Related papers: Interval Functions and their Integrals, by Ralph H…
Arzel\`a's bounded convergence theorem (1885) states that if a sequence of Riemann integrable functions on a closed interval is uniformly bounded and has an integrable pointwise limit, then the sequence of their integrals tends to the…
An integral on Euclidean space, equivalent to the Lebesgue integral, is constructed by extending the notion of Riemann sums. In contrast to the Henstock--Kurzweil and McShane integrals, the construction recovers the full measure-theoretic…
The Minkowski question mark function is a rich object which can be explored from the perspective of dynamical systems, complex dynamics, metric number theory, multifractal analysis, transfer operators, integral transforms, and as a function…
Additional integral inequalities are obtained for integrals of the differences of subharmonic functions by Borel measures on balls in a multidimensional Euclidean space. These integrals are still estimated from above through the Nevanlinna…
Integrating with respect to functions which are constant on intervals whose bounds are discontinuity points (of those functions) is frequent in many branches of Mathematics, specially in stochastic processes. For such functions and alike…
While the definition of a fractional integral may be codified by Riemann and Liouville, an agreed-upon fractional derivative has eluded discovery for many years. This is likely a result of integral definitions including numerous constants…
Weingarten functions provide a tool for computing Haar measure matrix integrals of polynomials in the matrix entries. An important property of Weingarten functions, is their particularly simple large $N$ limits. In 2017 Benoit Collins and…
No functions class for general measurable sets classes are known whose functions have the property of differentiability of integrals associated to such sets classes. In this paper,we give some subspaces of $L^s$ with $1<s<\infty$, whose…
In the paper Henstock, McShane, Birkhoff and variationally multivalued integrals are studied for multifunctions taking values in the hyperspace of convex and weakly compact subsets of a general Banach space X. In particular the existence of…
In this paper, the author establishes some Hadamard-type and Bullen-type inequalities for Lipschitzian functions via Riemann Liouville fractional integral. These results have some relationships with [K.-L. Tseng, S.-R. Hwang and K.-C. Hsu,…
We use the Baernstein star-function to investigate several questions about the integral means of the convolution of two analytic functions in the unit disc. The theory of univalent functions plays a basic role in our work.
We survey several non-absolutely convergent integrals, including the Henstock-Kurzweil and Pfeffer integrals, and use ideas from these theories to investigate the problem of multidimensional Young integration. We further present results on…
The paper concerns the investigation of nonconvex and nondifferentiable integral functionals on general Banach spaces, which may not be reflexive and/or separable. Considering two major subdifferentials of variational analysis, we derive…
Necessary and sufficient for $\int_a^bfg_n\to \int_a^bfg$ for all Henstock--Kurzweil integrable functions $f$ is that $g$ be of bounded variation, $g_n$ be uniformly bounded and of uniform bounded variation and, on each compact interval in…
The Three Gap Theorem, also known as the Steinhaus Conjecture, is a classical result on the combinatorics of the fractional part function, and has since been generalized in many ways. In this paper, we pose a new problem related to these…
It is of course well known that the usual definitions of Riemann integration and Riemann integrals are equivalent to simpler definitions which can be expressed in terms of just one sequence of partitions, using dyadic intervals or dyadic…
In the 1850s Weierstrass succeeded in solving the Jacobi inversion problem for the hyper-elliptic case, and claimed he was able to solve the general problem. At about the same time Riemann successfully applied the geometric methods that he…
In the theory of time scales, given $\mathbb{T}$ a time scale with at least two distinct elements, an integration theory is developed using ideas already well known as Riemann sums. Another, more daring, approach is to treat an integration…
We provide a proof of a conjecture made by Richard McIntosh in 1996 on the values of the Franel integrals, $$\int_0^1((ax))((bx))((cx))((ex))\,dx,$$ where $((x))$ is the first periodic Bernoulli function. Secondly, we extend our ideas to…
We remark a variant of the existence part of the fundamental theorem of calculus, which, together with the Lebesgue differentiation theorem, constitute a new proof that every Riemann-integrable function on a compact interval having limit…