Riemann–Stieltjes Integral Inequalities for Complex Functions Defined on Unit Circle

2019 ◽  
Author(s):  
Silvestru Sever Dragomir
Keyword(s):  
Unit Circle ◽  
Integral Inequalities ◽  
Stieltjes Integral ◽  
Complex Functions

Trapezoid type inequalities for complex functions defined on the unit circle with applications for unitary operators in Hilbert spaces

2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Sever S. Dragomir
Keyword(s):  
Unit Circle ◽  
Hilbert Spaces ◽  
Stieltjes Integral ◽  
Unitary Operators ◽  
Complex Functions ◽  
Continuous Complex ◽  
Complex Valued

AbstractSome trapezoid type inequalities for the Riemann–Stieltjes integral of continuous complex-valued integrands defined on the complex unit circle

scholarly journals New generalized trapezoid type inequalities for complex functions defined on unit circle and applications

2020 ◽  
Vol 72 (12) ◽  
pp. 1621-1632
Author(s):  
H. Budak
Keyword(s):  
Unit Circle ◽  
Bounded Variation ◽  
Quadrature Rule ◽  
Function Of Bounded Variation ◽  
Type Condition ◽  
Complex Functions

UDC 517.5 We establish new generalized trapezoid type inequalities for complex functions defined on unit circle via the function of bounded variation and the functions satisfying H¨older type condition. Using these results, quadrature rule formula is also provided.

scholarly journals Two Ostrowski type inequalities for the Stieltjes integral of monotonic functions

2007 ◽  
Vol 75 (2) ◽  
pp. 299-311 ◽  
Author(s):  
W.-S. Cheung ◽  
S. S. Dragomir
Keyword(s):  
Numerical Analysis ◽  
Trapezoidal Rule ◽  
Integral Inequalities ◽  
Stieltjes Integral ◽  
Hölder Continuous ◽  
Monotonic Functions ◽  
Dual Case

Two integral inequalities of Ostrowski type for the Stieltjes integral are given. The first is for monotonic integrators and Hölder continuous integrands while the second considers the dual case, that is, for monotonic integrands and Hölder continuous integrators. Applications for the mid-point inequality that are useful in the numerical analysis of Stieltjes integrals are exhibited. Some connections with the generalised trapezoidal rule are also presented.

scholarly journals Grüss Type Inequalities for Complex Functions Defined on Unit Circle with Applications for Unitary Operators in Hilbert Spaces

2015 ◽  
Vol 49 (1) ◽  
pp. 77-94 ◽  
Author(s):  
Silvestru Sever Dragomir
Keyword(s):  
Unit Circle ◽  
Hilbert Spaces ◽  
Unitary Operators ◽  
Complex Functions ◽  
Grüss Type Inequalities

Se proporcionan algunas desigualdades tipo Grüss para la integral de Riemann-Stieltjes de integrandos de valores continuos complejos definidos sobre el circulo unitario complejo C(0, 1) y varias subclases de integradores son dados. Aplicaciones naturales para funciones de operadores unitarios en espacios de Hilbert son proporcionadas.

scholarly journals Stable and Holomorphic Implementation of Complex Functions Using a Unit Circle-Based Transform

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Binesh Thankappan
Keyword(s):  
Unit Circle ◽  
Infinite Series ◽  
Complex Number ◽  
Complex Function ◽  
Unstable System ◽  
Base Complex ◽  
Base Unit ◽  
Complex Functions ◽  
Constituent Elements ◽  
The Given

A stable and holomorphic implementation of complex functions in ℂ plane making use of a unit circle-based transform is presented in this paper. In this method, any complex number or function can be represented as an infinite series sum of progressive products of a base complex unit and its conjugate only, where both are defined inside the unit circle. With each term in the infinite progression lying inside the unit circle, the sum ultimately converges to the complex function under consideration. Since infinitely large number of terms are present in the progression, the first element of which may be deemed as the base unit of the given complex number, it is addressed as complex baselet so that the complex number or function is termed as the complex baselet transform. Using this approach, various fundamental operations applied on the original complex number in ℂ are mapped to equivalent operations on the complex baselet inside the unit circle, and results are presented. This implementation has unique properties due to the fact that the constituent elements are all lying inside the unit circle. Out of numerous applications, two cases are presented: one of a stable implementation of an otherwise unstable system and the second case of functions not satisfying Cauchy–Riemann equations thereby not holomorphic in ℂ plane, which are made complex differentiable using the proposed transform-based implementation. Various lemmas and theorems related to this approach are also included with proofs.

Sign in / Sign up

Export Citation Format

Share Document