Dans cet article nous considerons l'inégalité

où les xi désignent les nombres réels qui remplissent les conditions suivantes

Cette inégalité est proposée par H. S. Shapiro [1].

1. For any positive integral n and any positive real x1,…,xn we write

clearly. It is known [1,2] that

for n ≦ 6, and further [4, 5, 6] that (5) is false for even n ≧ 14 and for odd n ≧ 53. Mordell [2] conjectured that (5) is false for all n ≧ 7, but recently [3] stated that computations indicated that (5) is true for n = 7 and gave some calculations in support of (5) for n = 7.

We define two functions F(p), G(p) in terms of a third function ψ(ζ) by the equations

A standard problem in the theory of dual integral equations is to determine the function ψ(ζ) such that

when the functions f(p), g(p) are prescribed.

Let {Si; i ε I} be a finite or infinite family of cancellative semigroups. Let U be a cancellative semigroup, and suppose that there exists, for each i in I, a monomorphism φi: u→ Si. We are interested in finding a semigroup T with the following properties.

(a) For each i in I, there is a monomorphism λi: Si → T such that uφiλi = uøjλi for all u ɛ U and all i, j in I. That is to say, there exists a monomorphism λ: U → T which equals øiλi for all i in I.

Siλi∩Sjλj = Uλ (i, j ε I; i ≠ j).

For i, j = 1, 2, …, let aij be real. A matrix A = (aij) will be called positive (A>0) or non-negative (A≧0) according as, for all i and j, aij>0 or aij≧0 respectively. Correspondingly, a real vector x = (x1, x2, …) will be called positive (x>0) or non-negative (x≧0) according as, for all i, xi>0 or x≧0. A matrix A is said to be bounded if ∥ Ax ∥ ≦M ∥ x ∥ holds for some constant M, 0 ≦ M < ∞, and all x in the Hilbert space H of real vectors x = (x1, x2, …) satisfying . The least such constant M is denoted by ∥ A ∥. If x and y belong to H, then (x, y) will denote as usual the scalar product Σxiyi. Whether or not x is in H, or A is bounded, y = Ax will be considered as defined by

whenever each of the series of (1) is convergent.

1. Introductory. The following two integrals will be established in § 2.

If m is a positive integer, if p ≧ q + 1 and if R(ar+kt) > 0 (r = 1, 2,…, p, t = 1, 2,…, m),

where co is 1 or e±in according as m is even or odd, the dash denotes that the factor sin (k,–k,)π does not appear and the asterisk that the parameter kt–kt + 1 is omitted. If pp ≧ qthe result holds if the integral is convergent.

1. Introduction. Recently, I gave an analogue [1] of the MacRobert's E-function [4[ in the form

where the symbol denotes that a similar expression with a and β interchanged is to be added to the expression following it. It has since then been generalized by N. Agarwal [2], who defined and studied the q-analogue of the generalized E-function. In this paper I give some further properties of the Eq-function.

1. In this note we consider the formal solution of the dual integral equations

where f(x) and g(x) are given and χ(x) is to be found. The direct solution of these equations has been given by Noble [1] but we shall show that they may be solved more easily if they are first reduced to a form in which g(x) ≡ 0.

It is known [1] that for a partial endomorphism μ of a group G that maps the subgroup A ⊆ G onto B ⊆ G. G to be extendable to a total endomorphism μ* of a supergroup G* ⊆ G such that μ an isomorphism on G*(μ*)m for some positive integer m, it is necessary and sufficient that there exist in G a sequence of normal subgroups

such that L1 ƞA is the kernel of μ and

for ι = 1, 2,…, m–1.

The question then arises whether these conditions could be simplified when the group G is abelian. In this paper it is shown not only that the conditions are simplified when Gis abelian but also that the extension group G*⊇G can be chosen as an abelian group.

This note is concerned with square matrices, denoted by capital letters, whose elements belong to aBoolean algebra with null element 0 and all element 1. Such matrices, which have important applicationsin the theory of electric circuits, can be compounded in the three following ways.

Let D be a bounded, closed, simply-connected domain whose boundary C consists of a finite number of analytic Jordan curves. Let γ be any analytic arc of C. Then we shall prove the following theorem.

Theorem 1. Let u(x, y) be harmonic in the interior of D and continuous on γ, and let ϱu(x, y)/ϱn=g(s) when (x, y) is on γ, where g(s) is an analytic function of arc-length s along γ. Then u(x, y) can be harmonically continued across γ.