LAUSR

Let N be a sufficiently large real number. In this paper, we prove that, for $$10$$ , the Diophantine inequality $$\begin{aligned} \big |p_1^c+p_2^c+p_3^c-N\big |<(\log N)^{-E} \end{aligned}$$ is solvable in prime variables $$p_1,p_2,p_3$$ such that each of the numbers $$p_i+2$$ for $$i=1,2,3$$ has at most $$[\frac{12626}{4865-4280c}]$$ prime factors counted with multiplicity. This result constitutes an improvement upon the previous result of Zhu (Proc Indian Acad Sci Math Sci 130(1):23, 2020).