We generalize some criteria of boundedness of L-index in joint variables for in a bidisc analytic functions. Our propositions give an estimate the maximum modulus on a skeleton in a bidisc and an estimate of (p + 1)th partial derivative by lower order partial derivatives (analogue of Hayman’s theorem).
Let Th be a triangulation of a bounded polygonal domain Ω ⊂ R2 , Lh the space of the functions from C(Ω) linear on the triangles from Th and Πh the interpolation operator from C(Ω) to Lh. For a unit vector z and an inner vertex a of Th, we describe the set of vectors of coefficients such that the related linear combinations of the constant derivatives ∂Πh(u)/∂z on the triangles surrounding a are equal to ∂u/∂z(a) for all polynomials u of the total degree less than or equal to two. Then we prove that, generally, the values of the so-called recovery operators approximating the gradient ∇u(a) cannot be expressed as linear combinations of the constant gradients ∇Πh(u) on the triangles surrounding a.