NMinimize[{axy (k - 1)/(2 kx^2 (y + z) + 2 xyz (1 + k) +
y^2 (2 z + (1 + k) x)),
x > 0 && y > 0 &&
z > 0 && (kb + kc - ka) xy + (kb - ka) xz + (b - a) yz + (b +
c) y^2 - dk^2 x^2 y - dk^2 x^2 z - 2 dkxyz - dkxy^2 - dy^2 z >
0 && (a + b + c) xy + (a + b) xz + (a + b) yz + (b + c) y^2 -
kdx^2 y - kdx^2 z - (1 + k) dxyz - dy^2 z - (1 + k) dxy^2/2 >
0 && -k (c + b - a) xy - k (b - a) xz - (b - a) yz - (b + c) y^2 +
kdx^2 y + kdx^2 z + (1 + k) dxyz + dy^2 z + (1 + k) dxy^2/2 >
0 && k (c + b - a) xy + k (b - a) xz + (b - a) yz + (b + c) y^2 +
kdx^2 y + kdx^2 z + (1 + k) dxyz + dy^2 z + (1 + k) dxy^2/2 >
0}, {x, y, z}]
y^2 (2 z + (1 + k) x)),
x > 0 && y > 0 &&
z > 0 && (kb + kc - ka) xy + (kb - ka) xz + (b - a) yz + (b +
c) y^2 - dk^2 x^2 y - dk^2 x^2 z - 2 dkxyz - dkxy^2 - dy^2 z >
0 && (a + b + c) xy + (a + b) xz + (a + b) yz + (b + c) y^2 -
kdx^2 y - kdx^2 z - (1 + k) dxyz - dy^2 z - (1 + k) dxy^2/2 >
0 && -k (c + b - a) xy - k (b - a) xz - (b - a) yz - (b + c) y^2 +
kdx^2 y + kdx^2 z + (1 + k) dxyz + dy^2 z + (1 + k) dxy^2/2 >
0 && k (c + b - a) xy + k (b - a) xz + (b - a) yz + (b + c) y^2 +
kdx^2 y + kdx^2 z + (1 + k) dxyz + dy^2 z + (1 + k) dxy^2/2 >
0}, {x, y, z}]