Let $f$ be a transcendental entire function. Set $$h(z)= z+ f(z)/f'(z)$$ and $$ F(z) = (z-a) f(z)$$. Then prove that for some $k>0,$ $$T(r, h)\thicksim k T(r, F'/F),$$ where $T$ is the Nevanlinna's characteristics function.
Attempt: $$F'(z)= (z-a)f'(z) + f(z)$$ Then $$F'(z)/F(z)= 1/(z-a) + f'(z)/f(z) = 1/(z-a) + 1/h(z) - z$$ Imples $$T/(r, F'/F) ≤ T(r, 1/z-a) + T(r, 1/h(z)-z) + \log 2$$ I am stuck here
We have $$ \frac{F'(z)}{F(z)} = \frac{1}{z-a} + \frac{1}{h(z)-z} \, . $$ If $F'/F$ and $h$ are both transcendental functions then $$ T(r, \frac{F'}{F}) \sim T(r, \frac{1}{h-z}) = T(r, h-z) + O(1) \sim T(r,h) $$ for $r \to \infty$.
Otherwise $F'/F$ and $h$ are both rational functions. Let their respective degrees be $m$ and $n$. Then $$ T(r, \frac{F'}{F}) \sim m \log(r) \\ T(r, h) \sim n \log(r) $$ and therefore $ T(r, \frac{F'}{F}) \sim \frac mn T(r, h) $.