A Diophantine Problem With Unlike Powers Of Primes

Let k be an integer with 4 <= k <= 6 and eta be any real number. Suppose that lambda(1),lambda(2),..,lambda(5) are nonzero real numbers, not all of them have the same sign, and lambda(1)/lambda(2) is irrational. It is proved that the inequality vertical bar lambda(1)p(1) + lambda(2)p(2)(2) + lambda(3)p(3)(3) + lambda(4)p(4)(4) + lambda(5)p(5)(k) + eta| < (max(1 <= j <= 5)p(j))(-sigma(k)) has infinitely many solutions in prime variables p(1), p(2), p(3), p(4), and p(5), where 0 < sigma(4) < 1/36, 0 < sigma(5) < 4/189, and 0 < sigma(6) < 1/54. This gives an improvement of the recent results.