On a system of two Diophantine inequalities with five prime variables

Abstract: Suppose that $c,d,\alpha,\beta$ are real numbers satisfying the inequalities $1<d<c<39/37$ and $1<\alpha<\beta<5^{1-d/c}$. In this paper, it is proved that, for sufficiently large real numbers $N_1$ and $N_2$ subject to $\alpha\leqslant N_2/N_1^{d/c}\leqslant\beta$, the following Diophantine inequalities system \begin{equation*} \begin{cases} \big|p_1^c+p_2^c+p_3^c+p_4^c+p_5^c-N_1\big|<\varepsilon_1(N_1) \ \big|p_1^d+p_2^d+p_3^d+p_4^d+p_5^d-N_2\big|<\varepsilon_2(N_2) \end{cases} \end{equation*} is solvable in prime variables $p_1,p_2,p_3,p_4,p_5$, where \begin{equation*} \begin{cases} \varepsilon_1(N_1)=N_1^{-(1/c)(39/37-c)}(\log N_1)^{201}, \ \varepsilon_2(N_2)=N_2^{-(1/d)(39/37-d)}(\log N_2)^{201}. \end{cases} \end{equation*} This result constitutes an improvement upon a series of previous results of Zhai [14] and Tolev [12].