A note on Diophantine approximation by unlike powers of primes

It is proved that if lambda(1), lambda(2), . . . , lambda(5) are nonzero real numbers, not all of the same sign and lambda(1)/lambda(2) is irrational, then for given real numbers eta and sigma, 0 < sigma < 5/252, 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)(5) + eta vertical bar < (max(1 <= j <= 5) p(j)(j))(-sigma) has infinitely many solutions in prime variables p(1), p(2), p(3), p(4), p(5). This result constitutes an improvement upon that of Liu for the range 0 < sigma < 5/288.