A Diophantine Inequality Involving Mixed Powers of Primes with a Specific Type

Let λ1,λ2,λ3 be nonzero real numbers, not all of the same sign; let λ1/λ2 be irrational; and let η be any real number. We investigate the solvability of the inequality |λ1p1+λ2p2+λ3p32+η|<(maxpj)−1/12+θ, θ>0 in the prime variables p1, p2, and p3. We require that p1+2 and p2+2 have no more than 20 prime factors, while p3+2 has no more than 42 prime factors.