Translation of a theorem in French

Question

As picture below, what is uno fonction ? I use google translate , it fail to translate it .

Answer 1

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)]