The subclasses S*(α, β) and C*(α, β) of T, the class of analytic and univalent functions of the form have been considered. Sharp results concerning coefficients, distortion of functions belonging to S*(α, β) and C*(α, β) are determined along with a representation formula for the functions in S*(α, β). Furthermore, it is shown that the classes S*(α, β) and C*(α,.β) are closed under arithmetic mean and convex linear combinations.

The main result in this paper is an existence theorem for the following complex nonlinear complementarity problem: find z such that

where S is a polyhedral cone in Cn, S* the polar cone, and g is a mapping from Cn into itself. It is shown that the above problem has a unique solution if the mapping g is continuous and strongly monotone on the polyhedral cone S.

If a finite group G has a ‘central’ involution t whose centralizer in G is (t) × H then, under certain conditions on H, G cannot be simple.

A strengthened form of the Kantorovich convergence theorem for the modified Newton's method is proved. The result is compared with previously known results.

There are two distinct approaches in the literature to framing a version of the law of the iterated logarithm for martingales. One involves norming by constants, using the martingale variance and the other involves norming by random variables, using the sums of conditional variances of the increments, given their past. In this paper a portmanteau approach is provided, still based on the Skorokhod representation of the martingale, but involving normalization by more general random variables. This extends the functional forms of all the previously existing results.

Let p, q, u, and v be any four positive integers, and let δ be a number in the interval 0 < δ ≤ 2. In one of his papers, Kurt Mahler, Bull. Austral. Math. Soc. 10 (1974), 325–335, proved that if q satisfies the inequalities

then

In this note, by a slightly different treatment of some inequalities in Mahler's paper, we easily obtain the same result with q only restricted by the first condition.

In this note the matrix equation A + WB + BTW + WCW = 0 is considered. A monotoneity result and an inertia theorem on the location of the eigenvalues of W and B + CW are proved.

If S is a set of positive integers which contains 1, 2, …, n–1, but not n or any multiple of n, where n ≥ 2, then

Here R is the field of real numbers, and ∥α∥ denotes the distance of α from the nearest integer.