Positive values of non-homogeneous quadratic forms of type (1,4): A conjecture of Bambah, Dumir and Hans-Gill

Let Q(x₁, ⋯,xₙ) be a real indefinite quadratic form of the type $(r,s)$, $n=r+s$, signature σ=r-s and determinant D≠ 0. Let Γr,n-r denote the infimum of all numbers Γ such that for any real numbers c₁, c₂ ,⋯, cₙ there exist integers x₁, x_ 2,⋯, xₙ satisfying 0< Q(x₁+c₁,x₂+c₂,⋯,xₙ+cₙ)≤ (Γ |D|)1/n. All the values of Γr,n-r are known except for Γ1,4. Earlier it was shown that 8≤ Γ1,4<12. It is conjectured that Γ1,4=8. Here we shall prove that Γ1,4=8, when (i) c₂ ≡ 0 1, (ii) c₂ ≡ 0 1, a≥ 1/2, where a is minima of positive definite ternary quadratic forms with determinant $4|D|$, and (iii) in some cases of c₂ ≡ 0 1, a< 1/2. We also obtain six critical forms for which the constant 8 is attained. In the remaining cases we prove that Γ1,4< 32/3.