus positive definite.
What is the trick with the first one?