Let \(1 < k < {7 \over 6},{\lambda _1},{\lambda _2},{\lambda _3}\) and λ4 be non-zero real numbers, not all of the same sign such that \({{{\lambda _1}} \over {{\lambda _2}}}\) is irrational and let ω be a real number. We prove that the inequality \(\left| {{\lambda _1}p_1^2 + {\lambda _2}p_2^2 + {\lambda _3}p_3^2 + {\lambda _4}p_4^k - \omega} \right| \le {\left({{{\max}_j}{p_j}} \right)^{- {{7 - 6k} \over {14k}} + \varepsilon}}\) has infinitely many solutions in prime variables p1, p2, p3, p4 for any ε > 0.
