A Diophantine Problem with Unlike Powers of Primes

Let k be an integer with 4 ≤ k ≤ 6 and η be any real number. Suppose that λ 1 , λ 2 , … , λ 5 are nonzero real numbers, not all of them have the same sign, and λ 1 / λ 2 is irrational. It is proved that the inequality λ 1 p 1 + λ 2 p 2 2 + λ 3 p 3 3 + λ 4 p 4 4 + λ 5 p 5 k + η < max 1 ≤ j ≤ 5 p j − σ k has infinitely many solutions in prime variables p 1 , p 2 , p 3 , p 4 ,  and  p 5 , where 0 < σ 4 < 1 / 36 , 0 < σ 5 < 4 / 189 , and 0 < σ 6 < 1 / 54 . This gives an improvement of the recent results.