The Product of Two Non‐Homogeneous Linear Forms

Now the types of F(z), f(z) are attained only along the directions of strongest growth. Consequently, at the points at which F(z) and Fv{z) satisfy an inequality of the form (9), the functions Fll(z)(fx ^ v) do not attain the type h of F(z). A similar property holds for/(l/z) and the/,,(l/z). It therefore follows that \F(z)\ and |/(l/z) | attain large values at corresponding points along each identical direction of strongest growth, given by a formula similar to (11), although the densities of such sets of points will, in general, be different along different directions of strongest growth. As already noted |̂ (e~)—f(z)\ is bounded in the neighbourhood of the point 2 = 0 and this remark completes the proof of the theorem.