On a ternary diophantine inequality with prime numbers of a special type II

Suppose that N is a sufficiently large real number and E is an arbitrarily large constant. In this paper, it is proved that, for \(1< c < \frac{7}{6}\) $1<c<\frac{7}{6}$ , the Diophantine inequality \(\begin{aligned} |p_1^c+p_2^c+p_3^c-N|<(\log N)^{-E} \end{aligned}\) $$\begin{array}{c}\hfill |p_1^c+p_2^c+p_3^c{-N|<(logN)}^{-E}\end{array}$$ is solvable in prime variables \(p_1,p_2,p_3\) $p_1,p_2,p_3$ so that each of the numbers \(p_i+2,\,i=1,2,3\) $p_i+2,i=1,2,3$ , has at most \(\big [3.43655+{\frac{12.12}{7-6c}}\big ]\) $[3.43655+\frac{12.12}{7-6c}]$ prime factors counted with multiplicity.