Two-Point Quadrature Rules for Riemann-Stieltjes Integrals with Lp-error estimates

TL;DR

This paper introduces new two-point quadrature rules for Riemann-Stieltjes integrals, providing sharp $L^p$ error estimates under Hölder and bounded variation conditions, enhancing numerical integration accuracy.

Contribution

The paper develops a general class of two-point quadrature rules for Riemann-Stieltjes integrals with proven dual formulas and sharp $L^p$ error bounds, advancing numerical integration methods.

Findings

Derived new two-point quadrature rules for Riemann-Stieltjes integrals.

Established sharp $L^p$ error estimates for the proposed rules.

Proved dual formulas under Hölder and bounded variation assumptions.

Abstract

In this work, we construct a new general two-point quadratre rules for the Riemann--Stieltjes integral $\int_a^b {f\left( t \right)du\left( t \right)}$, where the integrand $f$ is assumed to be satisfied with the H\"{o}lder condition on $[a,b]$ and the integrator $u$ is of bounded variation on $[a,b]$. The dual formulas under the same assumption are proved. Some sharp error $L^p$--Error estimates for the proposed quadrature rules are also obtained.

Equations

$${\int }_a^{b}f\left(t\right)du\left(t\right)\cong f\left(x\right)\left[u\left(b\right)-u\left(a\right)\right]\forall x\in [a,b].$$

$${\int }_a^{b}f\left(t\right)du\left(t\right)\cong \left[u\left(x\right)-u\left(a\right)\right]f\left(a\right)+\left[u\left(b\right)-u\left(x\right)\right]f\left(b\right).$$

$${\int }_a^{b}f\left(t\right)du\left(t\right)\cong \frac{f\left(a\right)+f\left(b\right)}{2}\left[u\left(b\right)-u\left(a\right)\right]\forall x\in [a,b].$$

$${\int }_a^{b}fdg\cong \left[G-g\left(a\right)\right]f\left(a\right)+\left[g\left(b\right)-G\right]f\left(b\right),$$

$${\int }_a^{b}f\left(t\right)dg\left(t\right)$$

$$+\left[g\left(b\right)-G\left(x,b\right)\right]f\left(b\right)$$

$${\int }_a^{b}f\left(t\right)du\left(t\right)\cong \left[u\left(\frac{a+b}{2}\right)-u\left(a\right)\right]f\left(x\right)+\left[u\left(b\right)-u\left(\frac{a+b}{2}\right)\right]f\left(a+b-x\right),$$

$$\Delta f_i=f\left(x_i\right)-f\left(x_{i-1}\right),$$

$$\underset{a}{\overset{b}{\bigvee }}(f;p)=\sup \left\{\sum (P):P\in \mathcal{P}\left(\left[a,b\right]\right)\right\},1\le p<\infty$$

$$\sum _{i=1}^n\mathrm{Osc}\left(f;\left[x_{i-1}^{\left(n\right)},x_i^{\left(n\right)}\right]\right)=\sum _{i=1}^n\left(\sup -\inf \right)f\left(t_i\right)\le M,t_i\in \left[x_{i-1}^{\left(n\right)},x_i^{\left(n\right)}\right],$$

$$\underset{a}{\overset{b}{\bigvee }}(f;\infty )=\sup \left\{\sum (P):P\in \mathcal{P}[a,b]\right\}:=\mathrm{Osc}\left(f;\left[a,b\right]\right),$$

$$\underset{a}{\overset{b}{\bigvee }}(f;\infty )=\underset{p\to \infty }{\lim }\underset{a}{\overset{b}{\bigvee }}(f;p)$$

$$=\mathrm{Osc}\left(f;\left[a,b\right]\right).$$

$$\mathrm{Osc}\left(f;\left[a,b\right]\right)\le \underset{a}{\overset{b}{\bigvee }}(f;p)\le \underset{a}{\overset{b}{\bigvee }}(f;1).$$

$${\mathcal{W}}_{\infty }(f)\subset {\mathcal{W}}_q(f)\subset {\mathcal{W}}_p(f)\subset {\mathcal{W}}_1(f)$$

$$\underset{a}{\overset{b}{\bigvee }}\left(f;p\right)={\left({\int }_a^b{\left|f^{\prime }\left(t\right)\right|}^{p}dt\right)}^{\frac{1}{p}}={\Vert f^{\prime }\Vert }_p,1\le p<\infty .$$

$$\left|{\int }_a^{b}w\left(t\right)d\nu \left(t\right)\right|\le {\Vert w\Vert }_{\infty }\cdot \mathrm{Osc}\left(\nu ;\left[a,b\right]\right)\le {\Vert w\Vert }_{\infty }\cdot \underset{a}{\overset{b}{\bigvee }}\left(\nu ;p\right),$$

$$\left|{\int }_a^{b}w\left(t\right)d\nu \left(t\right)\right|\le L{\left(b-a\right)}^{1-\frac{1}{p}}\cdot {\Vert w\Vert }_p,$$

$${\Vert w\Vert }_p={\left({\int }_a^b{\left|w\left(t\right)\right|}^{p}dt\right)}^{1/p},(1\le p<\infty ).$$

$${\int }_a^{b}f\left(s\right)du\left(s\right)={\mathcal{Q}}^{[a,b]}\left(f,u;t_0,x,t_1\right)+{\mathcal{R}}^{[a,b]}\left(f,u;t_0,x,t_1\right)$$

$${\mathcal{Q}}^{[a,b]}\left(f,u;t_0,x,t_1\right)=\left[u\left(x\right)-u\left(a\right)\right]f\left(t_0\right)+\left[u\left(b\right)-u\left(x\right)\right]f\left(t_1\right)$$

$${\mathcal{R}}^{[a,b]}\left(f,u;t_0,x,t_1\right):={\int }_a^{b}f\left(s\right)du\left(s\right)-\left[u\left(x\right)-u\left(a\right)\right]f\left(t_0\right)-\left[u\left(b\right)-u\left(x\right)\right]f\left(t_1\right)$$

$$\left|{\mathcal{R}}^{[a,b]}\left(f,u;t_0,x,t_1\right)\right|$$

$$\le H\max {\left\{\left[\frac{x-a}{2}+\left|t_0-\frac{a+x}{2}\right|\right],\left[\frac{b-x}{2}+\left|t_1-\frac{x+b}{2}\right|\right]\right\}}^r\cdot \underset{a}{\overset{b}{\bigvee }}\left(u;p\right)$$

$${\int }_a^x\left[f\left(t_0\right)-f\left(s\right)\right]du\left(s\right)+{\int }_x^b\left[f\left(t_1\right)-f\left(s\right)\right]du\left(s\right)$$

$$={\int }_a^{x}f\left(t_0\right)du\left(s\right)+{\int }_x^{b}f\left(t_1\right)du\left(s\right)-{\int }_a^{b}f\left(s\right)du\left(s\right)$$

$$=\left[u\left(x\right)-u\left(a\right)\right]f\left(t_0\right)+\left[u\left(b\right)-u\left(x\right)\right]f\left(t_1\right)-{\int }_a^{b}f\left(s\right)du\left(s\right)$$

$$=-{\mathcal{R}}^{[a,b]}\left(f,u;t_0,x,t_1\right),$$

$$\left|{\int }_c^{d}p\left(t\right)d\nu \left(t\right)\right|\le \underset{t\in \left[c,d\right]}{\sup }\left|p\left(t\right)\right|\underset{c}{\overset{d}{\bigvee }}\left(\nu \right).$$

$$\left|\left[u\left(x\right)-u\left(a\right)\right]f\left(t_0\right)+\left[u\left(b\right)-u\left(x\right)\right]f\left(t_1\right)-{\int }_a^{b}f\left(s\right)du\left(s\right)\right|$$

$$=\left|{\int }_a^x\left[f\left(t_0\right)-f\left(s\right)\right]du\left(s\right)+{\int }_x^b\left[f\left(t_1\right)-f\left(s\right)\right]du\left(s\right)\right|$$

$$\le \left|{\int }_a^x\left[f\left(t_0\right)-f\left(s\right)\right]du\left(s\right)\right|+\left|{\int }_x^b\left[f\left(t_1\right)-f\left(s\right)\right]du\left(s\right)\right|$$

$$\le \underset{s\in \left[a,x\right]}{\sup }\left|f\left(t_0\right)-f\left(s\right)\right|\cdot \underset{a}{\overset{x}{\bigvee }}\left(u;p\right)+\underset{s\in \left[x,b\right]}{\sup }\left|f\left(t_1\right)-f\left(s\right)\right|\cdot \underset{x}{\overset{b}{\bigvee }}\left(u;p\right).$$

$$\underset{s\in \left[a,x\right]}{\sup }\left|f\left(t_0\right)-f\left(s\right)\right|$$

$$=H\max \left\{{\left(x-t_0\right)}^r,{\left(t_0-a\right)}^r\right\}$$

Full text

1. Introduction

The number of proposed quadrature rules that provides approximation for the Riemann–Stieltjes integral ($\mathcal{RS}$–integral) $\int_{a}^{b}{f\left(t\right)du\left(t\right)}$ using derivatives or without using derivatives are very rare in comparison with the large number of methods available to approximate the classical Riemann integral $\int_{a}^{b}{f\left(t\right)dt}$.

The problem of introducing quadrature rules for $\mathcal{RS}$-integral $\int_{a}^{b}{fdg}$ was studied via theory of inequalities by many authors. Two famous real inequalities were used in this approach, which are the well known Ostrowski and Hermite-Hadamard inequalities and their modifications. For this purpose and in order to approximate the $\mathcal{RS}$-integral $\int_{a}^{b}{f\left(t\right)du\left(t\right)}$, a generalization of closed Newton-Cotes quadrature rules of $\mathcal{RS}$-integrals without using derivatives provides a simple and robust solution to a significant problem in the evaluation of certain applied probability models was presented by Tortorella in [32].

In 2000, Dragomir [16] introduced the Ostrowski’s approximation formula (which is of One-point type formula) as follows:

[TABLE]

Several error estimations for this approximation had been done in the works [15] and [16].

From different point of view, the authors of [17] (see also [11, 12]) considered the problem of approximating the Stieltjes integral $\int_{a}^{b}{f\left(t\right)du\left(t\right)}$ via the generalized trapezoid formula:

[TABLE]

Many authors have studied this quadrature rule under various assumptions of integrands and integrators. For full history of these two quadratures see [6] and the references therein.

Another trapezoid type formula was considered in [20], which reads:

[TABLE]

Some related results had been presented by the same author in [18] and [19]. For other connected results see [13] and [14].

In 2008, Mercer [27] introduced the following trapezoid type formula for the $\mathcal{RS}$-integral

[TABLE]

where $G=\frac{1}{{b-a}}\int_{a}^{b}{g\left(t\right)dt}$.

Recently, Alomari and Dragomir [4], proved several new error bounds for the Mercer–Trapezoid quadrature rule (1.1) for the $\mathcal{RS}$-integral under various assumptions involved the integrand $f$ and the integrator $g$.

Follows Mercer approach in [27], Alomari and Dragomir [10] introduced the following three-point quadrature formula:

[TABLE]

for all $a<x<b$, where $G\left({\alpha,\beta}\right):=\frac{1}{{\beta-\alpha}}\int_{\alpha}^{\beta}{g\left(t\right)dt}$.

Several error estimations of Mercer’s type quadrature rules for $\mathcal{RS}$-integral under various assumptions about the function involved have been considered in [4] and [7].

Motivated by Guessab-Schmeisser inequality (see [22]) which is of Ostrowski’s type, Alomari in [5] and [9] presented the following approximation formula for $\mathcal{RS}$-integrals:

[TABLE]

for all $x\in\left[{a,\frac{{a+b}}{2}}\right]$. For other related results see [6]. For different approaches variant quadrature formulae the reader may refer to [1], [8], [21] and [28].

Among others the $L^{\infty}$-norm gives the highest possible degree of precision; so that it is recommended to be ‘almost’ the norm of choice. However, in some cases we cannot access the $L^{\infty}$-norm, so that $L^{p}$-norm ($1\leq p<\infty$) is considered to be a variant norm in error estimations.

In this work, several $L^{p}$-error estimates ($1\leq p<\infty$) of general two and three points quadrature rules for Riemann-Stieltjes integrals are presented. The presented proofs depend on new triangle type inequalities for $\mathcal{RS}$-integrals.

Let $f$ be defined on $[a,b]$. If $P:=\left\{{x_{0},x_{1},\cdots,x_{n}}\right\}$ is a partition of $[a,b]$, write

[TABLE]

for $i=1,2,\cdots,n$. A function $f$ is said to be of bounded $p$-variation if there exists a positive number $M$ such that $\left({\sum\limits_{i=1}^{n}{\left|{\Delta f_{i}}\right|^{p}}}\right)^{\frac{1}{p}}\leq M$, $(1\leq p<\infty)$ for all partition of $[a,b]$, (see [26]).

Let $f$ be of bounded $p$-variation on $[a,b]$, and let $\sum(P)$ denote the sum $\left({\sum\limits_{i=1}^{n}{\left|{\Delta f_{i}}\right|^{p}}}\right)^{\frac{1}{p}}$ corresponding to the partition $P$ of $[a,b]$. The number

[TABLE]

is called the total $p$–variation of $f$ on the interval $[a,b]$, where $\mathcal{P}{\left(\left[a,b\right]\right)}$ denotes the set of all partitions of $[a,b]$. For $p=1$ it is the usual variation of $f(x)$ that was introduced by Jordan (see [24], [25]). For very constructive systematic study of Jordan variation we recommend the interested reader to refer to [29].

In special case, we define the variation of order $\infty$ of $f$ along $[a,b]$ in the classical sense, i.e., if there exists a positive number $M$ such that

[TABLE]

for all partition of $[a,b]$, then $f$ is said to be of bounded $\infty$–variation on $[a,b]$. The number

[TABLE]

is called the oscillation of $f$ on $[a,b]$. Equivalently, we may define the oscillation of $f$ as, (see [23]):

[TABLE]

Let $\mathcal{W}_{p}$ denotes the class of all functions of bounded $p$-variation $(1\leq p\leq\infty)$. For an arbitrary $p\geq 1$ the class $\mathcal{W}_{p}$ was firstly introduced by Wiener in [30], where he had shown that $\mathcal{W}_{p}$ can only have discontinuities of the first kind. More generally, if $f$ is a real function of bounded $p$-variation on an interval $[a,b]$, then:

• •

$f$ is bounded, and

[TABLE]

This fact follows by Jensen’s inequality applied for $h(p)=\bigvee_{a}^{b}(f;p)$ which is log-convex and decreasing for all $p>1$. Moreover, the inclusions

[TABLE]

are valid for all $1<p<q<\infty$, (see [31]).

• •

$f$ is continuous except at most on a countable set.

• •

$f$ has one-sided limits everywhere (limits from the left everywhere in $(a,b]$, and from the right everywhere in $[a,b)$;

• •

The derivative $f^{\prime}(x)$ exists almost everywhere (i.e. except for a set of measure zero).

• •

If $f\left(x\right)$ is differentiable on $\left[a,b\right]$, then

[TABLE]

Lemma 1**.**

[2*]**

Fix $1\leq p<\infty$. Let $f,g:[a,b]\to\mathbb{R}$ be such that $f$ is continuous on $[a,b]$ and $g$ is of bounded $p$–variation on $[a,b]$. Then the Riemann–Stieltjes integral $\int_{a}^{b}{f\left(t\right)dg\left(t\right)}$ exists and the inequality:*

[TABLE]

*holds. The constant $`1$’ in the both inequalities is the best possible. ***

Lemma 2**.**

[2]** Let $1\leq p<\infty$. Let $w,\nu:[a,b]\to\mathbb{R}$ be such that is $w\in L^{p}[a,b]$ and $\nu$ has a Lipschitz property on $[a,b]$. Then the inequality

[TABLE]

holds and the constant $`1$’ in the right hand side is the best possible, where

[TABLE]

In this paper, we establish two–point of Ostrowski’s integral inequality for the Riemann-Stieltjes integral $\int_{a}^{b}{f\left(t\right)du\left(t\right)}$, where $f$ is assumed to be of $r$-$H$-Hölder type on $[a,b]$ and $u$ is of bounded variation on $[a,b]$, are given. The dual formulas under the same assumption are proved. Some sharp error $L^{p}$–Error estimates for the proposed quadrature rules are also obtained.

2. The Results

Consider the quadrature rule

[TABLE]

where $\mathcal{Q}^{[a,b]}\left(f,u;t_{0},x,t_{1}\right)$ is the quadrature formula

[TABLE]

for all $a\leq t_{0}\leq x\leq t_{1}\leq b$.

Hence, the remainder term $\mathcal{R}^{[a,b]}\left(f,u;t_{0},x,t_{1}\right)$ is given by

[TABLE]

The following Two-point Ostrowski’s inequality for Riemann-Stieltjes integral holds.

Theorem 1**.**

Let $f:[a,b]\rightarrow\mathbb{R}$ be Hölder continuous of order $r$, $(0<r\leq 1)$, and $u:[a,b]\rightarrow\mathbb{R}$ is a mapping of bounded $p$-variation $(1\leq p\leq\infty)$ on $[a,b]$. Then we have the inequality

[TABLE]

for all $a\leq t_{0}\leq x\leq t_{1}\leq b$. Furthermore, the first half of each max-term is the best possible in the sense that it cannot be replaced by a smaller one, for all $r\in(0,1]$.

Proof.

Using the integration by parts formula for Riemann–Stieltjes integral, we have

[TABLE]

It is well known that if $p:[c,d]\to\mathbb{R}$ is continuous and $\nu:[c,d]\to\mathbb{R}$ is of $p$-bounded variation ($1\leq p<\infty$), then the Riemann-Stieltjes integral $\int_{c}^{d}{p(t)d\nu(t)}$ exists and the following inequality holds:

[TABLE]

Applying the inequality \reftagform@2.5 for $\nu(t)=u(t)$, $p(t)=f\left(t_{0}\right)-f\left(s\right)$, for all $s\in\left[{a,x}\right]$; and then for $p(t)=f\left({t_{1}}\right)-f\left(s\right)$, $\nu(t)=u(t)$ for all $t\in\left({x,b}\right]$, we get

[TABLE]

As $f$ is of $r$-$H$-Hölder type, we have

[TABLE]

and

[TABLE]

Therefore, by \reftagform@2.6, we have

[TABLE]

To prove the sharpness of the constant $\frac{1}{2^{r}}$ for any $r\in(0,1]$, assume that \reftagform@2.4 holds with a constant $C>0$, that is,

[TABLE]

Choose $f(t)=t^{r}$, $r\in(0,1]$, $t\in[0,1]$ and $u:[0,1]\to[0,\infty)$ given by

[TABLE]

As

[TABLE]

it follows that $f$ is $r$-$H$-Hölder type with the constant $H=1$.

By using the integration by parts formula for Riemann-Stieltjes integrals, we have:

[TABLE]

and $\bigvee_{0}^{1}(u;p)=1$. Consequently, by \reftagform@2.7, we get

[TABLE]

For $t_{0}=\frac{x}{2}$ and $t_{1}=x=1$ we get $\frac{1}{{2^{r}}}\leq C^{r}$, which implies that $C\geq\frac{1}{2}$.

It remains to prove the second part, so we consider

[TABLE]

therefore as we have obtained previously

[TABLE]

Consequently, by \reftagform@2.4, we get

[TABLE]

For $t_{0}=x=0$ and $t_{1}=\frac{1}{2}$ we get $\frac{1}{{2^{r}}}\leq C^{r}$, which implies that $C\geq\frac{1}{2}$, and the theorem is completely proved.

∎

The following inequalities are hold:

Corollary 1**.**

Let $f$ and $u$ as in Theorem 1. In 2.4 choose

1. (1)

$t_{0}=a$* and $t_{1}=b$, then we get the following trapezoid type inequality*

[TABLE]

or equivalently, we may write using parts formula for Riemann-Stieltjes integral

[TABLE]

The constant $\frac{1}{2}$ is the best possible for all $r\in(0,1]$. 2. (2)

$x=\frac{a+b}{2}$, then we get the following mid-point type inequality

[TABLE]

The constant $\frac{1}{4}$ is the best possible for all $r\in(0,1]$. For instance, setting $t_{0}=y$ and $t_{1}=a+b-y$, we get

[TABLE]

for all $y\in\left[{a,\frac{a+b}{2}}\right]$. 3. (3)

$t_{0}=\frac{a+x}{2}$* and $t_{1}=\frac{x+b}{2}$, then*

[TABLE]

Both constants $\frac{1}{2^{r}}$ and $\frac{1}{2}$ are the best possible for all $r\in(0,1]$.

Corollary 2**.**

Let $f$ be a Hölder continuous function of order $r$ $(0<r\leq 1)$, on $[a,b]$, and $g:[a,b]\to\mathbb{R}$ is continuous on $[a,b]$. Then we have the inequality

[TABLE]

for all $a\leq t_{0}\leq x\leq t_{1}\leq b$, where $\left\|g\right\|_{p}=\left({\int_{a}^{b}{\left|{g\left(t\right)}\right|^{p}dt}}\right)^{1/p}$.

Proof.

Define the mapping $u:[a,b]\to\mathbb{R}$, $u(t)=\int_{a}^{t}{g(s)ds}$. Then $u$ is differentiable on $(a,b)$ and $u^{\prime}(t)=g(t)$. Using the properties of the Riemann-Stieltjes integral, we have

[TABLE]

and

[TABLE]

which gives the required result. ∎

Theorem 2**.**

Let $1\leq p<\infty$. Let $f,u:[a,b]\to\mathbb{R}$ be such that is $f\in L^{p}[a,b]$ and $u$ has a Lipschitz property on $[a,b]$. If $f$ is $r$–$H$–Hölder continuous, then the inequality

[TABLE]

holds for all $p>1$ and $r\in\left(0,1\right]$.

Proof.

From Lemma 2 we have

[TABLE]

which proves the required result. ∎

Corollary 3**.**

Let $f$ and $u$ as in Theorem 2. In \reftagform@2.8 choose

1. (1)

$t_{0}=a$* and $t_{1}=b$, then we get the following trapezoid type inequality*

[TABLE]

or equivalently, we may write using parts formula for Riemann-Stieltjes integral

[TABLE] 2. (2)

$x=\frac{a+b}{2}$, then we get the following mid-point type inequality

[TABLE]

For instance, setting $t_{0}=y$ and $t_{1}=a+b-y$, we get

[TABLE]

for all $y\in\left[{a,\frac{a+b}{2}}\right]$. 3. (3)

$t_{0}=\frac{3a+b}{4}$, $x=\frac{a+b}{2}$ and $t_{1}=\frac{a+3b}{4}$, then

[TABLE]

Now, let $I$ be a real interval such that $[a,b]\subseteq I^{\circ}$ the interior of $I$, $a,b\in\mathbb{R}$ with $a<b$. Consider $\mathfrak{U}^{p}(I)$ ($p>1$) be the space of all positive $n$-th differentiable functions $f$ whose $n$-th derivatives $f^{(n)}$ is positive locally absolutely continuous on $I^{\circ}$ with $\int_{a}^{b}{\left({f^{(n)}\left(t\right)}\right)^{p}dt}<\infty,$ and $f^{(n)}(a)=f^{(n)}(b)=0$.

$L^{p}$-error estimates for Riemann–Stieltjes $\int_{a}^{b}{f\left(t\right)du\left(t\right)}$ where $f$ belongs to $\mathfrak{U}^{p}(I)$ is considered in the following result.

Theorem 3**.**

Let $1\leq p<\infty$. Let $f,u:[a,b]\to\mathbb{R}$ be such that is $f\in\mathfrak{U}^{p}(I)$ and $u$ has a Lipschitz property on $[a,b]$. If $f$ is $r$–$H$–Hölder continuous, then the inequality holds for all $p>1$ and $r\in\left(0,1\right]$.

[TABLE]

Proof.

As in the proof of Theorem 2, we have by Lemma 2

[TABLE]

which proves the required result, where we have used that fact that if $h\in\mathfrak{U}^{p}(I)$ then for all $\xi\in(a,b)$ we have

[TABLE]

In case $n=1$, the inequality \reftagform@2.10 is sharp, see [3]. ∎

Remark 1**.**

If $f\in\mathfrak{U}^{p}(I)$ and $f^{(n)}$ is bounded on $I$, so that as $p\to\infty$ in \reftagform@2.9, then since $\mathop{\lim}\limits_{p\to\infty}\frac{{p\sin\left({\frac{\pi}{p}}\right)}}{{\sqrt[p]{{p-1}}}}=\pi$, therefore we have

[TABLE]

In what follows we observe several general quadrature rules for the Riemann–Stieltjes integral $\int_{a}^{b}{f\left(t\right)du\left(t\right)}$ where $f$ is $n$-times differentiable whose derivatives belongs ton $L^{p}([a,b])$. To the best of our knowledge, this is the first time of such result concerning Riemann–Stieltjes integral without using interpolation.

Corollary 4**.**

Let $f$ and $u$ as in Theorem 3. In \reftagform@2.9 choose

1. (1)

$t_{0}=a$* and $t_{1}=b$, then we get the following trapezoid type inequality*

[TABLE]

or equivalently, we may write using parts formula for Riemann-Stieltjes integral

[TABLE] 2. (2)

$x=\frac{a+b}{2}$, then we get the following mid-point type inequality

[TABLE]

For instance, setting $t_{0}=y$ and $t_{1}=a+b-y$, we get

[TABLE]

for all $y\in\left[{a,\frac{a+b}{2}}\right]$. 3. (3)

$t_{0}=\frac{3a+b}{4}$, $x=\frac{a+b}{2}$ and $t_{1}=\frac{a+3b}{4}$, then

[TABLE]

3. The dual assumptions

In this section, $L^{p}$-error estimates of Two-point quadrature rules for the Riemann–Stieltjes integral $\int_{a}^{b}{f\left(t\right)du\left(t\right)}$, where the integrand $f$ is of bounded variation on $[a,b]$ and the integrator $u$ is assumed to be satisfied the Hölder condition on $[a,b]$.

Theorem 4**.**

Let $u:[a,b]\rightarrow\mathbb{R}$ be a Hölder continuous of order $r$, $(0<r\leq 1)$, and $f:[a,b]\rightarrow\mathbb{R}$ is a mapping of bounded $p$-variation $(1\leq p\leq\infty)$ on $[a,b]$. Then we have the inequality

[TABLE]

for all $a\leq t_{0}\leq x\leq t_{1}\leq b$. Furthermore, the constant $1$ is the best possible in the sense that it cannot be replaced by a smaller one, for all $r\in(0,1]$.

Proof.

Using the integration by parts formula for Riemann–Stieltjes integral, we have

[TABLE]

Adding these identities, we get

[TABLE]

Applying the triangle inequality on the above identity and then use Lemma 1, for each term separately, we get

[TABLE]

As $u$ is of $r$-$H$–Hölder type, we have

[TABLE]

[TABLE]

and

[TABLE]

Therefore, by \reftagform@3.3, we have

[TABLE]

To prove the sharpness of the constant $1$ for any $r\in(0,1]$, assume that \reftagform@3.1 holds with a constant $C>0$, that is,

[TABLE]

Choose $u(t)=t^{r}$, $r\in(0,1]$, $t\in[0,1]$ and $f:[0,1]\to[0,\infty)$ given by

[TABLE]

As

[TABLE]

it follows that $u$ is $r$-$H$-Hölder type with the constant $H=1$.

By using the integration by parts formula for Riemann-Stieltjes integrals, we have:

[TABLE]

and $\bigvee_{0}^{1}(f;p)=1$. Consequently, by \reftagform@3.4, we get

[TABLE]

Assume first

[TABLE]

so that we get $1\leq C$.

Now, assume that

[TABLE]

choose $t_{1}=1-t_{0}$, so that we get $1\leq C$.

Finally, we assume that

[TABLE]

Define $f:[0,1]\to[0,\infty)$ given by

[TABLE]

Clearly, $\bigvee_{0}^{1}(f;p)=2$. Therefore, for $t_{0}=0$ and $t_{1}=1$, so that we get $1\leq C\left(\frac{1}{2}+\left|x-\frac{1}{2}\right|\right)^{r}2^{1/p}$. Choosing $x=\frac{1}{2}$ and $r=\frac{1}{p}$ or $p=\frac{1}{r}$, it follows that $1\leq C\left(\frac{1}{2}\right)^{r}2^{r}$, i.e., $C\geq 1$. Hence, the inequality \reftagform@3.1 is sharp, and the theorem is completely proved.

∎

Theorem 5**.**

Let $1\leq p<\infty$. Let $f,u:[a,b]\to\mathbb{R}$ be such that is $u\in L^{p}[a,b]$ and $f$ has a Lipschitz property on $[a,b]$. If $u$ is $r$-$H$–Hölder continuous, then the inequality

[TABLE]

holds for all $p>1$ and $r\in\left(0,1\right]$ with constant $H>0$.

Proof.

As in the proof of Theorem 4, we have by Lemma 2

[TABLE]

Simple computations yield that

[TABLE]

[TABLE]

and

[TABLE]

Combining these equalities with the last inequality above we get the required result. ∎

Corollary 5**.**

Let $1\leq p<\infty$. Let $f,u:[a,b]\to\mathbb{R}$ be such that is $u\in L^{p}[a,b]$ and $f$ has a Lipschitz property on $[a,b]$. If $u$ is $r$-$H$–Hölder continuous, then the inequality

[TABLE]

holds for all $p>1$ and $r\in\left(0,1\right]$ with constant $H>0$.

Proof.

Setting $u(t)=t^{r}$, $t\in[a,b]$, $r\in(0,1]$, in Theorem 5 we get the required result. ∎

Corollary 6**.**

Let $1\leq p<\infty$. Let $f,u:[a,b]\to\mathbb{R}$ be such that is $u\in L^{p}[a,b]$ and $f$ has a Lipschitz property on $[a,b]$. If $u$ is $K$-Lipschitz continuous on $[a,b]$, then the inequality

[TABLE]

holds for all $p>1$ and constant $K>0$.

Proof.

Setting $r=1$c in Corollary 5, we get the required result. ∎

Remark 2**.**

The inequalities \reftagform@3.6 and \reftagform@3.7 generalize the recent result(s) in [2].