Letp denote the family of functions regular in E{z:|z| < 1} and with positive real part there. We propose to study, in this article, the subclass p2a1 of p whose functions P(z) have pre-assigned second coefficient 2a1. In what follows we may assume, without loss in generality, that a1 is real and non-negative. This assumption will be made throughout. As is well known [2], 0 ≦ a1 ≦ 1. In Theorem 1 we derive a generalization of Zmorovic's theorem 1, [3]. determine the radius of univalence and starlikeness of the class of functions

The notion of weak compactness plays an important role in normed linear space theory. However in many instances norm completeness of the space is an important assumption for obtaining relevant theorems. This note generalises the concept of weak compactness for subsets of normed linear spaces and obtains generalisations of known theorems including that of Eberlein [2]. We use the methods and techniques of “non-standard analysis” in the proofs; several useful non-standard results for normed linear spaces are given.

In [3], Ligh proved that every distributively generated Boolean near-ring is a ring, and he gave an example to which the above fact can not be extended. That is, let G be an additive group and let the multiplication on G be defined by xy = y for all x, y in G. Ligh called this Boolean near-ring G a general Boolean near-ring. near-ring. Then in [4], Ligh called Ra β-near-ring if for each x in R, x2 = x and xyz = yxz for all x, y, z in R, and he proved that the structure of a β-near-ring is “very close” to that of a usual Boolean ring. We note that general Boolean near-rings and Boolean semirings as defined in [5] are β-near-rings. The purpose of this paper is to generalize the structure theorem on β-near-rings given by Ligh in [4] to a broader class of near-rings.

The results described herein were obtained in connection with the development of some lattice theoretic machinery needed by Randall and Foulis in their treatment of the so-called “logic of empirical science” (See [3] for an introduction to this subject). The final product, however, turned out to be of sufficient interest to warrant its independent presentation.

The concept of integrals of fractional order of a function f, defined by if Reα > 0, can be extended to generalised functions in the framework of the theory of convolution of distributions. The resulting theory [2, Chap. I §5.5] is very satisfactory for many purposes but there are circumstances in which it is not suitable. Such circumstances arise in particular if it is necessary to multiply, before or after integratrion, by non-integral powers of the variable. Pointwise multiplication by fractional powers of the independent variable does not make sense in the theory of distributions.

A group which is minimally generated by n generators and defined by n relations is said to have zero deficiency. The class of finite groups known to have zero deficiency is small, consisting of cyclic groups, certain metacyclic groups [4] and classes of groups given in [1], [2] and [3].

In his paper [1], Thron introduced a concept of lattice-equivalence of topological spaces. Let C(X) denote the lattice of all closed sets of a topological space X. Two topological spaces X and Y are said to be lattice-equivalent if there exists a lattice-isomorphism between C(X) and C(Y). It is clear that for any continuous function f: X → Y, the induced map ψf: C(Y) → C(X), defined by ψ(F)=f−1(F), is a lattice-homomorphism. Furthermore, if h: X→ Y is a homeomorphism then ψh: C(Y) → C(X) is a lattice-isomorphism. Thron proved among others that for TD-spaces X and Y, any lattice-isomorphism: C(Y) → C(X) can be induced by a homeomorphism f: X → Y in the above way.

Let τ be any cardinal number. Edwin Hewitt [3] has defined a topological space (X, J) to be τ-maximal if δ(J) ≥ τ and δ(J') < τ whenever J' is a topology for X which is strictly stronger that J (Δ denotes dispersion character, the least cardinality of a nonempty open set). The notion of an ℵ0-maximal space was introduced independently by Katětov [4].

Lattices of congruences have been studied by many authors. Hall [4] has shown that the lattice of congruences on a semilattice satisfy a certain chain condition which is a natural extension to arbitrary lattices of the Jordan-Dedekind chain condition for finite lattices.

In Theorem 1, we shall discuss some properties of semifinite measure, that is, the measure μ on a ring R of sets with the property that, for every E in R, μ(E) is equal to the least upper bound of μ(F) where F runs over sets such that F is in R (F ⊂ E) and μ(F) < ∞. Let σ(R) be the σ-ring generated by R. To prove Theorem 2 we shall use the uniqueness theorem in Luther's paper [2], which is stated as a lemma in this paper.

Central Sidon sets and central ∧p sets are defined and equivalent characterizations are given. It is shown that a central Sidon set with an upper bound on the degrees of its elements is a ∧p set (1 < p < ∞ ). The bound on the degrees is shown to be necessary by an example.

A Room squareR of order 2n is a way of arranging 2n objects (usually 1,2,…, 2n) in a square array R of side 2n – 1 so that:

(i) every cell of the array is empty or contains two objects;

(ii) each unordered pair of objects occurs once in R

(iii) every row and column of R contains one copy of each object.

In this article the concept of Dedekind semidomain is defined, and it is shown that certain structures of this kind are Noetherian, have integral closure, and have the property that their prime k-ideals are maximal. The second section provides the appropriate transportation theorems, while the third gives the main result concerning Dedekind semidomains. Examples are given throughout the paper to show that the hypothesies of certain theorems in the paper cannot be greatly weakened.

In this paper we shall be interested in the Weierstrass transform defined by (1.1.) converging (conditionally) for x in some interval, where (1.2) .

If A and B are locally convex topological vector spaces, and B has certain additional structure, then the space L(A, B) of all continuous linear mappings of A into B is characterized, within isomorphism, as the inductive limit of a family of spaces, whose elements are functions, or measures. The isomorphism is topological if L(A, B) is given a particular topology, defined in terms of the seminorms which define the topologies of A and B. The additional structure on B enables L(A, B) to be constructed, using the duals of the normed spaces obtained by giving A the topology of each of its seminorms separately.

Gleason [3] proved that in the category G of compact Hausdorff spaces and continuous maps, the projective objects are precisely the extremally disconnected spaces contained in the category. Strauss [7] generalised this and proved that in the category G of regular Hausdorif spaces and perfect maps the projective objects are again precisely the extremally disconnected ones. Observe that Gleason's category is a full subcategory of Strauss's category.