 Solution

Here the iteration index is a positive integer: $1\le m<n$. If ${\vartheta }^m(\lambda )=\lambda$, then ${\vartheta }^m$ fixes $F$ and the generator, so it fixes all of $K$. Every element of $K$ would then solve $T^{p^m}-T=0$. A nonzero polynomial of degree $p^m$ over a field has at most $p^m$ roots, contradicting $|K|=p^n>p^m$. Thus ${\vartheta }^m(\lambda )\ne \lambda$. The printed inequality alone omits positivity of $m$; at $m=0$ the claimed inequality would be false.