TITLE: Translation of a theorem in French
QUESTION [6 upvotes]: As picture below, what is uno fonction ?  I use google translate , it fail to translate it .

REPLY [17 votes]: Translation:

Lemma 4.1. Let $\varphi(t)$ be a function defined for $t \ge k_0$ non-negative and non-decreasing such that if $h > k \ge k_0$ we have: $$\varphi(h) \le \frac{c}{(h-k)^\alpha} [\varphi(k)]^\beta$$
  $c,\alpha,\beta$ being positive constants.  
Then, (i) if $\beta > 0$ we have ...
  where

[similarly for cases (ii) and (iii)]