Volume 25, Issue 3 - 3

A normalized analytic function f ( z ) = z + a 2 z 2 + ··· (| z | < 1) is said to be in U (resp. P (2)) if for | z | < 1, It is known that P (2) ≠⊆ U ≠⊆ S , where S denotes the set of all normalized analytic functions that are univalent in | z | < 1. In this paper, we prove a general result which implies that We also show that if f ∊ S , then one has r −1 f ( rz ) ∊ P (2) for 0 < r < r 0 , where r 0 = 0.60629, correctly rounded to six decimal places, is the unique root of the equation 2 r 8 − 9 r 6 + 10 r 4 − 8 r 2 + 2 = 0.

The paper investigates the length L m of the m -th instability interval of the Hill equation (1 + ∊ A ( x )) y ″ + ∊ B ( x ) y ′ + (λ + ∊ C ( x )) y = 0 with A ( x ), B ( x ), C ( x ) being trigonometric polynomials. The leading term in the expansion of L m in powers of the perturbation parameter ∊ is found. The results are extensions of earlier work of Levy and Keller (1963).

The main result in this paper is that the solution operator for the bi-Laplace problem with zero Dirichlet boundary conditions on a bounded smooth 2d-domain can be split in a positive part and a possibly negative part which both satisfy the zero boundary condition. Moreover, the positive part contains the singularity and the negative part inherits the full regularity of the boundary. Such a splitting allows one to find a priori estimates for fourth order problems similar to the one proved via the maximum principle in second order elliptic boundary value problems. The proof depends on a careful approximative fill-up of the domain by a finite collection of limaçons. The limaçons involved are such that the Green function for the Dirichlet bi-Laplacian on each of these domains is strictly positive.