(untitled) (Annual of Sofia University St. Kliment Ohridski. Faculty of Mathematics and Informatics, 2015)
We show that for B > 1 and for some constants λ_i, i = 1, 2, 3 subject to certain assumptions, there are infinitely many prime triples p_1, p_2, p_3 satisfying the inequality λ_1p_1 + λ_2p_2 + λ_3p_3 + η< [log(max p_j )]^-B and such that p_1 + 2, p_2 + 2 and p_3 + 2 have no more than 8 prime factors. The proof uses Davenport - Heilbronn adaption of the circle method together with a vector sieve method.