  We prove that if a super-Poincar\'e inequality is satisfied by an
infinitesimal generator $-A$ of a symmetric contracting semigroup then it
implies a corresponding super-Poincar\'e inequality for $-g(A)$ with any
Bernstein function $g$. We also study the converse statement. We deduce similar
results for the Nash-type inequality. Our results applied to fractional powers
of $A$ and to $\log(I+A)$ and thus generalize some results of Biroli and
Maheux, and Wang 2007. We provide several examples.
