Diophantine approximation with one prime, two squares of primes and one kth power of a prime

Abstract Let 1 < k 14 / 5 1\lt k\lt 14\hspace{-0.08em}\text{/}\hspace{-0.08em}5 , λ , 2 3 {\lambda }_{1},{\lambda }_{2},{\lambda }_{3} and 4 }_{4} be non-zero real numbers, not all of the same sign such that }_{1}\hspace{-0.08em}\text{/}\hspace{-0.08em}{\lambda }_{2} is irrational let ω \omega a number. We prove inequality ∣ p + − ≤ ( max ) ψ ε | }_{1}{p}_{1}+{\lambda }_{2}{p}_{2}^{2}+{\lambda }_{3}{p}_{3}^{2}+{\lambda }_{4}{p}_{4}^{k}-\omega \le {\left(\max \left({p}_{1},{p}_{2}^{2},{p}_{3}^{2},{p}_{4}^{k}))}^{-\psi \left(k)+\varepsilon } has infinitely many solutions in prime variables {p}_{1},{p}_{2},{p}_{3},{p}_{4} for any > 0 \varepsilon \gt 0 where = min 28 \psi \left(k)=\min \left(\frac{1}{14},\frac{14-5k}{28k}\right) .