An asymmetric inequality for non-homogeneous ternary quadratic forms

LetQ(x,y,z) be an indefinite ternary quadratic form of type (2,1) and determinantD(<0). Let 0≤t≤1/3 and (f(t) = frac{4}{{(1 + t)^2 (1 + 5t)}}) . Then given any real numbersx ₀,y ₀,z ₀ there exist integersx,y,z satisfying $$ - t(f(t)|D|)^{{1 mathord{left/ {vphantom {1 3}} right. kern-nulldelimiterspace} 3}}< Q (x + x_0 ,y + y_0 ,z + z_0 ) leqslant (f(t)|D|)^{{1 mathord{left/ {vphantom {1 3}} right. kern-nulldelimiterspace} 3}} $$ All the cases when equality holds are also obtained.