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Let $c > 1$ and $0 < \gamma < 1$ be real, with $c \notin \mathbb N$. We studythe solubility of the Diophantine inequality \[ \left| p_1^c + p_2^c + \dots +p_s^c - N \right| < \varepsilon \] in Piatetski-Shapiro primes $p_1, p_2,\dots, p_s$ of index $\gamma$---that is, primes of the form $p = \lfloorm^{1/\gamma} \rfloor$ for some $m \in \mathbb N$.

A hybrid of two theorems of Piatetski-Shapiro