A Diophantine inequality with one prime of the form

Let 1 < c < (300s + 31)/(200s + 64) be fixed. Assume that N > 0 is a large enough number and ε > 0 is an arbitrarily small constant. This paper establish that the Diophantine inequality |p1c + … + psc − N | < ε has a solution in prime numbers p1,…,ps, where s ≥ 7 and s is a natural number and p1 can be expressed as p1 = m2 + n2 + 1 for some integers m,n.