  For nonnegative integers $n$ and $k$, we introduce in this paper a new class
of $(n,k)$-quasi-*-paranormal operators satisfying
  $$||T^{1+n}(T^{k}x)||^{1/(1+n)}||T^{k}x||^{n/(1+n)} \geq
  ||T^*(T^{k}x)|| \makebox{\ for all} x \in H.$$
  This class includes the class of $n$-*-paranormal operators and the class of
$(1,k)$-quasi-*-paranormal operators contains the class of $k$-quasi-*-class
$A$ operators. We study basic properties of $(n,k)$-quasi-*-paranormal
operators: (1) inclusion relations and examples; (2) a matrix representation;
(3) joint (approximate) point spectrum and single valued extension property.
