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 ${\mathrm{\Delta }}_{c}f=f(z+c)-f(z)$ 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 $$N(r,\frac{1}{f^{\mathrm{\prime }}-a})=S(r,f),$$ then the finite difference ${\mathrm{\Delta }}_{c}f$ cannot attain the value $ac$ significantly more often $$N(r,\frac{1}{{\mathrm{\Delta }}_{c}f-ac})=S(r,f)\mathrm{.}$$ Additional applications of the estimate above include a new type of a second main theorem, deficiency relations between ${\mathrm{\Delta }}_{c}f$ and $f^{\mathrm{\prime }}$ and new Clunie and Mohon’ko type lemmas.