Diophantine approximation with a quaternary problem

Let $1<k<7\mathrm{/}6$, ${\lambda }_1,{\lambda }_2,{\lambda }_3$ and ${\lambda }_4$ be non-zero real numbers, not all of the same sign such that ${\lambda }_1\mathrm{/}{\lambda }_2$ is irrational and let $\omega$ be a real number. We prove that the inequality $\mathrm{\mid }{\lambda }_1{p}_1^2+{\lambda }_2{p}_2^2+{\lambda }_3{p}_3^2+{\lambda }_4{p}_4^k-\omega \mathrm{\mid }\le ({\max }_j{p}_j{)}^{-\frac{7-6k}{14k}+\epsilon }$ has infinitely many solutions in prime variables $p_1,p_2,p_3,p_4$ for any $\epsilon >0$.