One Diophantine inequality with unlike powers of prime variables

Abstract

Let [Formula: see text] be an integer with [Formula: see text] and [Formula: see text] be any real number. Suppose that [Formula: see text] are nonzero real numbers, not all of the same sign and [Formula: see text] is irrational. It is proved that the inequality [Formula: see text] has infinitely many solutions in prime variables [Formula: see text], where [Formula: see text] for [Formula: see text], and [Formula: see text] for [Formula: see text]. This gives an improvement of the recent result.