A new proximity function estimate on the quotient of the difference and the derivative of a meromorphic function
It is shown that, under certain assumptions on the growth and value distribution of a meromorphic function $f(z)$, \begin{equation*} m\left(r,\frac{Δ_cf - ac}{f' - a}\right)=S(r,f'), \end{equation*} where $Δ_c f=f(z+c)-f(z)$ and $a,c\in\mathbb{C}$. This estimate implies a lower bound for the Nevanlinna ramification term in terms of the difference operator with an arbitrary shift. As a consequence it follows, for instance, that if $f$ is an entire function of hyper-order $<1$ whose derivative does not attain a value $a\in\mathbb{C}$ often $$N\left(r,\frac{1}{f'-a}\right)=S(r,f),$$ then the finite difference $Δ_c f$ cannot attain the value $ac$ significantly more often $$N\left(r,\frac{1}{Δ_c f-ac}\right)=S(r,f).$$ Additional applications of the estimate above include a new type of a second main theorem, deficiency relations between $Δ_cf$ and $f'$ and new Clunie and Mohon'ko type lemmas.