How to prove this inequality involving trigamma functions?

While solving a problem I succeeded to reduce it to the following inequality:$$\forall \{a,b,z\in\mathbb R_+,\ a\ne b\}:\quad 0<\frac1{a-b}\int_0^\infty\frac{t(a^2e^{-azt}-b^2e^{-bzt})}{1-e^{-t}}dt<\frac1z,$$which holds by numerical evidence. The integral can be presented as a linear combination of trigamma functions.

The left inequality is rather simple since the integral can be expressed as$$\frac1{a-b}\sum_{k=1}^\infty\left[\frac1{\left(z+\frac ka\right)^2}-\frac1{\left(z+\frac kb\right)^2}\right]$$but I have problems with the right one. I will appreciate any hint. A proof by elementary means is preferred.