Open Access

Sharp error bounds in approximating the Riemann-Stieltjes integral by a generalised trapezoid formula and applications

Journal of Inequalities and Applications20132013:53

DOI: 10.1186/1029-242X-2013-53

Received: 18 July 2012

Accepted: 22 January 2013

Published: 18 February 2013

Abstract

Sharp error bounds in approximating the Riemann-Stieltjes integral a b f ( t ) d u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq1_HTML.gif with the generalised trapezoid formula f ( b ) [ u ( b ) 1 b a a b u ( s ) d s ] + f ( a ) [ 1 b a a b u ( s ) d s u ( a ) ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq2_HTML.gif are given for various pairs ( f , u ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq3_HTML.gif of functions. Applications for weighted integrals are also provided.

MSC: 26D15, 26D10, 41A55.

Keywords

Riemann-Stieltjes integral trapezoid rule integral inequalities weighted integrals

1 Introduction

In [1], in order to approximate the Riemann-Stieltjes integral a b f ( t ) d u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq4_HTML.gif by the generalised trapezoid formula
[ u ( b ) u ( x ) ] f ( b ) + [ u ( x ) u ( a ) ] f ( a ) , x [ a , b ] , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equ1_HTML.gif
(1.1)
the authors considered the error functional
T ( f , u ; a , b ; x ) : = a b f ( t ) d u ( t ) [ u ( b ) u ( x ) ] f ( b ) [ u ( x ) u ( a ) ] f ( a ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equ2_HTML.gif
(1.2)
and proved that
| T ( f , u ; a , b ; x ) | H [ 1 2 ( b a ) + | x a + b 2 | ] r a b ( f ) , x [ a , b ] , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equ3_HTML.gif
(1.3)

provided that f : [ a , b ] R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq5_HTML.gif is of bounded variation on [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq6_HTML.gif and u is of r-H-Hölder type, that is, u : [ a , b ] R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq7_HTML.gif satisfies the condition | u ( t ) u ( s ) | H | t s | r https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq8_HTML.gif for any t , s [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq9_HTML.gif, where r ( 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq10_HTML.gif and H > 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq11_HTML.gif are given.

The dual case, namely, when f is of q-K-Hölder type and u is of bounded variation, has been considered by the authors in [2] in which they obtained the bound:
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equ4_HTML.gif
(1.4)

for any x [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq12_HTML.gif.

The case where f is monotonic and u is of r-H-Hölder type, which provides a refinement for (1.3), and respectively the case where u is monotonic and f of q-K-Hölder type were considered by Cheung and Dragomir in [3], while the case where one function was of Hölder type and the other was Lipschitzian was considered in [4]. For other recent results in estimating the error T ( f , u ; a , b , x ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq13_HTML.gif for absolutely continuous integrands f and integrators u of bounded variation, see [5] and [6].

The main aim of the present paper is to investigate the error bounds in approximating the Stieltjes integral by a different generalised trapezoid rule than the one from (1.1) in which the value u ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq14_HTML.gif, x [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq12_HTML.gif is replaced with the integral mean 1 b a a b u ( s ) d s https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq15_HTML.gif. Applications in approximating the weighted integrals a b h ( t ) f ( t ) d t https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq16_HTML.gif are also provided.

2 Representation results

We consider the following error functional T g ( f ; u ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq17_HTML.gif in approximating the Riemann-Stieltjes integral a b f ( t ) d u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq18_HTML.gif by the generalised trapezoid formula:
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equ5_HTML.gif
(2.1)
If we consider the associated functions Φ f https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq19_HTML.gif, Γ f https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq20_HTML.gif and Δ f https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq21_HTML.gif defined by
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equa_HTML.gif
and
Δ f ( t ) : = f ( b ) f ( t ) b t f ( t ) f ( a ) t a , t ( a , b ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equb_HTML.gif
then we observe that
Φ f ( t ) = 1 b a Γ f ( t ) = ( b t ) ( t a ) b a Δ f ( t ) for any  t ( a , b ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equ6_HTML.gif
(2.2)

The following representation result can be stated.

Theorem 1 Let f , u : [ a , b ] R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq22_HTML.gif be bounded on [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq6_HTML.gif and such that the Riemann-Stieltjes integral a b f ( t ) d u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq4_HTML.gif and the Riemann integral a b u ( t ) d t https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq23_HTML.gif exist. Then we have the identities

T g ( f ; u ) = a b Φ f ( t ) d u ( t ) = 1 b a a b Γ f ( t ) d u ( t ) = 1 b a a b ( b t ) ( t a ) Δ f ( t ) d u ( t ) = D ( u ; f ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equ7_HTML.gif
(2.3)
where
D ( u ; f ) = a b u ( t ) d f ( t ) [ f ( b ) f ( a ) ] 1 b a a b u ( t ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equ8_HTML.gif
(2.4)

Proof

Integrating the Riemann-Stieltjes integral by parts, we have
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equc_HTML.gif

and the first equality in (2.3) is proved.

The second and third identity is obvious by the relation (2.2).

For the last equality, we use the fact that for any g , h : [ a , b ] R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq24_HTML.gif bounded functions for which the Riemann-Stieltjes integral a b h ( t ) d g ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq25_HTML.gif and the Riemann integral a b g ( t ) d t https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq26_HTML.gif exist, we have the representation (see, for instance, [7])
D ( g ; h ) = a b Φ h ( t ) d g ( t ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equ9_HTML.gif
(2.5)

The proof is now complete. □

In the case where u is an integral, the following identity can be stated.

Corollary 1 Let p , h : [ a , b ] R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq27_HTML.gif be continuous on [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq6_HTML.gif and f : [ a , b ] R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq5_HTML.gif be Riemann integrable. Then we have the identity
T g ( f ; a p h ) = 1 b a [ f ( b ) a b ( t a ) p ( t ) h ( t ) d t + f ( a ) a b ( b t ) p ( t ) h ( t ) d t ] a b p ( t ) f ( t ) h ( t ) d t = a b Φ f ( t ) p ( t ) h ( t ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equ10_HTML.gif
(2.6)

Proof Since p and h are continuous, the function u ( t ) = a t p ( s ) h ( s ) d s https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq28_HTML.gif is differentiable and u ( t ) = p ( t ) h ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq29_HTML.gif for each t ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq30_HTML.gif.

Integrating by parts, we have
a b u ( t ) d t = ( a t p ( s ) h ( s ) d s ) t | a b a b t p ( t ) h ( t ) d t = b a b p ( s ) h ( s ) d s a b t p ( t ) h ( t ) d t = a b ( b t ) p ( t ) h ( t ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equd_HTML.gif
Since
u ( b ) 1 b a a b u ( t ) d t = a b p ( t ) h ( t ) d t 1 b a a b ( b t ) p ( t ) h ( t ) d t = 1 b a a b ( t a ) p ( t ) h ( t ) d t , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Eque_HTML.gif

then, by the definition of T g https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq31_HTML.gif in (2.1), we deduce the first part of (2.6).

The second part of (2.6) follows by (2.3). □

Remark 1 In the particular case p ( t ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq32_HTML.gif, t [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq33_HTML.gif, we have the equality
T g ( f ; a h ) = 1 b a [ f ( b ) a b ( t a ) h ( t ) d t + f ( a ) a b ( b t ) h ( t ) d t ] a b f ( t ) h ( t ) d t = a b Φ f ( t ) h ( t ) d t = 1 b a a b Γ f ( t ) h ( t ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equ11_HTML.gif
(2.7)

3 Some inequalities for f-convex

The following result concerning the nonnegativity of the error functional T g ( ; ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq34_HTML.gif can be stated.

Theorem 2 If u is monotonic nonincreasing and f : [ a , b ] R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq35_HTML.gif is such that the Riemann-Stieltjes integral a b f ( t ) d u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq1_HTML.gif exists and
f ( b ) f ( t ) b t f ( t ) f ( a ) t a for any t ( a , b ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equ12_HTML.gif
(3.1)
then T g ( f ; u ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq36_HTML.gif or, equivalently,
f ( b ) [ u ( b ) 1 b a a b u ( t ) d t ] + f ( a ) [ 1 b a a b u ( t ) d t u ( a ) ] a b f ( t ) d u ( t ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equ13_HTML.gif
(3.2)

A sufficient condition for (3.1) to hold is that f is convex on [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq6_HTML.gif.

Proof The condition (3.1) is equivalent with the fact that Δ f ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq37_HTML.gif for any t ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq30_HTML.gif and then, by the equality
T g ( f ; u ) = 1 b a a b ( b t ) ( t a ) Δ f ( t ) d u ( t ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equf_HTML.gif

we deduce that T g ( f ; u ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq36_HTML.gif.

If f is convex, then
t a b a f ( b ) + b t b a f ( a ) f [ ( t a b a ) b + ( b t b a ) a ] = f ( t ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equg_HTML.gif

which shows that Φ f ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq38_HTML.gif, namely, the condition (3.1) is satisfied. □

Corollary 2 Let p , h : [ a , b ] R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq27_HTML.gif be continuous on [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq6_HTML.gif and f : [ a , b ] R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq5_HTML.gif be Riemann integrable. If p ( t ) h ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq39_HTML.gif for any t [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq40_HTML.gif and f satisfies (3.1) or, sufficiently, f is convex on [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq6_HTML.gif, then
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equ14_HTML.gif
(3.3)

We are now able to provide some new results.

Theorem 3 Assume that p and h are continuous and synchronous (asynchronous) on ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq41_HTML.gif, i.e.,
( p ( t ) p ( s ) ) ( h ( t ) h ( s ) ) ( ) 0 for any t , s [ a , b ] . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equ15_HTML.gif
(3.4)

If f satisfies (3.1) and is Riemann integrable on [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq42_HTML.gif (or sufficiently, f is convex on [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq6_HTML.gif), then

T g ( f ; a p ) T g ( f ; a h ) ( ) T g ( f ; a 1 ) T g ( f ; a p h ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equ16_HTML.gif
(3.5)
where
T g ( f ; a 1 ) = f ( a ) + f ( b ) 2 ( b a ) a b f ( t ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equ17_HTML.gif
(3.6)

Proof

We use the Čebyšev inequality
a b α ( t ) d t a b α ( t ) p ( t ) h ( t ) d t ( ) a b α ( t ) p ( t ) d t a b α ( t ) h ( t ) d t , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equ18_HTML.gif
(3.7)

which holds for synchronous (asynchronous) functions p, h and nonnegative α for which the involved integrals exist.

Now, on applying the Čebyšev inequality (3.7) for α ( t ) = Φ f ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq43_HTML.gif and utilising the representation result (2.6), we deduce the desired inequality (3.5). □

We also have the following theorem.

Theorem 4 Assume that f : [ a , b ] R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq5_HTML.gif is Riemann integrable and satisfies (3.1) (or sufficiently, f is concave on [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq6_HTML.gif). Then, for p , h : [ a , b ] R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq44_HTML.gif continuous, we have
| T g ( f ; a p h ) | sup t [ a , b ] | h ( t ) | T g ( f ; a | p | ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equ19_HTML.gif
(3.8)
and
| T g ( f ; a p h ) | [ T g ( f ; a | p | α ) ] 1 α [ T g ( f ; a | h | β ) ] 1 β , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equ20_HTML.gif
(3.9)
where α > 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq45_HTML.gif, 1 α + 1 β = 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq46_HTML.gif. In particular, we have
| T g ( f ; a p h ) | 2 T g ( f ; a | p | 2 ) T g ( f ; a | h | 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equ21_HTML.gif
(3.10)

Proof

Observe that
| T g ( f ; a p h ) | = | a b Φ f ( t ) p ( t ) h ( t ) d t | a b | Φ f ( t ) p ( t ) h ( t ) | d t = a b Φ f ( t ) | p ( t ) | | h ( t ) | d t sup t [ a , b ] | h ( t ) | a b Φ f ( t ) | p ( t ) | d t = sup t [ a , b ] | h ( t ) | T g ( f ; a | p | ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equh_HTML.gif

and the inequality (3.8) is proved.

Further, by the Hölder inequality, we also have
| T g ( f ; a p h ) | a b Φ f ( t ) | p ( t ) | | h ( t ) | d t ( a b Φ f ( t ) | p ( t ) | α d t ) 1 α ( a b Φ f ( t ) | h ( t ) | β d t ) 1 β = [ T g ( f ; a | p | α ) ] 1 α [ T g ( f ; a | h | β ) ] 1 β https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equi_HTML.gif

for α > 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq45_HTML.gif, 1 α + 1 β = 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq46_HTML.gif, and the theorem is proved. □

Remark 2 The above result can be useful for providing some error estimates in approximating the weighted integral a b h ( t ) f ( t ) d t https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq47_HTML.gif by the generalised trapezoid rule
1 b a [ f ( b ) a b ( t a ) h ( t ) d t + f ( a ) a b ( b t ) h ( t ) d t ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equj_HTML.gif
as follows:
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equ22_HTML.gif
(3.11)

provided f satisfies (3.1) and is Riemann integrable (or sufficiently, convex on [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq6_HTML.gif), which is continuous on [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq48_HTML.gif.

If h ( t ) = | w ( t ) | 1 β https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq49_HTML.gif, t [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq40_HTML.gif, then for some f, we also have
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equ23_HTML.gif
(3.12)

with α > 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq45_HTML.gif, 1 α + 1 β = 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq46_HTML.gif.

Finally, we can state the following Jensen type inequality for the error functional T g ( f ; a b h ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq50_HTML.gif.

Theorem 5 Assume f : [ a , b ] R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq5_HTML.gif is Riemann integrable and satisfies (3.1) (or sufficiently, f is convex on [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq6_HTML.gif), while h : [ a , b ] R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq51_HTML.gif is continuous. If F : R R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq52_HTML.gif is convex (concave), then
F ( T g ( f ; a b h ) T g ( f ; a b 1 ) ) ( ) T g ( f ; a b F h ) T g ( f ; a b 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equ24_HTML.gif
(3.13)

Proof

By the use of Jensen’s integral inequality, we have
F ( a b Φ f ( t ) h ( t ) d t a b Φ f ( t ) d t ) ( ) a b Φ f ( t ) F ( h ( t ) ) d t a b Φ f ( t ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equ25_HTML.gif
(3.14)
Since, by the identity (2.6), we have
a b Φ f ( t ) F ( h ( t ) ) d t = T g ( f ; a b F h ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equk_HTML.gif

then (3.14) is equivalent with the desired result (3.13). □

4 Sharp bounds via Grüss type inequalities

Due to the identity (2.3), in which the error bound T g ( f ; u ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq53_HTML.gif can be represented as D ( u ; f ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq54_HTML.gif, where
D ( u ; f ) = a b u ( t ) d f ( t ) [ f ( b ) f ( a ) ] 1 b a a b u ( t ) d t , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equl_HTML.gif

is a Grüss type functional introduced in [8], any sharp bound for D ( u ; f ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq54_HTML.gif will be a sharp bound for T g ( f ; u ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq17_HTML.gif.

We can state the following result.

Theorem 6 Let f , u : [ a , b ] R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq22_HTML.gif be bounded functions on [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq6_HTML.gif.

(i) If there exist constants n, N such that n u ( t ) N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq55_HTML.gif for any t [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq40_HTML.gif, u is Riemann integrable and f is K-Lipschitzian ( K > 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq56_HTML.gif), then
| T g ( f ; u ) | 1 2 K ( N n ) ( b a ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equ26_HTML.gif
(4.1)

The constant 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq57_HTML.gif is best possible in (4.1).

(ii) If f is of bounded variation and u is S-Lipschitzian ( S > 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq58_HTML.gif), then
| T g ( f ; u ) | 1 2 S ( b a ) a b ( f ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equ27_HTML.gif
(4.2)

The constant 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq57_HTML.gif is best possible in (4.2)

(iii) If f is monotonic nondecreasing and u is S-Lipschitzian, then
| T g ( f ; u ) | 1 2 S ( b a ) [ f ( b ) f ( a ) P ( f ) ] 1 2 S ( b a ) [ f ( b ) f ( a ) ] , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equ28_HTML.gif
(4.3)
where
P ( f ) = 4 ( b a ) 2 a b ( t a + b 2 ) f ( t ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equm_HTML.gif

The constant 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq57_HTML.gif is best possible in both inequalities.

(iv) If f is monotonic nondecreasing and u is of bounded variation and such that the Riemann-Stieltjes integral a b f ( t ) d u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq59_HTML.gif exists, then
| T g ( f ; u ) | [ f ( b ) f ( a ) Q ( f ) ] a b ( u ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equ29_HTML.gif
(4.4)
where
Q ( f ) : = 1 b a a b sgn ( t a + b 2 ) f ( t ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equn_HTML.gif

The inequality (4.4) is sharp.

(v) If f is continuous and convex on [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq6_HTML.gif and u is of bounded variation on [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq6_HTML.gif, then
| T g ( f ; u ) | 1 4 [ f ( b ) f + ( a ) ] a b ( u ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equ30_HTML.gif
(4.5)

The constant 1 4 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq60_HTML.gif is sharp (if f ( b ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq61_HTML.gif and f + ( a ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq62_HTML.gif are finite).

(vi) If f : [ a , b ] R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq5_HTML.gif is continuous and convex on [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq6_HTML.gif and u is monotonic nondecreasing on [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq6_HTML.gif, then
0 T g ( f ; u ) 2 f ( b ) f + ( a ) b a a b ( t a + b 2 ) u ( t ) d t { 1 2 [ f ( b ) f + ( a ) ] max { | u ( a ) | , | u ( b ) | } ( b a ) ; 1 ( q + 1 ) 1 / q [ f ( b ) f + ( a ) ] u p ( b a ) 1 / q if  p > 1 , 1 p + 1 q = 1 ; [ f ( b ) f + ( a ) ] u 1 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equ31_HTML.gif
(4.6)

The constants 2 and 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq57_HTML.gif are best possible in (4.6) (if f ( b ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq61_HTML.gif and f + ( a ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq62_HTML.gif are finite).

Proof The inequality (4.1) follows from the inequality (2.5) in [8] applied to D ( u ; f ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq54_HTML.gif, while (4.2) comes from (1.3) of [9]. The inequalities (4.3) and (4.4) follow from [7], while (4.5) and (4.6) are valid via the inequalities (2.8) and (2.1) from [10] applied to the functional D ( u ; f ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq63_HTML.gif. The details are omitted. □

If we consider the error functional in approximating the weighted integral a b h ( t ) f ( t ) d t https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq16_HTML.gif by the generalised trapezoid formula,
1 b a [ f ( b ) a b ( t a ) h ( t ) d t + f ( a ) a b ( b t ) h ( t ) d t ] , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equo_HTML.gif
namely (see also (2.7)),
E ( f ; h ) : = T g ( f ; a b h ) = 1 b a [ f ( b ) a b ( t a ) h ( t ) d t + f ( a ) a b ( b t ) h ( t ) d t ] a b h ( t ) f ( t ) d t , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equ32_HTML.gif
(4.7)

then the following corollary provides various sharp bounds for the absolute value of E ( f ; h ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq64_HTML.gif.

Corollary 3 Assume that f and u are Riemann integrable on [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq65_HTML.gif.

(i) If there exist constants γ, Γ such that γ a t h ( s ) d s Γ https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq66_HTML.gif for each t [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq33_HTML.gif, and f is K-Lipschitzian on [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq6_HTML.gif, then
| E ( f ; h ) | 1 2 K ( Γ γ ) ( b a ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equ33_HTML.gif
(4.8)

The constant 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq57_HTML.gif is best possible in (4.8).

(ii) If f is of bounded variation and | h ( t ) | M https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq67_HTML.gif for each t [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq40_HTML.gif, then
| E ( f ; h ) | 1 2 M ( b a ) a b ( f ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equ34_HTML.gif
(4.9)

The constant 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq57_HTML.gif is best possible in (4.9).

(iii) If f is monotonic nondecreasing and | h ( t ) | M https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq68_HTML.gif, t [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq40_HTML.gif, then
| E ( f ; h ) | 1 2 M ( b a ) [ f ( b ) f ( a ) P ( f ) ] 1 2 M ( b a ) [ f ( b ) f ( a ) ] , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equ35_HTML.gif
(4.10)

where P ( f ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq69_HTML.gif is defined in Theorem  6. The constant 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq57_HTML.gif is sharp in both inequalities.

(iv) If f is monotonic nondecreasing and a b | h ( t ) | d t < https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq70_HTML.gif, then
| E ( f ; h ) | [ f ( b ) f ( a ) Q ( f ) ] a b | h ( t ) | d t , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equ36_HTML.gif
(4.11)

where Q ( f ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq71_HTML.gif is defined in Theorem  6. The inequality (4.11) is sharp.

(v) If f is continuous and convex on [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq6_HTML.gif and a b | h ( t ) | d t < https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq72_HTML.gif, then
| E ( f ; h ) | 1 4 [ f ( b ) f + ( a ) ] a b | h ( t ) | d t . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equ37_HTML.gif
(4.12)

The constant 1 4 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq60_HTML.gif is sharp (if f ( b ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq61_HTML.gif and f + ( a ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq62_HTML.gif are finite).

(vi) If f : [ a , b ] R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq5_HTML.gif is continuous and convex on [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq6_HTML.gif and h ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq73_HTML.gif for t [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq40_HTML.gif, then
0 E ( f ; h ) f ( b ) f + ( a ) b a a b ( b t ) ( t a ) h ( t ) d t { 1 2 [ f ( b ) f + ( a ) ] ( b a ) a b h ( t ) d t ; 1 ( q + 1 ) 1 / q [ f ( b ) f + ( a ) ] [ a b ( a t h ( s ) d s ) p d t ] 1 p ( b a ) 1 / q if  p > 1 , 1 p + 1 q = 1 ; [ f ( b ) f + ( a ) ] a b ( b t ) h ( t ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equ38_HTML.gif
(4.13)

The first inequality in (4.13) is sharp (if f ( b ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq74_HTML.gif and f + ( a ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq62_HTML.gif are finite).

Proof We only prove the first inequality in (4.13).

Utilising the inequality (4.6) for u ( t ) = a t h ( s ) d s https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_IEq75_HTML.gif, we get
0 E ( f ; h ) 2 f ( b ) f + ( a ) b a a b ( t a + b 2 ) a t h ( s ) d s d t . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equ39_HTML.gif
(4.14)
However, on integrating by parts, we have
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-53/MediaObjects/13660_2012_Article_507_Equp_HTML.gif

The rest of the inequality is obvious. □

Declarations

Acknowledgements

Most of the work for this article was undertaken while the first author was at Victoria University, Melbourne Australia.

Authors’ Affiliations

(1)
Department of Mathematics and Statistics, La Trobe University
(2)
Mathematics, College of Engineering & Science, Victoria University
(3)
School of Computational and Applied Mathematics, University of the Witwatersrand

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This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.