The value ofσin the inequality|λ1 p 1 +λ2 p 2 +λ3 p k3 +η|< (maxp j )-σ is improved by applying Davenport-Heilbronn method,where k is an integer with k ≥4,η is an any real number,andλ1 ,λ2 ,λ3 are nonzero real numbers,not all of the same sign,withλ1/λ2 being irra-tional.It is proved that the better value ofσ is obtained,where 0<σ<1/2(2 k +1 +1)for k =4 and 0<σ<2k -1/(2 k +1 +1)k (k +1)for k ≥5.%采用Davenport-Heilbronn方法改进不等式|λ1 p 1+λ2 p 2+λ3 pk3+η|<(maxp j)-σ 中σ的值,其中k是大于或等于4的正整数,η是任意给定的实数,λ1,λ2,λ3是非零实数,不全同号,并且λ1/λ2是无理数,得出了更好的σ 值是:当k=4时,0<σ<1/2(2k+1+1);当k≥5时,0<σ<2k-1/(2k+1+1)k(k+1).
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