Two new inequalities for Riemann--Stieltjes integral are introduced functions of bounded $p$-variation and H\"{o}lder continuous integrators.

Utilising the Beesack version of the Darst-Pollard inequality, some error bounds for approximating the Riemann-Stieltjes integral are given. Some applications related to the trapezoid and mid-point quadrature rules are provided.

We study the process of integration on time scales in the sense of Riemann-Stieltjes. Analogues of the classical properties are proved for a generic time scale, and examples are given.

In this paper we present a formula relating Stieltjes numbers γn (1.3) and ηn (1.4). Using it we derive an explicit formula for the oscillating part of Li’s numbers ∼ λn (3.1) which are connected with the Riemann hypothesis.

Error bounds in approximating the Riemann-Stieltjes integral in terms of some moments of the integrand are given. Applications for p convex functions and in approximating the Finite Foureir Transform are pointed out as well. 1. Introduction In order to approximate the Riemann-Stieltjes integral R b a f (t) du (t) with the arguably simpler expression (1.1) u (b) u (a) b a Z b a f (t) dt; where R...

We study the existence and multiplicity of positive solutions a Riemann-Liouville fractional differential equation with r-Laplacian operator singular nonnegative nonlinearity dependent on integrals, subject to nonlocal boundary conditions containing various derivatives Riemann-Stieltjes integrals. use Guo–Krasnosel’skii fixed point theorem in proof our main results.

Thisstudy considered the Ruin problem with an income process stationaryindependent increments. The characterization is obtained which general forthe probability of r(y), that asset a firm will never bezero whenever initial level y. aim this studyis also to determine r(y) = P{T < ¥ | Y(0)= y}, If we let T inf{t ≥ 0; Y(t)< 0}, A condition necessary and sufficient studied for adistribution one – d...

1. Assume everything you know from the notes on Riemann-Stieltjes integration. (a) Show that if g : [0, 1]→ R is Riemann-integrable on [0, 1] and φ : [a, b]→ R is continuous on an interval [a, b] ⊇ g[[0, 1]], then φ ◦ g is Riemann-integrable on [0, 1]. (b) Assume that f : [0, 1]→ R is a function for which f ′(t) exists at each t ∈ [0, 1] (one-sidedly at endpoints) and that the Riemann integral ...