Diophantine Inequality With Prime Variables And Mixed Powers

In this paper,we mainly study the Diophantine inequality problem with one prime,one square of prime,two cubed of primes and one k-th power of a prime,where k is a real number or an integer,by means of Davenport-Heilbronn's improved Hardy-Littlewood method,we obtain the following results.Theorem 1 Assume that k₁ is a real number and 1<k₁<8/3,?₁,?₂,?₃,?₄,?₅ are non-zero real numbers,not all of the same sign,that ?₁/?₂ is irrational and let ? be a real number.The inequality |?₁p₁+?₂p₂²+?₃p₃³+?₄p₄³+?₅p₅^k₁+?|<(maxp_j)^{-1/16(8-3k₁/k₁)+?} has infinitely many solutions in primes variables p₁,p₂,p₃,p₄,p₅ for any ?>0.Theorem 2 Assume that k₂ is an integer and k₂?3,?₁,?₂,?₃,?₄,?₅ are non-zero real numbers,not all of the same sign,that ?₁/?₂ is irrational and let ? be a real number ?(k₂)=min(2^s(k₂)-1,1/2s(k₂)(s(k₂)+1)),s(k₂)=[k₂+1/2],,The inequality|?₁p₁+?₂p₂²+?₃p-3³+?₄p₄³+?₅p₅^k₂+?|<(max p_j)^{-1/16?(k₂)+?} has infinitely many solutions in primes variables p₁,p₂,p₃,p₄,p₅ for any ?>0.