It is proved that if λ₁, λ₂, ···, λ₇are nonzero real numbers, not all of the same sign and not all in rational ratios, then for any given real numbers η and σ, 0 < σ <1/16, the inequality |λ₁x₁²+ λ₂x₂²+∑7 i=3λ_ix_i⁴+ η| <（ max1≤i≤7|x_i|）^-σhas infinitely many solutions in positive integers λ₁, λ₂, ···, λ₇ Similar result is proved for |λ₁x₁²+ λ₂x₂²+ λ₃x₃²+ λ₄x₄⁴+ λ₅x₅⁴+ λ₆x₆⁴+ η| <（ max1≤i≤6|x_i|）^-σ.These results constitute an improvement upon those of Shi and Li. 

It is proved that if λ₁, λ₂, ···, λ₇are nonzero real numbers, not all of the same sign and not all in rational ratios, then for any given real numbers η and σ, 0 < σ <1/16, the inequality |λ₁x₁²+ λ₂x₂²+∑7 i=3λ_ix_i⁴+ η| <（ max1≤i≤7|x_i|）^-σhas infinitely many solutions in positive integers λ₁, λ₂, ···, λ₇ Similar result is proved for |λ₁x₁²+ λ₂x₂²+ λ₃x₃²+ λ₄x₄⁴+ λ₅x₅⁴+ λ₆x₆⁴+ η| <（ max1≤i≤6|x_i|）^-σ.These results constitute an improvement upon those of Shi and Li.