Nevanlinna second main theorem ($(q-2)T_f\le {\sum }_{j=1}^q\overline{N}(r;a_j)+S_f$)

For distinct fixed target values and a nonconstant meromorphic function, the truncated second main theorem has the displayed form, with $S_f=O({\log }^{+}{T}_f+\log r)$ outside a set of finite linear measure. The truncated Nevanlinna counting function retains only one unit at each preimage. For finite-order functions the error is $O(\log r)$; for transcendental functions it is $o(T_f)$ along the nonexceptional radii.

For a nonconstant meromorphic function on the unit disc and $q\ge 3$ distinct spherical values, this version of the Nevanlinna second main theorem holds outside a set $E$ with ${\int }_{E}dr/(1-r)<\infty$, as $r\uparrow 1$. The truncated Nevanlinna counting function counts distinct preimages rather than multiplicity. The boundary-distance error is essential and need not be small relative to the characteristic. It follows from the disk Nevanlinna logarithmic derivative lemma and the growth lemma in the variable $\log (1/(1-r))$. The disk error estimate and weighted exceptional-set argument appear in mathweb.tifr.res.in/Documents/Publications/Lectures/tifr17.pdf .

If a transcendental meromorphic function had only finitely many simple preimages over each of five distinct finite values, their simple counting functions would be $O(\log r)$. The displayed multiplicity inequality and the Nevanlinna first main theorem would bound the truncated sum by $5T_f/2+O(\log r)$. The Nevanlinna second main theorem requires at least $3T_f$ up to its $o(T_f)$ error, a contradiction. Thus at least one value has infinitely many simple preimages; four other values can remain totally ramified.