Diophantine approximation by Piatetski-Shapiro primes

Let $[\cdot ]$ be the floor function. In this paper we show that whenever $\eta$ is real, the constants ${\lambda }_i$ satisfy some necessary conditions, then for any fixed $1<c<38\mathrm{/}37$ there exist infinitely many prime triples $p_1,p_2,p_3$ satisfying the inequality \begin{equation*} |\lambda_1p_1 + \lambda_2p_2 + \lambda_3p_3+\eta|<(\max p_j)^{{\frac{37c-38}{26c}}}(\log\max p_j)^{10} \end{equation*} and such that $p_i=[n_i^c]$, $i=1,2,3$.