Let $g(t), w(t): \mathbb{R}^+ \rightarrow \mathbb{R}^+$ two real-valued, positive functions. They are two density generators in the sense that, the functions $p_g(t)$ and $p_w(t)$ defiend as:
$$ p_g(t) = \delta_{m,g}^{-1}t^{m-1}g(t) \quad p_w(t) = \delta_{m,w}^{-1}t^{m-1}w(t) $$
are probability density functions and
$$ \delta_{m,g} \triangleq \int_{0}^{\infty}t^{m-1}g(t)dt, \quad\delta_{m,w} \triangleq \int_{0}^{\infty}t^{m-1}w(t)dt. $$
Both $g(t)$ and $w(t)$ satisfy the following property:
$$ \frac{\delta_{m+1,g}}{\delta_{m,g}} = m, \quad \frac{\delta_{m+1,w}}{\delta_{m,w}} = m. $$
Let $w'(t)$ be the first derivative of $w(t)$.
I have to prove that: $$ \int_{0}^{\infty}\delta_{m,g}^{-1}t^{m}\frac{w'(t)g(t)}{w(t)}dt = -m. $$
Any suggestions are welcome!
Thanks!