  We consider three different families of Vafa-Witten invariants of $K3$
surfaces. In each case, the partition function that encodes the Vafa-Witten
invariants is given by combinations of twisted Dedekind $\eta$-functions. By
utilising known properties of these $\eta$-functions, we obtain exact formulae
for each of the invariants and prove that they asymptotically satisfy all
higher-order Tur\'{a}n inequalities.
