A new characterization of the Poisson distribution

In this note we show that an infinitely divisible (i.d.) distribution function $F$ is Poisson if and only if it satisfies the conditions $F(+0)>0$, for any $0<\varepsilon<1$
$$\int_{-\infty}^{1-\varepsilon}\frac{|x|}{1+|x|}\,dF=0,$$
and for any $0<\alpha<1$
$$\int_0^\infty e^{\alpha x\ln(x+1)}\,dF<\infty$$