PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 92(106) (2012), 145–156 DOI: 10.2298/PIM1206145A

ON APPROXIMATION OF THE RIEMANN–STIELTJES INTEGRAL AND APPLICATIONS

Mohammad Wajeeh Alomari

Communicated by Gradimir Milovanović

Abstract. Several inequalities of Grüss type for the Stieltjes integral with various type of integrand and integrator are introduced. Some improvements inequalities are proved. Applications to the approximation problem of the Riemann–Stieltjes integral are also pointed out.

1. Introduction

In 2002, Guessab and Schmeisser [3], incorporate the mid-point and the trapezoid inequality together, and they proved the following companion of Ostrowski’s inequality:

Theorem 1. Assume that the function f : [a, b] → ℝ is of r-H-Hölder type, where r ∈ (0, 1] and H > 0 are given, i.e., |f(t) − f(s)| ⩽ H|t − s|^{r}, for any t, s ∈ [a, b]. Then, for each x ∈ [a, (a + b)/2], one has the inequality

(1.1)  | \frac{f(x) + f(a + b − x)}{2} − \frac{1}{b − a} ∫_{a}^{b}f(t) dt|

⩽ H [ \frac{2^{r+1} (x − a)^{r+1} + (a + b − 2x)^{r+1}}{2^{r}(r + 1)(b − a)} ] .

This inequality is sharp for each admissible x. Equality is obtained if and only if f = ±Hf_{∗} + c, with c ∈ ℝ and

f_{∗}(t) = \{ \begin{matrix}(x − t)^{r}, & a ⩽ t ⩽ x \\ t(t − x)^{r}, & x ⩽ t ⩽ (a + b)/2 \\ f_{∗}(a + b − x), & (a + b)/2 ⩽ t ⩽ b \end{matrix}

In [11] Dragomir proved the following companion of the Ostrowski inequality for mappings of bounded variation.

2010 Mathematics Subject Classification: Primary 26D15, 26D20; Secondary 41A55.

Key words and phrases: Ostrowski’s inequality, bounded variation.

Theorem 2. Let f : [a, b] → ℝ be a mapping of bounded variation on [a, b]. Then we have the inequalities:

(1.2) | \frac{f(x) + f(a + b − x)}{2} − \frac{1}{b − a} ∫_{a}^{b}f(t) dt| ⩽ [ \frac{1}{4} + | \frac{x − (3a + b)/4}{b − a} |] ⋁_{a}^{b} (f),

for any x ∈ [a, (a + b)/2], where ⋁ _{a}^{b}(f) denotes the total variation of f on [a, b]. The constant 1/4 is best possible.

Also, Dragomir in [12] proved some companions of Ostrowski’s integral inequality for absolutely continuous mappings. Among others, our interest is incorporated in the following result:

Theorem 3. Let f : I ⊂ ℝ → ℝ be an absolutely continuous function on [a, b] such that f′ ∈ L_{∞} [a, b]. Then for all x ∈ [a, (a + b)/2] we have the inequality

(1.3)  | \frac{f(x) + f(a + b − x)}{2} − \frac{1}{b − a} ∫_{a}^{b}f(t) dt|

⩽ [ \frac{1}{8} + 2 ( \frac{x − (3a + b)/4}{b − a} ) ^{2}] (b − a) ∥f′∥_{∞} .

By Guessab–Schmeisser functional we mean the functional

GS(f; u) := ∫_{a}^{(a+b)/2} \frac{f(x) + f(a + b − x)}{2} du(x) − \frac{u((a + b)/2) − u(a)}{b − a} ∫_{a}^{b}f(t) dt,

provided that the Stieltjes integral ∫_{a}^{b} \frac{1}{2}(f(x)+f(a+b−x)) du(x), and the Riemann integral ∫_{a}^{b}f(t) dt exist.

Motivated by Guessab–Schmeisser companion of Ostrowski’s inequality (1.1), the author of this paper, has established the functional GS(f; u) in [1], and he has proved the following results in estimating GS(f; u).

Theorem 4. Let f : [a, b] → ℝ be an r-H-Hölder type mapping on [a, b], where r and H > 0 are given, and u : [a, b] → ℝ be a mapping of bounded variation on [a, b]. Then the following inequality holds

(1.4)  |GS(f; u)| ⩽ \frac{H}{r + 1} (b − a)^{r} ⋁_{a}^{(a+b)/2} (u).

Theorem 5. Let f : [a, b] → ℝ be an r-H-Hölder type mapping on [a, b], and u : [a, b] → ℝ be an L-Lipschitzian mapping on [a, b], where r and H, L > 0 are given. Then the following inequality holds

(1.5)  |GS(f; u)| ⩽ \frac{LH}{(r + 1)(r + 2)} (b − a)^{r+1}

In this paper we point out several bounds for the functional GS(f; u) with various type of integrand and integrator. Improvements bounds for GS(f; u) are proved.

Finally, we apply the obtained results to approximate the Riemann–Stieltjes integral

∫_{a}^{(a+b)/2} \frac{f(x) + f(a + b − x)}{2} du(x)

in terms of the Riemann integral ∫_{a}^{b}f(t) dt

2. The case of bounded variation integrators

2.1. The case of bounded variation integrands.

Theorem 6. Let u : [a, b] → ℝ be a mapping of bounded variation on [a, b] and f : [a, b] → ℝ be continuous and of bounded variation on [a, b]. Then we have the inequality:

(2.1)  |GS(f; u)| ⩽ \frac{1}{2} ⋁_{a}^{b} (f) ⋁_{a}^{(a+b)/2} (u).

Proof. Using the fact that for a continuous function p : [a, b] → ℝ and a function ν : [a, b] → ℝ of bounded variation, one has the inequality

(2.2)  |∫_{a}^{b}p(t) dν(t)| ⩽ sup_{t∈[a,b]} |p(t)| ⋁_{a}^{b} (ν).

As u is of bounded variation on [a, b] and f is continuous, by (2.2) we have

|GS(f; u)| = |∫_{a}^{(a+b)/2} [ \frac{f(x) + f(a + b − x)}{2} − \frac{1}{b − a} ∫_{a}^{b}f(t) dt ]du(x)|

⩽ sup_{x∈[a,(a+b)/2]} | \frac{f(x) + f(a + b − x)}{2} − \frac{1}{b − a} ∫_{a}^{b}f(t) dt| ⋁_{a}^{(a+b)/2} (u).

Since f is of bounded variation, then using the companion of Ostrowski type inequality (1.2), we may state that

sup_{x∈[a,(a+b)/2]} | \frac{f(x) + f(a + b − x)}{2} − \frac{1}{b − a} ∫_{a}^{b}f(t) dt|

⩽ sup_{x∈[a,(a+b)/2 ]} [ \frac{1}{4} + | \frac{x − (3a + b)/4}{b − a} |] ⋁_{a}^{b} (f) ⩽ \frac{1}{2} ⋁_{a}^{b} (f).

It follows that

|GS(f; u)| ⩽ sup_{x∈[a,(a+b)/2]} | \frac{f(x) + f(a + b − x)}{2} − \frac{1}{b − a} ∫_{a}^{b}f(t) dt| ⋁_{a}^{(a+b)/2} (u)

⩽ \frac{1}{2} ⋁_{a}^{b} (f) ⋁_{a}^{(a+b)/2} (u),

Remark 1. If ⋁ _{a}^{(a+b)/2} (u) = ⋁ _{(a+b)/2}^{b} (u), then (2.1) becomes

|GS(f; u)| ⩽ \frac{1}{4} ⋁_{a}^{b} (f) ⋁_{a}^{b} (u)

Corollary 1. Let u be as in Theorem 6.

(1) If f : [a, b] → ℝ be an L-Lipschitzian mapping on [a, b], then

|GS(f; u)| ⩽ \frac{1}{2}L(b − a) ⋁_{a}^{(a+b)/2} (u).

(2) If f ∈ C^{(1)}[a, b], then |GS(f; u)| ⩽ \frac{1}{2} ⋁_{a}^{(a+b)/2} (u) · ∥f′∥_{1,[a,b]} .

(3)If f : [a, b] → ℝ be a monotonic mapping, then

|GS(f; u)| ⩽ \frac{1}{2} ⋁_{a}^{(a+b)/2} (u) · |f(b) − f(a)|,

where ∥·∥_{1} is the L_{1} norm, namely ∥f′∥_{1,[a,b]} := ∫_{a}^{b}|f′(t)|dt.

Corollary 2. Let f be as in Theorem 6.

(1) If u : [a, b] → ℝ be an K-Lipschitzian mapping on [a, b], then

|GS(f; u)| ⩽ \frac{1}{4}K(b − a) ⋁_{a}^{b} (f).

(2) If u ∈ C^{(1)}[a, b], then |GS(f; u)| ⩽ \frac{1}{2} ⋁_{a}^{b} (f) · ∥u′∥_{1,[a,(a+b)/2 ]} .

(3) If u : [a, b] → ℝ is a monotonic mapping, then

|GS(f; u)| ⩽ \frac{1}{2} ⋁_{a}^{b} (f) · |u( \frac{a + b}{2} ) − u(a) |,

where ∥ · ∥_{1} is the L_{1} norm, namely ∥u′∥_{1,[a,(a+b)/2 ]} := ∫_{a}^{(a+b)/2} |u′(t)|dt.

Remark 2. In Corollary1, we have the following cases:

(1) If f is L-Lipschitzian mapping on [a, b] and (a) u is K-Lipschitzian mapping on [a, b], then

(2.3)  |GS(f; u)| ⩽ \frac{1}{4}KL(b − a)^{2}.

(b) u ∈ C^{(1)}[a, b], then

(2.4)  |GS(f; u)| ⩽ \frac{1}{2}L(b − a) ∥u′∥_{1,[a,(a+b)/2 ]}.

(c) u is monotonic on [a, b], then

(2.5)  |GS(f; u)| ⩽ \frac{1}{2}L(b − a) |u ((a + b)/2) − u(a) |.

(2) If f ∈ C^{(1)}[a, b], and (a) u is K-Lipschitzian mapping on [a, b], then

(2.6)  |GS(f; u)| ⩽ \frac{1}{4}K(b − a)∥f′∥_{1,[a,b]}.

(b) u ∈ C^{(1)}[a, b], then we have the inequality:

(2.7)  |GS(f; u)| ⩽ \frac{1}{2}∥f′∥_{1,[a,b]}∥u′∥_{1,[a,(a+b)/2 ]}.

(c) u is monotonic on [a, b], then

(2.8)  |GS(f; u)| ⩽ \frac{1}{2}∥f′∥_{1,[a,b]} |u( \frac{a + b}{2} ) − u(a) |.

(3) If f is monotonic on [a, b], and (a) u is K-Lipschitzian mapping on [a, b], then

(2.9)  |GS(f; u)| ⩽ \frac{1}{4}K(b − a)|f(b) − f(a)|.

(b) u ∈ C^{(1)}[a, b], then

(2.10)  |GS(f; u)| ⩽ \frac{1}{2}|f(b) − f(a)|∥u′∥_{1,[a,(a+b)/2 ]}.

(c) u is monotonic on [a, b], then

(2.11)  |GS(f; u)| ⩽ \frac{1}{2}|f(b) − f(a)| |u( \frac{a + b}{2} ) − u(a) |.

Remark 3. In Corollary 2 we have the following cases:

(1) If u is K-Lipschitzian mapping on [a, b] and (a) f is L-Lipschitzian mapping on [a, b], then inequality (2.3) holds.

(b) f ∈ C^{(1)}[a, b], then inequality (2.6) holds.

(c) f is monotonic on [a, b], then inequality (2.9) holds.

(2) If u ∈ C^{(1)}[a, b], and (a) f is L-Lipschitzian mapping on [a, b], then inequality (2.4) holds.

(b) f ∈ C^{(1)}[a, b], then inequality (2.7) holds.

(c) f is monotonic on [a, b], then inequality (2.10) holds.

(3) If u is monotonic on [a, b], and (a) f is L-Lipschitzian mapping on [a, b], then inequality (2.5) holds.

(b) f ∈ C^{(1)}[a, b], then inequality (2.8) holds.

(c) f is monotonic on [a, b], then inequality (2.11) holds.

2.2. The case of r-H -Hölder type integrands.

Theorem 7. Let u : [a, b] → ℝ be a mapping of bounded variation on [a, b] and f : [a, b] → ℝ be of r-H-Hölder type mapping on [a, b]. Then

(2.12)  |GS(f; u)| ⩽ \frac{H}{2^{r}(r + 1)} (b − a)^{r} ⋁_{a}^{(a+b)/2} (u).

Proof. As u is of bounded variation on [a, b] and f is of r-H-Hölder type on [a, b], by (2.2) we have

|GS(f; u)| = | ∫_{a}^{(a+b)/2} [ \frac{f(x) + f(a + b − x)}{2} − \frac{1}{b − a} ∫_{a}^{b}f(t) dt]du(x) |

⩽ sup_{x∈[a,(a+b)/2 ]} | \frac{f(x) + f(a + b − x)}{2} − \frac{1}{b − a} ∫_{a}^{b}f(t) dt| ⋁_{a}^{(a+b)/2} (u).

Using the companion of Ostrowski’s type inequality (1.1), we may state that

sup_{x∈[a,(a+b)/2 ]} | \frac{f(x) + f(a + b − x)}{2} − \frac{1}{b − a} ∫_{a}^{b}f(t) dt|

⩽ H sup_{x∈[a,(a+b)/2]} [ \frac{2^{r+1} (x − a)^{r+1} + (a + b − 2x)^{r+1}}{2^{r}(r + 1)(b − a)} ] ⩽ H \frac{(b − a)^{r}}{2^{r}(r + 1)} .

It follows that

|GS(f; u)| ⩽ sup_{x∈[a,(a+b)/2 ]} | \frac{f(x) + f (a + b − x )}{2} − \frac{1}{b − a} ∫_{a}^{b}f(t) dt| ⋁_{a}^{(a+b)/2} (u)

⩽ H \frac{(b − a)^{r}}{2^{r}(r + 1)} ⋁_{a}^{(a+b)/2} (u),

and the theorem is proved. □

Remark 4. Inequality (2.12) improves inequality (1.4) by the constant \frac{1}{2^{r}} , and therefore, (2.12) is better than (1.4).

Corollary 3. Let u : [a, b] → ℝ be a mapping of bounded variation on [a, b] and f : [a, b] → ℝ be of L-Lipschitzian type mapping on [a, b]. Then

|GS(f; u)| ⩽ \frac{1}{4}L(b − a) ⋁_{a}^{(a+b)/2} (u).

Corollary 4. Let f : [a, b] → ℝ be of r-H-Hölder type mapping on [a, b].

(1) If u is K-Lipschitzian on [a, b], then |GS(f; u)| ⩽ \frac{HK}{2^{r+1} (r+1)} (b − a)^{r+1} .

(2) If u ∈ C^{(1)}[a, b], then |GS(f; u)| ⩽ \frac{H}{2^{r} (r+1)} (b − a)^{r} ∥u′∥_{1,[a,(a+b)/2 ]}.

(3) If u is monotonic on [a, b], then |GS(f; u)| ⩽ \frac{H}{2^{r} (r+1)} (b − a)^{r} |u( \frac{a+b}{2} ) − u(a) |.

Therefore, we may deduce the following result.

Corollary 5. Let f : [a, b] → ℝ be L-Lipschitzian mapping on [a, b].

(1) If u is K-Lipschitzian on [a, b], then |GS(f; u)| ⩽ \frac{1}{8}KL(b − a)^{2}.

(2) If u ∈ C^{(1)}[a, b], then |GS(f; u)| ⩽ \frac{1}{4}L(b − a) ∥u′∥_{1,[a,(a+b)/2]} .

(3) If u is monotonic on [a, b], then |GS(f; u)| ⩽ \frac{1}{4}L(b − a)|u( \frac{a+b}{2} ) − u(a) |.

2.3. The case of absolutely continuous integrands.

Theorem 8. Let u : [a, b] → ℝ be a mapping of bounded variation on [a, b] and f : [a, b] → ℝ is absolutely continuous on [a, b]. Then

|GS(f; u)| ⩽ \frac{1}{4}(b − a)∥f′∥_{∞,[a,b]} ⋁_{a}^{(a+b)/2} (u).

Proof. As u is of bounded variation on [a, b] and f is continuous, by (2.2) we have

|GS(f; u)| = | ∫_{a}^{(a+b)/2} [ \frac{f(x) + f(a + b − x)}{2} − \frac{1}{b − a} ∫_{a}^{b}f(t) dt]du(x)|

⩽ sup_{x∈[a,(a+b)/2 ]} | \frac{f(x) + f(a + b − x)}{2} − \frac{1}{b − a} ∫_{a}^{b}f(t) dt| ⋁_{a}^{(a+b)/2} (u).

Since f is absolutely continuous on [a, b], then using the companion of Ostrowski type inequality (1.3), we may state that

sup_{x∈[a,(a+b)/2 ]} | \frac{f(x) + f(a + b − x)}{2} − \frac{1}{b − a} ∫_{a}^{b}f(t) dt|

⩽ sup_{x∈[a,(a+b)/2 ]} [ \frac{1}{8} + 2( \frac{x − (3a + b)/4}{b − a} )^{2}] (b−a)∥f′∥_{∞,[a,b]} ⩽ \frac{1}{4}(b−a)∥f′∥_{∞,[a,b]} .

It follows that

|GS(f; u)| ⩽ sup_{x∈[a,(a+b)/2 ]} | \frac{f(x) + f(a + b − x)}{2} − \frac{1}{b − a} ∫_{a}^{b}f(t) dt| ⋁_{a}^{(a+b)/2} (u)

⩽ \frac{1}{4}(b − a)∥f′∥_{∞,[a,b]} ⋁_{a}^{(a+b)/2} (u),

and the theorem is proved. □

Corollary 6. Let f be as in Theorem 8.

(1) If u is K-Lipschitzian on [a, b], then |GS(f; u)| ⩽ \frac{1}{8}K(b − a)^{2}∥f′∥_{∞,[a,b]} .

(2) If u ∈ C^{(1)}[a, b], then |GS(f; u)| ⩽ \frac{1}{4}(b − a)∥f′∥_{∞,[a,b]} · ∥u′∥_{1,[a,(a+b)/2 ]}.

(3) If u is monotonic on [a, b], then |GS(f; u)| ⩽ \frac{1}{4}(b−a)∥f′∥_{∞,[a,b]} ·|u( \frac{a+b}{2} ) − u(a) |.

3. The case of Lipschitzian integrators

3.1. The case of bounded variation integrands.

Theorem 9. Let u : [a, b] → ℝ be an K-Lipschitzian on [a, b] and f : [a, b] → ℝ be of bounded variation on [a, b]. Then |GS(f; u)| ⩽ \frac{3}{16} K(b − a) ⋁ _{a}^{b}(f).

Proof. It is well-known that for a Riemann integrable function p : [a, b] → ℝ and L-Lipschitzian function ν : [a, b] → ℝ, one has the inequality

(3.1)  | ∫_{a}^{b}p(t)dν(t)| ⩽ L ∫_{a}^{b}|p(t)|dt.

Therefore, as u is K-Lipschitzian on [a, b], by (3.1) we have

|GS(f; u)| = | ∫_{a}^{(a+b)/2} [ \frac{f(x) + f(a + b − x)}{2} − \frac{1}{b − a} ∫_{a}^{b}f(t) dt]du(x)|

⩽ K ∫_{a}^{(a+b)/2} | \frac{f(x) + f(a + b − x)}{2} − \frac{1}{b − a} ∫_{a}^{b}f(t) dt|dx

Since f is of bounded variation, then using the companion of Ostrowski type inequality (1.2), we may state that

∫_{a}^{(a+b)/2} | \frac{f(x) + f(a + b − x)}{2} − \frac{1}{b − a} ∫_{a}^{b}f(t) dt|dx

⩽ ⋁_{a}^{b} (f) · ∫_{a}^{(a+b)/2} [ \frac{1}{4} + | \frac{x − (3a + b)/4}{b − a} |] dx ⩽ \frac{3}{16}(b − a) ⋁_{a}^{b} (f).

It follows that

|GS(f; u)| ⩽ K ∫_{a}^{(a+b)/2} | \frac{f(x) + f(a + b − x)}{2} − \frac{1}{b − a} ∫_{a}^{b}f(t) dt|dx

⩽ \frac{3}{16}K(b − a) ⋁_{a}^{b} (f),

and the theorem is proved. □

Corollary 7. Let u be as in Theorem 9.

(1) If f is L-Lipschitzian on [a, b], then |GS(f; u)| ⩽ \frac{3}{16} KL(b − a)^{2}.

(2) If f ∈ C^{(1)}[a, b], then |GS(f; u)| ⩽ \frac{3}{16} K(b − a)∥f′∥_{1,[a,b]}.

(3) If f is monotonic on [a, b], then |GS(f; u)| ⩽ \frac{3}{16} K(b − a) · |f(b) − f(a)|.

3.2. The case of r-H -Hölder type integrands.

Theorem 10. Let u : [a, b] → ℝ be an K-Lipschitzian on [a, b] and f : [a, b] → ℝ be of r-H-Hölder type mapping on [a, b]. Then

(3.2)  |GS(f; u)| ⩽ KH \frac{(b − a)^{r+1}}{2^{r}(r + 1)(r + 2)} .

Proof. As u is K-Lipschitzian on [a, b] and f is continuous, by (3.1) we have

|GS(f; u)| ⩽ K ∫_{a}^{(a+b)/2} | \frac{f(x) + f(a + b − x)}{2} − \frac{1}{b − a} ∫_{a}^{b}f(t) dt|dx

Using the companion of Ostrowski type inequality (1.1), we may state that

∫_{a}^{(a+b)/2} | \frac{f(x) + f(a + b − x)}{2} − \frac{1}{b − a} ∫_{a}^{b}f(t) dt|dx

⩽ H ∫_{a}^{(a+b)/2} [ \frac{2^{r+1} (x − a)^{r+1} + (a + b − 2x)^{r+1}}{2^{r}(r + 1)(b − a)} ] dx

⩽ H \frac{(b − a)^{r+1}}{2^{r}(r + 1)(r + 2)} .

It follows that

|GS(f; u)| ⩽ K ∫_{a}^{(a+b)/2} | \frac{f(x) + f(a + b − x)}{2} − \frac{1}{b − a} ∫_{a}^{b}f(t) dt|dx

⩽ KH \frac{(b − a)^{r+1}}{2^{r}(r + 1)(r + 2)} ,

and the theorem is proved. □

Remark 5. Inequality (3.2) improves inequality (1.5) by the constant \frac{1}{2^{r}} , and therefore (3.2) is better than (1.5).

Corollary 8. Let u be as in Theorem 10, and f : [a, b] → ℝ be of L-Lipschitzian type mapping on [a, b]. Then |GS(f; u)| ⩽ \frac{1}{12} KL(b − a)^{2}.

3.3. The case of absolutely continuous integrands.

Theorem 11. Let u : [a, b] → ℝ be a mapping of bounded variation on [a, b] and f : [a, b] → ℝ be absolutely continuous on [a, b]. Then

|GS(f; u)| ⩽ \frac{1}{12}K(b − a)^{2}∥f′∥_{∞,[a,b]} .

Proof. As u is K-Lipschitzian on [a, b] and f is continuous, by (3.1) we have

|GS(f; u)| ⩽ K ∫_{a}^{(a+b)/2} | \frac{f(x) + f(a + b − x)}{2} − \frac{1}{b − a} ∫_{a}^{b}f(t) dt|dx

Using the companion of Ostrowski’s type inequality (1.3), we may state that

∫_{a}^{(a+b)/2} | \frac{f(x) + f(a + b − x)}{2} − \frac{1}{b − a} ∫_{a}^{b}f(t) dt|dx

⩽ (b−a)∥f′∥_{∞,[a,b]} ∫_{a}^{(a+b)/2} [ \frac{1}{8} + 2( \frac{x − (3a + b)/4}{b − a} )^{2}] dx ⩽ \frac{1}{12}(b−a)^{2}∥f′∥_{∞,[a,b]} .

It follows that

|GS(f; u)| ⩽ K ∫_{a}^{(a+b)/2} | \frac{f(x) + f(a + b − x)}{2} − \frac{1}{b − a} ∫_{a}^{b}f(t) dt|dx

⩽ \frac{1}{12}K(b − a)^{2}∥f′∥_{∞,[a,b]} ,

4. A Numerical quadrature formula for the Riemann–Stieltjes integral

In this section, we use the results from the previous sections to approximate the Riemann–Stieltjes integral ∫_{a}^{(a+b)/2} [ \frac{f(x)+f(a+b−x)}{2} ] du(x), in terms of the Riemann

integral ∫_{a}^{b}f(t) dt.

Theorem 12. Let f, u be as in Theorem 6 and consider

I_{h} := {a = x_{0} < x_{1} < ⋯< x_{n−1} < x_{n} = b} ,

be a partition of [a, b]. Denote h_{i} = x_{i+1} − x_{i}, i = 1, 2, …n − 1. Then we have

(4.1)  ∫_{a}^{(a+b)/2} \frac{f(x) + f(a + b − x)}{2} du(x) = A_{n}(f, u, I_{h}) + R_{n}(f, u, I_{h})

where,

(4.2)  A_{n}(f, u, I_{h}) = ∑_{i=0}^{n−1} \frac{u ((x_{i+1} + x_{i})/2) − u(x_{i})}{h_{i}} × ∫_{x_{i}}^{(x_{i+1} +x_{i} )/2} f(t) dt

and the remainder R_{n}(f, u, I_{h}) satisfies the estimation

|R_{n}(f, u, I_{h})| ⩽ \frac{1}{2} · max_{i=\overline{0,n−1}} \{ ⋁_{x_{i}}^{x_{i+1}} (f)\} ⋁_{a}^{(a+b)/2} (u).

Proof. Applying Theorem 6 on the intervals [x_{i}, x_{i+1} ], i = 1, 2, ⋯ n − 1, we get

| ∫_{x_{i}}^{(x_{i+1} +x_{i} )/2} \frac{f(x) + f(a + b − x)}{2} du(x)

− \frac{u ((x_{i+1} + x_{i})/2) − u(x_{i})}{h_{i}} ∫_{x_{i}}^{(x_{i+1} +x_{i} )/2} f(t) dt| ⩽ \frac{1}{2} ⋁_{x_{i}}^{x_{i+1}} (f) ⋁_{x_{i}}^{(x_{i+1} +x_{i} )/2} (u).

Summing the above inequality over i from 0 to n − 1 and using the generalized triangle inequality, we deduce that

|∫_{a}^{(a+b)/2} \frac{f(x) + f(a + b − x)}{2} du(x) − A_{n}(f, u, I_{h}) | ⩽ \frac{1}{2} ∑_{i=0}^{n−1} [ ⋁_{x_{i}}^{x_{i+1}} (f) ⋁_{x_{i}}^{(x_{i+1} +x_{i} )/2} (u)]

= \frac{1}{2} · max_{i=\overline{0,n−1}} \{ ⋁_{x_{i}}^{x_{i+1}} (f)\} ∑_{i=0}^{n−1} ⋁_{x_{i}}^{(x_{i+1} +x_{i} )/2} (u)

= \frac{1}{2} · max_{i=\overline{0,n−1}} \{ ⋁_{x_{i}}^{x_{i+1}} (f)\} ⋁_{a}^{(a+b)/2} (u),

Theorem 13. Let f, u be as in Theorem 10 and I_{h} as above. Then

∫_{a}^{(a+b)/2} \frac{f(x) + f(a + b − x)}{2} du(x) = A_{n}(f, u, I_{h}) + R_{n}(f, u, I_{h})

where, A_{n}(f, u, I_{h}) is defined in (4.2) and the remainder R_{n}(f, u, I_{h}) satisfies the estimation

|R_{n}(f, u, I_{h})| ⩽ \frac{KH}{2^{r}(r + 1)(r + 2)} [ν(h)]^{r} (b − a)

where, ν(h) = max_{i=\overline{0,n−1}} \{h_{i}\}.

Proof. Applying Theorem 10 on the intervals [x_{i}, x_{i+1} ], i = 1, 2, …n − 1, we get

| ∫_{x_{i}}^{(x_{i+1} +x_{i} )/2} \frac{f(x) + f(a + b − x)}{2} du(x)

− \frac{u ((x_{i+1} + x_{i})/2) − u(x_{i})}{h_{i}} ∫_{x_{i}}^{(x_{i+1} +x_{i} )/2} f(t) dt| ⩽ \frac{KH}{2^{r}(r + 1)(r + 2)} h_{i}^{r+1} .

Summing the above inequality over i from 0 to n − 1 and using the generalized triangle inequality, we deduce that

| ∫_{a}^{(a+b)/2} \frac{f(x) + f(a + b − x)}{2} du(x) − A_{n}(f, u, I_{h})| ⩽ \frac{KH}{2^{r}(r + 1)(r + 2)} ∑_{i=0}^{n−1} h_{i}^{r+1}

⩽ \frac{KH}{2^{r}(r + 1)(r + 2)} [ max_{i=\overline{0,n−1}} \{h_{i}\}]^{r} ∑_{i=0}^{n−1} h_{i}

⩽ \frac{KH}{2^{r}(r + 1)(r + 2)} [ν(h)]^{r} (b − a),

and the theorem is proved. □

Remark 6. In order to approximate the Riemann–Stieltjes integral (4.1), one may state several interesting error estimations for the remainder R_{n}(f, u, I_{h}) under various assumptions using the inequalities from Sections 2 and 3. We omit the details.

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Department of Mathematics (Received 21 02 2012) Faculty of Science, Jerash University 26150 Jerash, Jordan

 mwomath@gmail.com