For the class of meromorphically starlike functions of prescribed order, the concept of type has been introduced. A characterization of meromorphically starlike functions of order α and type β has been obtained when the coefficients in its Laurent series expansion about the origin are all positive. This leads to a study of coefficient estimates, distortion theorems, radius of convexity estimates, integral operators, convolution properties et cetera for this class. It is seen that the class considered demonstrates, in some respects, properties analogous to those possessed by the corresponding class of univalent analytic functions with negative coefficients.

The first property occurred in the investigation of directly representable varieties, and was named C2 by R. McKenzie, the second one is new. Our analysis is independent of injectivity. However, in the forthcoming second part of this paper we are going to prove that varieties with enough injectives satisfy both properties, and shall use intensively the results proved here.

Varieties with enough injectives satisfy the congruence extension property (CEP). We investigate the CEP in modular varieties by using the techniques developed in the first part of the present paper. As corollaries, we obtain the results of B. Davey and J. Kollár for the congruence distributive case as well as the description of all varieties of groups and rings with CEP, given by B. Biró, E.W. Kiss and P.P. Pálfy.

The main aim of the paper is to examine the properties of the integral operator , α ∈ C, defined on some classes of functions univalent and convex in the disc |z| < 1. As special cases we obtain results of Kim and Merkes and of Kumar and Shukla.

Let A and B be closed subalgebras of Cr(K) whose direct sum is Cr(K) According to a result of Stephen Fisher the set of common zeros of elements of B is a retract of K. We give examples to show that this result is not correct.

In this paper we show that a number of existence theorems for the Cauchy problem of ordinary differential equations in Banach spaces are only apparent generalizations of the previous ones.

A semigroup S is said to be monotone if its binary operation is a monotone function from S × S into S. This paper utilizes some of the known algebraic structure of Clifford semigroups, semigroups which are unions of groups, to study topological Clifford semigroups which are monotone. It is shown that such semigroups are preserved under products, homomorphisms, and, under certain conditions, closures. Necessary and sufficient conditions for monotonicity of groups, paragroups, bands, compact orthodox Clifford semigroups, and compact bands of groups are developed.

This paper is an endeavour to improve upon the work begun in an earlier paper with the same title. We prove a general theorem on the summability |R, exp((log ω)β+1), ρ| of the series ∑ {sn(x)−s}/n, where {sn(x)} is the sequence of partial sums at a point x of the Fourier series of a Lebesgue integrable 2π-periodic function and s is a suitable constant. While the theorem improves upon the main result contained in the previous paper, corollaries to it include recent results due to Chandra and Yadava.

Explicit expressions are given for the noncentral moments of the multivariate normal. Finding the general moment is shown to be equivalent to finding the general derivative of the density of the multivariate normal, that is to finding an expression for the multivariate Hermite polynomial.

Let L be a closed linear space of continuous real-valued functions, containing constants, on a compact Hausdorff space Ω. This paper gives some new criteria for a closed subset E of Ω to be an L-interpolation set, or more generally for L|E to be uniformly closed or simplicial, in terms of distances between certain compact convex subsets of the state space of L. These criteria involve the facial structure of the state space and hence are of a geometric nature. The results sharpen some standard results of Glicksberg.

There exists in the literature several notions of joint spectra which can be generalized to joint Browder spectra. The purpose of this note is to show that various notions of joint Browder spectra coincide for a special class of operators.

In the study of torsion-free abelian groups of finite rank the notions of irreducibility, field of definition and E-ring have played significant rôles. These notions are tied together in the following theorem of R. S. Pierce:

THEOREM. Let R be a ring whose additive group is torsion free finite rank irreducible and let Γ be the centralizer of QR as a QE(R) module. Then Γ is the unique smallest field of definition of R. Moreover, Γ ∩ R is an E-ring, in fact, it is a maximal E-subring of R.

In this paper we consider extensions of Pierce's result to the infinite rank case. This leads to the concept of local irreducibility for torsion free groups.

In Theorem 2 of [1] conclusion (c) should read:

(c) lim(log[Φn(m)/(m])/m=log λn, where λn is the (unique) m→∞ positive (and the largest in absolute value) zero of the x2n+1 − x2n+1 − 1.