  We prove an inequality between the $L^{\infty}$-norm of the contact
Hamiltonian of a positive loop of contactomorphims and the minimal Reeb period.
This implies that there are no small positive loops on hypertight or Liouville
fillable contact manifolds. Non-existence of small positive loops for
overtwisted 3-manifolds was proved by Casals-Presas-Sandon in [CPS16].
  As corollaries of the inequality we deduce various results. E.g. we prove
that certain periodic Reeb flows are the unique minimizers of the
$L^\infty$-norm. Moreover, we establish $L^\infty$-type contact systolic
inequalities in the presence of a positive loop.
