A Diophantine Inequality With Prime Variables Mixed Power

In this article mainly studied the a power of a prime number,two power of two prime numbers,and the power of a prime number k diophantine inequalities problems.On the basis of existing research,by using Davenport-Heilbronn method recent developments in exponential sum estimation and enlarged major arc the following results are obtained.Let k be an integer with k?3,v is an arbitrarily given real number,let?(k)=min(2s(k)-1,1/2(s(k)+1)),wheres(k)=[k+1/2],here[x]represents the maximum integer not exceeding the defined x.Suppose that ?1,?2,?3,?4 are non-zero real number,not all negative,and ?2/?3 is irrational and algebraic.It is proved that the Diophantine inequality|?1p1+?2p22+?3p32+?4p4k-v<(max pj)-?has infinitely many solutions in prime variables p1,p2,p3,p4,where ?=-1/8?(k)+2?.