Mean values and Frullani integrals

Function Values as Boundary Integrals

Generalized Abstracted Mean Values

In this article, the author introduces the generalized abstracted mean values which extend the concepts of most means with two variables, and researches their basic properties and monotonicities.

Boundary Values of Cauchy Type Integrals

Results by A. G. Poltoratskĭı and A. B. Aleksandrov about nontangential boundary values of pseudocontinuable H2-functions on sets of zero Lebesgue measure are used for the study of operators on L2-spaces on the unit circle. For an arbitrary bounded operator X acting from one such L2-space to another and having the property that the commutator of it with multiplication by the independent variabl...

The First Mean Value Theorem for Integrals

For simplicity, we use the following convention: X is a non empty set, S is a σ-field of subsets of X, M is a σ-measure on S, f , g are partial functions from X to R, and E is an element of S. One can prove the following three propositions: (1) If for every element x of X such that x ∈ dom f holds f(x) ≤ g(x), then g − f is non-negative. (2) For every set Y and for every partial function f from...

Mean values of multiplicative functions

Let f(n) be a totally multiplicative function such that |f(n)| ≤ 1 for all n, and let F (s) = ∑∞ n=1 f(n)n−s be the associated Dirichlet series. A variant of Halász’s method is developed, by means of which estimates for ∑N n=1 f(n)/n are obtained in terms of the size of |F (s)| for s near 1 with 1. The result obtained has a number of consequences, particularly concerning the zeros of the p...