A sharp companion of Ostrowski’s inequality for the Riemann–Stieltjes integral and applications

Mohammad W. Alomari

Displaying similar documents to “A sharp companion of Ostrowski’s inequality for the Riemann–Stieltjes integral and applications”

Riemann-Stieltjes Integral

Keiko Narita, Kazuhisa Nakasho, Yasunari Shidama (2016)

Formalized Mathematics

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In this article, the definitions and basic properties of Riemann-Stieltjes integral are formalized in Mizar [1]. In the first section, we showed the preliminary definition. We proved also some properties of finite sequences of real numbers. In Sec. 2, we defined variation. Using the definition, we also defined bounded variation and total variation, and proved theorems about related properties. In Sec. 3, we defined Riemann-Stieltjes integral. Referring to the way of the article [7],...

The Basic Existence Theorem of Riemann-Stieltjes Integral

Kazuhisa Nakasho, Keiko Narita, Yasunari Shidama (2016)

Formalized Mathematics

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In this article, the basic existence theorem of Riemann-Stieltjes integral is formalized. This theorem states that if f is a continuous function and ρ is a function of bounded variation in a closed interval of real line, f is Riemann-Stieltjes integrable with respect to ρ. In the first section, basic properties of real finite sequences are formalized as preliminaries. In the second section, we formalized the existence theorem of the Riemann-Stieltjes integral. These formalizations are...

Φ V[h] and Riemann-Stieltjes integration

N. Paul Schembari, Michael Schramm (1990)

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A strengthening of the Nyman-Beurling criterion for the Riemann hypothesis

Luis Báez-Duarte (2003)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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According to the well-known Nyman-Beurling criterion the Riemann hypothesis is equivalent to the possibility of approximating the characteristic function of the interval $(0, 1]$ in mean square norm by linear combinations of the dilations of the fractional parts $\{1 / a x\}$ for real $a$ greater than $1$ . It was conjectured and established here that the statement remains true if the dilations are restricted to those where the $a$ ’s are positive integers. A constructive sequence of such approximations is given. ...

Uniformity of holomorphic families of non-homeomorphic planar Riemann surfaces

Sachiko Hamano (2014)

Annales Polonici Mathematici

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We show the variation formula for the Schiffer span s(t) for moving Riemann surfaces R(t) with $t \in B = t \in ℂ | | t | < ρ$ , and apply it to show the simultaneous uniformization of moving planar Riemann surfaces of class $O_{A D}$ .

$C^{k}$ -estimates for the Cauchy-Riemann equations on certain weakly pseudoconvex domains

Jerzy Ryczaj (1987)

Colloquium Mathematicae

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Functional equations stemming from numerical analysis

Tomasz Szostok

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Always when a numerical method gives exact results an interesting functional equation arises. And, since no regularity is assumed, some unexpected solutions may appear. Here we deal with equations constructed in this spirit. The vast majority of this paper is devoted to the equation $\sum_{i = 0}^{l} {(y - x)}^{i} [f_{1, i} (α_{1, i} x + β_{1, i} y) + \dots + f_{k_{i}, i} (α_{k_{i}, i} x + β_{k_{i}, i} y)] = 0 (1)$ and its particular cases. We use Sablik’s lemma to prove that all solutions of (1) are polynomial functions. Since a continuous polynomial function is an ordinary polynomial, the crucial problem throughout...

< $> 𝒜 <$ >-Schemes and Zariski-Riemann Spaces

Satoshi Takagi (2012)

Rendiconti del Seminario Matematico della Università di Padova

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Variation of the argument of the Riemann ξ function on vertical lines

Xiannan Li (2009)

Acta Arithmetica

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Ergodic Universality Theorems for the Riemann Zeta-Function and other $L$ -Functions

Jörn Steuding (2013)

Journal de Théorie des Nombres de Bordeaux

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We prove a new type of universality theorem for the Riemann zeta-function and other $L$ -functions (which are universal in the sense of Voronin’s theorem). In contrast to previous universality theorems for the zeta-function or its various generalizations, here the approximating shifts are taken from the orbit of an ergodic transformation on the real line.

$Ω_{\pm}$ -results of the error term in the mean square formula of the Riemann zeta-function in the critical strip

Yuk-Kam Lau, Kai-Man Tsang (2001)

Acta Arithmetica

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Fermat’s method of quadrature

Jaume Paradís, Josep Pla, Pelegrí Viader (2008)

Revue d'histoire des mathématiques

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The of Fermat (c. 1659), besides containing the first known proof of the computation of the area under a higher parabola, $\int x^{+ m / n} d x$ , or under a higher hyperbola, $\int x^{- m / n} d x$ —with the appropriate limits of integration in each case—has a second part which was mostly unnoticed by Fermat’s contemporaries. This second part of the is obscure and difficult to read. In it Fermat reduced the quadrature of a great number of algebraic curves in implicit form to the quadrature of known curves: the higher parabolas...

Lower bounds of discrete moments of the derivatives of the Riemann zeta-function on the critical line

Thomas Christ, Justas Kalpokas (2013)

Journal de Théorie des Nombres de Bordeaux

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We establish unconditional lower bounds for certain discrete moments of the Riemann zeta-function and its derivatives on the critical line. We use these discrete moments to give unconditional lower bounds for the continuous moments $I_{k, l} (T) = \int_{0}^{T} {| ζ^{(l)} (\frac{1}{2} + i t) |}^{2 k} d t$ , where $l$ is a non-negative integer and $k \geq 1$ a rational number. In particular, these lower bounds are of the expected order of magnitude for $I_{k, l} (T)$ .

On subspaces of Riemann-Otsuki space.

Nadj, Djerdji F. (1981)

Publications de l'Institut Mathématique. Nouvelle Série

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Dirichlet series induced by the Riemann zeta-function

Jun-ichi Tanaka (2008)

Studia Mathematica

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The Riemann zeta-function ζ(s) extends to an outer function in ergodic Hardy spaces on $^{ω}$ , the infinite-dimensional torus indexed by primes p. This enables us to investigate collectively certain properties of Dirichlet series of the form $(a_{p}, s) = \prod_{p} {(1 - a_{p} p^{- s})}^{- 1}$ for $a_{p}$ in $^{ω}$ . Among other things, using the Haar measure on $^{ω}$ for measuring the asymptotic behavior of ζ(s) in the critical strip, we shall prove, in a weak sense, the mean-value theorem for ζ(s), equivalent to the Lindelöf hypothesis.

Henstock-Kurzweil integral on $BV$ sets

Jan Malý, Washek Frank Pfeffer (2016)

Mathematica Bohemica

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The generalized Riemann integral of Pfeffer (1991) is defined on all bounded $BV$ subsets of $ℝ^{n}$ , but it is additive only with respect to pairs of disjoint sets whose closures intersect in a set of $σ$ -finite Hausdorff measure of codimension one. Imposing a stronger regularity condition on partitions of $BV$ sets, we define a Riemann-type integral which satisfies the usual additivity condition and extends the integral of Pfeffer. The new integral is lipeomorphism-invariant and closed with respect...