I have a function $f$ defined on $]0,+\infty[$ such that: $$ \begin{cases} f(1)=0\\ \forall z \in ]0,1], \quad f(z) \in i\mathbb{R}\\ \forall z \in [1,+\infty[, \quad f(z) \in \mathbb{R} \end{cases} $$
I also have the existence of a function $\Psi$ such that: $$ \Psi \in \mathcal{C}^1([-1,+\infty[)\cap\mathcal{C}^2(]-1,+\infty[) \;\text{and }$$ $$\forall z\in [-1,+\infty[ \;\Psi(z)=\int_1^{+ \infty}\mathrm{d}y\;f(y)\frac{z}{y}\frac{1}{y+z}$$ Is there anything we can say about f or its derivatives ?
I know an example of such f: $$f(z)=[\ln z]^{3/2}$$