We prove that if a super-Poincar\'e inequality is satisfied by an
infinitesimal generator βA of a symmetric contracting semigroup then it
implies a corresponding super-Poincar\'e inequality for βg(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.Comment: submitted. 23p. no figure. Title slightly changed. Results and text
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