DIOPHANTINE INEQUALITY WITH FOUR SQUARES AND ONE kTH POWER OF PRIMES

Let k be an integer with $k{\geq}3$. Define $h(k)=[{\frac{k+1}{2}}]$, ${\sigma}(k)={\min}\(2^{h(k)-1},\;{\frac{1}{2}}h(k)(h(k)+1)\)$. Suppose that ${\lambda}_1,{\ldots},{\lambda}_5$ are non-zero real numbers, not all of the same sign, satisfying that ${\frac{{\lambda}_1}{{\lambda}_2}}$ is irrational. Then for any given real number ${\eta}$ and ${\varepsilon}>0$, the inequality $${\mid}{\lambda}_1p^2_1+{\lambda}_2p^2_2+{\lambda}_3p^2_3+{\lambda}_4p^2_4+{\lambda}_5p^k_5+{\eta}{\mid}<({\max_{1{\leq}j{\leq}5}}p_j)^{-{\frac{3}{20{\sigma}(k)}}+{\varepsilon}}$$ has infinitely many solutions in prime variables $p_1,{\ldots},p_5$. This gives an improvement of the recent results.