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In my paper under the same title, to which I shall refer in future as I‡, I generalized the Vitali covering principle from the case of Lebesgue measure to the case of any non-negative additive function. This allowed me to establish the relative differentiation of additive functions. The convergent sequences of sets in this generalized form of the covering principle were restricted to sequences of concentric circles, and therefore the differentiation arrived at was that in the symmetrical sense. In the present paper, I extend the principle to the case of any regular convergent sequences of covering sets; and then establish the relative differentiation of additive functions in the general sense, and in particular the differentiation of indefinite integrals with respect to any measure function. This problem has a complete solution. It is established that indefinite integrals are differentiable at almost all points. In the case of the general measure function, it is not true that the derivative is equal to the integrand at almost all points, but necessary and sufficient conditions are given under which this is true.

For the group of real numbers R, an exponential monomial is defined as a function of the form xr(−ixz), for some non-negative integer r and some complex number z. Similarly, an exponential polynomial is a function P(x) exp (−ixz), for a polynomial P. In a now famous paper ((15)), Schwartz proved that every closed translation invariant subspace (variety) of the space of continuous functions on R is determined by the exponential monomials it contains. His techniques do not generalize to groups other than R as they use the theory of functions of one complex variable. A shorter proof of this result, using the Carleman transform of a function, was given by Kahane in his thesis ((9)). Ehrenpreis ((5)) proved results similar to those of Schwartz for certain varieties in the space of analytic functions of n complex variables, and Malgrange ((13)) proved the related result that any solution in ℰ(Rn) (for the notation see (16)) of the homogeneous convolution equation μ*f = 0, for some μ∈ℰ′, belongs to the closure of the exponential polynomial solutions of the equation.

In this paper we investigate the following problem.We suppose given a sequence of complex values wn, defined for n = 0, 1, 2, …, and for n = ∞, and such thatwhile at least one wn differs from zero and ∞. We consider functions f(z), which are regular in | z | < 1, and take none of the sequence of values wn, and we investigate the effect of this restriction on the rate of growth of the function, as given by the maximum modulus

In a recent paper (1) the authors considered some generalizations of Cauchy's inequality. The method of approach was by the construction of certain convex functions (where by a convex function we mean a function f(x) satisfyingfor every pair of unequal values x1 and x2). For example, it was shown that if (a), (b) are the sets of non-negative real numbers a1, …,an; bl, …, bn, the functionwhich had been used earlier by Callebaut(2), is a convex function with minimum at x = 0, except if the sets (a) and (b) are proportional, in which case the function is a constant. This function takes the value (σab)2 when x = 0 and (σa2) (Σb2) when x = 1, and the property of convexity gives Cauchy's inequality.

Let σs(n) denote the sum of the sth powers of the divisors of n,and let , where ζ(s) is the Riemann ζfunction. Ramanujan, in his paper “On certain arithmetical functions*”, proves that, ifandthen , whenever r and s are odd positive integers. He conjectures that the error term on the right of (1·1) is of the formfor every ε > 0, and that it is not of the form . He further conjectures thatfor all positive values of r and s; and this conjecture has recently been proved to be correct.

In two classical papers (1, 2) J. M. Whittaker introduced the study of integral functions bounded at the lattice points m + in(m, n = 0, ± 1, …,). He succeeded in showing (cf. also G. Polya(3)) that an integral function of at most the minimum type of order 2 uniformly bounded at the lattice points was necessarily constant. This result was improved almost simultaneously by A. Pflüger(5) and V. Ganapathy Iyer(11), who showed that the result was true also for functions of type Kz) function shows that theirs is a best possible result in this direction.