APPROXIMATING THE RIEMANN-STIELTJES INTEGRAL VIA A ČEBYŠEV TYPE FUNCTIONAL

SEVER S. DRAGOMIR

Abstract. Some new sharp upper bounds for the absolute value of the error functional D (f, u) in approximating the Riemann-Stieltjes integral ∫_{a}^{b} f (t) du (t) by the quantity [u (b) − u (a)] · \frac{1}{b−a} ∫_{a}^{b} f (t) dt are given.

1. Introduction

In order to approximate the Riemann-Stieltjes integral ∫_{a}^{b} f (t) du (t) by the simpler quantity

[u (b) − u (a)] · \frac{1}{b − a} ∫_{a}^{b} f (t) dt,

provided that both integrals exist, Dragomir and Fedotov introduced in [8] the following error functional of Čebyšev type

(1.1)  D (f; u) = ∫_{a}^{b} f (t) du (t) − [u (b) − u (a)] · \frac{1}{b − a} ∫_{a}^{b} f (t) dt,

and pointed out the following sharp upper bound for |D (f; u)| , namely

(1.2)  |D (f; u)| ≤ \frac{1}{2}L (M − m) (b − a) ,

provided the integrator u : [a, b] → ℝ is L−Lipschitzian on [a, b] , i.e., |u (x) − u (y)| ≤ L |x − y| for any x, y ∈ [a, b] and the integrand f : [a, b] → ℝ is Riemann integrable on [a, b] and satisfies the boundedness condition

(1.3)  −∞ < m ≤ f (x) ≤ M < ∞ for a.e. x ∈ [a, b] .

The multiplicative constant \frac{1}{2} in (1.2) is best possible in the sense that it cannot be replaced by a smaller constant.

In the follow-up paper [9], the authors provided a different bound, namely

(1.4)  |D (f; u)| ≤ \frac{1}{2}K (b − a) ⋁_{a}^{b} (u) ,

provided that f is is K−Lipschitzian and u is of bounded variation on [a, b] .

Date: November 27, 2007.

1991 Mathematics Subject Classification. 26D15, 41A55.

Key words and phrases. Riemann-Stieltjes integral, Quadrature rules, Grüss type inequalities, Čebyšev functional, Error bounds.

The result (1.4) was improved in [4] for the case of monotonic nondecreasing functions. We have shown in this case that

(1.5)  |D (f; u)| ≤ \frac{1}{2}K (b − a) [u (b) − u (a) − K (u)]

(≤ \frac{1}{2}K (b − a) [u (b) − u (a)]) ,

where

K (u) := \frac{4}{(b − a)^{2}} ∫_{a}^{b} u (t) (t − \frac{a + b}{2}) dt ≥ 0.

In (1.5) the constant \frac{1}{2} is best possible in both inequalities.

For other sharp bounds on the error functional D (f; u), see the recent papers [6], [7], and [10]. For other inequalities for the Riemann-Stieltjes integral, see [1] and [2].

The main aim of this paper is to further investigate the error functional D (f; u) . Two representations are given. These are applied to obtain some inequalities for D (f; u) which improve earlier results.

Applications for the classical Čebyšev functional C (f, g) , where

(1.6) C (f, g) := \frac{1}{b − a} ∫_{a}^{b} f (t) g (t) dt − \frac{1}{b − a} ∫_{a}^{b} f (t) dt · \frac{1}{b − a} ∫_{a}^{b} g (t) dt,

and f, g are integrable and belonging to different classes of functions, are also provided.

2. Representation Results

For a function g : [a, b] → ℝ, consider the generalised trapezoid error transform Φ_{g} : [a, b] → ℝ given by

(2.1)  Φ_{g} (t) := \frac{1}{b − a} [(b − t) g (a) + (t − a) g (b)] − g (t) , t ∈ [a, b]

and if g is Lebesgue integrable, the Ostrowski transform, which is the error of approximating the function by its integral mean, defined by:

(2.2)  Θ_{g} (t) := g (t) − \frac{1}{b − a} ∫_{a}^{b} g (s) ds, t ∈ [a, b] .

We also define the kernel Q : [a, b]^{2} → ℝ,

(2.3)  Q (t, s) := \{ \begin{matrix}t − b & if a ≤ s ≤ t ≤ b, \\ t − a & if a ≤ t < s ≤ b. \end{matrix}

The following representation result in terms of Θ_{g} and Q may be stated:

Lemma 1. If f, u : [a, b] → ℝ are bounded functions and such that the Riemann-Stieltjes integral ∫_{a}^{b} f (t) du (t) and the Riemann integral ∫_{a}^{b} f (t) dt exist, then we have the representation:

(2.4)  D (f; u) = ∫_{a}^{b} Θ_{f} (s) du (s) = \frac{1}{b − a} ∫_{a}^{b} (∫_{a}^{b} Q (t, s) df (t)) du (s) .

Proof. We have by the definition of Q and integrating by parts in the Riemann-Stieltjes integral that

\frac{1}{b − a} ∫_{a}^{b} (∫_{a}^{b} Q (t, s) df (t)) du (s)

= \frac{1}{b − a} ∫_{a}^{b} [∫_{a}^{s} (t − a) df (t) + ∫_{s}^{b} (t − b) df (t)] du (s)

= \frac{1}{b − a} ∫_{a}^{b} [f (t) (t − a) |_{a}^{s} − ∫_{a}^{s} f (t) dt + (t − b) f (t) |_{s}^{b} − ∫_{s}^{b} f (t) dt] du (s)

= \frac{1}{b − a} ∫_{a}^{b} [f (s) (s − a) − ∫_{a}^{s} f (t) dt + (b − s) f (s) − ∫_{s}^{b} f (t) dt] du (s)

= ∫_{a}^{b} Θ_{f} (s) du (s)

and the second inequality is proved.

The first identity is obvious by the definition of D (f; u) . □

The following corollary can be stated about the representation of the Čebyšev functional C (f, g) defined in (1.6).

Corollary 1. Assume that f, g : [a, b] → ℝ are Riemann integrable on [a, b] , then

(2.5)  C (f, g) = \frac{1}{b − a} ∫_{a}^{b} Θ_{f} (s) g (s) ds

= \frac{1}{(b − a)^{2}} ∫_{a}^{b} (∫_{a}^{b} Q (t, s) df (t)) g (s) ds.

Proof. It is well known (see for instance [3, Theorem 7.33, p. 162] that if g is Riemann integrable and u (t) = ∫_{a}^{t} g (s) ds , then for any f a Riemann integrable function we have that the Riemann-Stieltjes integral ∫_{a}^{b} f (t) du (t) exists and ∫_{a}^{b} f (t) du (t) = ∫_{a}^{b} f (t) g (t) dt. Therefore, we have D (f; u) = (b − a) C (f, g) and

∫_{a}^{b} (∫_{a}^{b} Q (t, s) df (t)) du (s) = ∫_{a}^{b} (∫_{a}^{b} Q (t, s) df (t)) g (s) ds.
□

The second representation of D (f; u) is incorporated in

Lemma 2. With the assumptions in Lemma 1, we have

(2.6)  D (f; u) = ∫_{a}^{b} Φ_{u} (t) df (t) = \frac{1}{b − a} ∫_{a}^{b} (∫_{a}^{b} Q (t, s) du (s)) df (t) ,

where Q is defined by (2.3).

Proof. By the Fubini type theorem for the Riemann-Stieltjes integral (see for instance [3, Theorem 7.41, p. 167]) we have that

∫_{a}^{b} (∫_{a}^{b} Q (t, s) du (s)) df (t) = ∫_{a}^{b} (∫_{a}^{b} Q (t, s) df (t)) du (s) ,

and the equality between the first and the last term in (2.6) is proved. Now, observe that

∫_{a}^{b} Q (t, s) du (s) = ∫_{a}^{t} (t − b) du (s) + ∫_{t}^{b} (t − a) du (s)
= (t − b) [u (t) − u (a)] + (t − a) [u (b) − u (t)]
= (b − a) Φ_{u} (t) ,

for any t ∈ [a, b] , and then integrating over f (t) , we deduce the second inequality in (2.6). □

Corollary 2. Assume that f and g are Riemann integrable on [a, b] , then we have

(2.7) C (f, g) = \frac{1}{b − a} ∫_{a}^{b} Φ̃_{g} (t) df (t) = \frac{1}{(b − a)^{2}} ∫_{a}^{b} (∫_{a}^{b} Q (t, s) g (s) ds) df (t) ,

where

(2.8)  Φ̃_{g} (t) = Φ_{∫_{a}^{·} g} (t) = \frac{t − a}{b − a} ∫_{a}^{b} g (s) ds − ∫_{a}^{t} g (s) ds, t ∈ [a, b] .

3. Bounds in the Case when u is of Bounded Variation

The following lemma is of interest in itself.

Lemma 3. If p : [a, b] → ℝ is continuous on [a, b] and v : [a, b] → ℝ is of bounded variation on [a, b] , then

(3.1)  |∫_{a}^{b} p (t) dv (t)| ≤ ∫_{a}^{b} |p (t)| d ⋁_{a}^{t} (v)

≤ [⋁_{a}^{b} (v)]^{\frac{1}{q}} \{∫_{a}^{b} |p (t)|^{p} d [⋁_{a}^{t} (v)]\}^{\frac{1}{p}}

≤ max_{t∈[a,b]} |p (t)| ⋁_{a}^{b} (v) ,

where p > 1, \frac{1}{p} + \frac{1}{q} = 1.

Proof. Since the Stieltjes integral ∫_{a}^{b} p (t) dv (t) exists, then for any division I_{n} : a = t_{0} < t_{1} < ⋯ < t_{n−1} < t_{n} = b with the norm v (I_{n}) := max_{i∈{0,...,n−1}} (t_{i+1} − t_{i}) → 0 and for any intermediate points ξ_{i} ∈ [t_{i}, t_{i+1}] , i ∈ {0, …, n − 1} we have

(3.2)  |∫_{a}^{b} p (t) dv (t)| = | lim_{v(I_{n})→0} ∑_{i=0}^{n−1} p (ξ_{i}) [v (t_{i+1}) − v (t_{i})]|

≤ lim_{v(I_{n})→0} ∑_{i=0}^{n−1} |p (ξ_{i})| |v (t_{i+1}) − v (t_{i})| .

However,

(3.3)  |v (t_{i+1}) − v (t_{i})| ≤ ⋁_{t_{i}}^{t_{i+1}} (v) = ⋁_{a}^{t_{i+1}} (v) − ⋁_{a}^{t_{i}} (v) ,

RIEMANN-STIELTJES INTEGRAL VIA A ČEBYŠEV TYPE FUNCTIONAL 5

for any i ∈ {0, …, n − 1} and by (3.3) we have

|∫_{a}^{b} p (t) dv (t)| ≤ lim_{v(I_{n})→0} ∑_{i=0}^{n−1} |p (ξ_{i})| [ ⋁_{a}^{t_{i+1}} (v) − ⋁_{a}^{t_{i}} (v)]

= ∫_{a}^{b} |p (t)| d [⋁_{a}^{t} (v)] ,

and the last Riemann-Stieltjes integral exists since |p| is continuous and ⋁_{a}^{·} (v) is monotonic nondecreasing.

The last part follows from the following Hölder type inequality

(3.4)  |∫_{a}^{b} g (t) dv (t)| ≤ [v (b) − v (a)]^{\frac{1}{q}} [∫_{a}^{b} |g (t)|^{p} dv (t)]^{\frac{1}{p}} ,

where p > 1, \frac{1}{p} + \frac{1}{q} = 1, that holds for any continuous function g : [a, b] → ℝ and any monotonic nondecreasing function v : [a, b] → ℝ. The details are omitted. □

The following result holds. Theorem 1. Assume that f, u : [a, b] → ℝ are of bounded variation and such that the Riemann-Stieltjes integral ∫_{a}^{b} f (t) du (t) exists. Then

(3.5) |D (f; u)| ≤ \frac{1}{b − a} [∫_{a}^{b} ⋁_{a}^{s} (f) (2s − a − b) d (⋁_{a}^{s} (u))

+2 ∫_{a}^{b} (⋁_{a}^{s} (u) · ⋁_{a}^{s} (f)) ds − ⋁_{a}^{b} (u) ∫_{a}^{b} (⋁_{a}^{s} (f)) ds]

≤ \frac{1}{b − a} ∫_{a}^{b} ⋁_{a}^{s} (f) (2s − a − b) d (⋁_{a}^{s} (u))

+ \frac{1}{b − a} ∫_{a}^{b} (⋁_{a}^{s} (u) · ⋁_{a}^{s} (f)) ds

≤ \frac{1}{b − a} ∫_{a}^{b} ⋁_{a}^{s} (f) (2s − a − b) d (⋁_{a}^{s} (u)) + ⋁_{a}^{b} (u) · ⋁_{a}^{b} (f) .

Proof. Utilising the identity (2.4) and the first inequality in (3.1) we have:

(3.6) |D (f; u)| ≤ \frac{1}{b − a} ∫_{a}^{b} |∫_{a}^{b} Q (t, s) df (t)| d (⋁_{a}^{s} (u))

= \frac{1}{b − a} ∫_{a}^{b} |∫_{a}^{s} (t − a) df (t) + ∫_{s}^{b} (t − b) df (t)| d (⋁_{a}^{s} (u))

≤ \frac{1}{b − a} ∫_{a}^{b} [|∫_{a}^{s} (t − a) df (t)| + |∫_{s}^{b} (t − b) df (t)|] d (⋁_{a}^{s} (u))

=: I.

Since f is of bounded variation, then by the same inequality in (3.1) we have

|∫_{a}^{s} (t − a) df (t)| ≤ ∫_{a}^{s} (t − a) d (⋁_{a}^{t} (f))

= ⋁_{a}^{s} (f) · (s − a) − ∫_{a}^{s} (⋁_{a}^{t} (f)) dt

and

|∫_{s}^{b} (t − b) df (t)| ≤ ∫_{s}^{b} (t − b) d ⋁_{s}^{t} (f) = ∫_{s}^{b} (⋁_{s}^{t} (f)) dt

= ∫_{s}^{b} [⋁_{a}^{t} (f) − ⋁_{a}^{s} (f)] dt

= ∫_{s}^{b} (⋁_{a}^{t} (f)) dt − (b − s) ⋁_{a}^{s} (f)

which gives that

I ≤ \frac{1}{b − a} ∫_{a}^{b} [⋁_{a}^{s} (f) (s − a) − ∫_{a}^{s} (⋁_{a}^{t} (f)) dt (3.7)

+ ∫_{s}^{b} (⋁_{a}^{t} (f)) dt − (b − s) ⋁_{a}^{s} (f)] d (⋁_{a}^{s} (u))

= \frac{1}{b − a} ∫_{a}^{b} [⋁_{a}^{s} (f) (2s − a − b) − ∫_{a}^{s} (⋁_{a}^{t} (f)) dt

+ ∫_{s}^{b} (⋁_{a}^{t} (f)) dt] d (⋁_{a}^{s} (u))

= \frac{1}{b − a} ∫_{a}^{b} (2s − a − b) ⋁_{a}^{s} (f) d (⋁_{a}^{s} (u))

+ \frac{1}{b − a} ∫_{a}^{b} [∫_{a}^{b} (⋁_{a}^{t} (f)) dt − 2 ∫_{a}^{s} (⋁_{a}^{t} (f)) dt] d (⋁_{a}^{s} (u))

= \frac{1}{b − a} ∫_{a}^{b} (2s − a − b) ⋁_{a}^{s} (f) d (⋁_{a}^{s} (u)) + \frac{1}{b − a} ∫_{a}^{b} (⋁_{a}^{t} (f)) dt · ⋁_{a}^{b} (u)

− \frac{2}{b − a} ∫_{a}^{b} (∫_{a}^{s} (⋁_{a}^{t} (f)) dt) d (⋁_{a}^{s} (u)) .

However, integrating by parts in the Riemann-Stieltjes integral we have

∫_{a}^{b} (∫_{a}^{s} (⋁_{a}^{t} (f)) dt) d (⋁_{a}^{s} (u))

= ∫_{a}^{s} (⋁_{a}^{t} (f)) dt · ⋁_{a}^{s} (u)|_{a}^{b} − ∫_{a}^{b} ⋁_{a}^{s} (u) · ⋁_{a}^{s} (f) ds

= ⋁_{a}^{b} (u) · ∫_{a}^{b} (⋁_{a}^{t} (f)) dt − ∫_{a}^{b} ⋁_{a}^{s} (u) · ⋁_{a}^{s} (f) ds

Inserting this value in the expression of I from (3.7) we deduce the first inequality in (3.5).

The other inequalities are obvious. □

The following result may be stated as well.

Theorem 2. If u : [a, b] → ℝ is of bounded variation and f : [a, b] → ℝ is L−Lipschitzian, then

(3.8) |D (f; u)| ≤ L [\frac{1}{2} (b − a) ⋁_{a}^{b} (u) − \frac{2}{b − a} ∫_{a}^{b} (⋁_{a}^{s} (u)) (s − \frac{a + b}{2}) ds]

≤ \frac{1}{2}L (b − a) ⋁_{a}^{b} (u) .

The constant \frac{1}{2} is sharp in both inequalities.

Proof. It is well known that if p : [α, β] → ℝ is L−Lipschitzian and v : [α, β] → ℝ is Riemann integrable, then the Riemann-Stieltjes integral ∫_{α}^{β} p (s) dv (s) exists and

|∫_{α}^{β} p (s) dv (s)| ≤ L ∫_{α}^{β} |p (s)| ds. Utilising this property, we then have

|∫_{a}^{s} (t − a) df (t)| ≤ L ∫_{a}^{s} (t − a) dt = \frac{L}{2} (s − a)^{2} ,

|∫_{s}^{b} (t − b) df (t)| ≤ L ∫_{s}^{b} (b − t) dt = \frac{L}{2} (b − s)^{2} .

Therefore, by the relation (3.6) we have

I ≤ \frac{L}{2 (b − a)} ∫_{a}^{b} [(b − s)^{2} + (s − a)^{2}] d (⋁_{a}^{s} (u))

= \frac{L}{2 (b − a)} [[(b − s)^{2} + (s − a)^{2}] ⋁_{a}^{s} (u)|_{a}^{b} − 2 ∫_{a}^{b} ⋁_{a}^{s} (u) (2s − a − b) ds]

= \frac{L}{2 (b − a)} [(b − a)^{2} ⋁_{a}^{b} (u) − 4 ∫_{a}^{b} ⋁_{a}^{s} (u) (s − \frac{a + b}{2}) ds]

and the first inequality in (3.8) is proved.

To prove the last part, we use the Čebyšev inequality which states that for two nondecreasing functions g and h,

\frac{1}{b − a} ∫_{a}^{b} g (s) h (s) ds ≥ \frac{1}{b − a} ∫_{a}^{b} g (s) ds · \frac{1}{b − a} ∫_{a}^{b} h (s) ds.

Then

\frac{1}{b − a} ∫_{a}^{b} ⋁_{a}^{s} (u) (s − \frac{a + b}{2}) ds

≥ \frac{1}{b − a} ∫_{a}^{b} (⋁_{a}^{s} (u)) ds · \frac{1}{b − a} ∫_{a}^{b} (s − \frac{a + b}{2}) ds

and since ∫_{a}^{b} (s − \frac{a+b}{2}) ds = 0, the inequality is proved.

For the sharpness of the constant, we consider the functions f (t) = t − \frac{a+b}{2}, t ∈ [a, b] and u : [a, b] → ℝ defined by

u (t) := \{ \begin{matrix}1 & if t = a \\ 0 & if t ∈ (a, b) \\ 1 & if t = b. \end{matrix}

Then f is Lipschitzian with L = 1 and u is of bounded variation on [a, b] . We have

⋁_{a}^{s} (u) = 1, s ∈ (a, b) and ⋁_{a}^{b} (u) = 2. Also,

D (f; u) = ∫_{a}^{b} f (t) du (t) − \frac{u (b) − u (a)}{b − a} ∫_{a}^{b} f (t) dt
= ∫_{a}^{b} f (t) du (t)
= f (t) u (t) |_{a}^{b} − ∫_{a}^{b} u (t) df (t) = b − a

and

∫_{a}^{b} (⋁_{a}^{s} (u)) (s − \frac{a + b}{2}) ds = ∫_{a}^{b} (s − \frac{a + b}{2}) ds = 0.

Replacing the values in (3.8) we get in all sides the same quantity b − a. This shows that the constant \frac{1}{2} is best possible in both inequalities. □

Remark 1. The inequality between the first and last term in (3.8) was firstly discovered by Dragomir and Fedotov in [9] where they also showed the sharpness of the constant \frac{1}{2}.

The following result may be stated as well.

Theorem 3. Assume that u : [a, b] → ℝ is of bounded variation and f : [a, b] → ℝ is monotonic nondecreasing and such that the Riemann-Stieltjes integral ∫_{a}^{b} f (t) du (t) exists. Then,

(3.9) |D (f; u)| ≤ \frac{1}{b − a} [∫_{a}^{b} (2s − a − b) f (s) d (⋁_{a}^{s} (u))

+ 2 ∫_{a}^{b} (⋁_{a}^{s} (u)) f (s) ds − ∫_{a}^{b} f (s) ds · ⋁_{a}^{b} (u)] .

Proof. It is well known that if the Stieltjes integrals ∫_{α}^{β} p (t) dv (t) and ∫_{α}^{β} |p (t)| dv (t) exist and v is monotonic nondecreasing on [α, β] , then

|∫_{α}^{β} p (t) dv (t)| ≤ ∫_{α}^{β} |p (t)| dv (t) .

Utilising this property we then have

|∫_{a}^{s} (t − a) df (t)| ≤ ∫_{a}^{s} (t − a) df (t) = (s − a) f (s) − ∫_{a}^{s} f (t) dt

and

|∫_{s}^{b} (t − b) df (t)| ≤ ∫_{s}^{b} (t − b) df (t) = ∫_{s}^{b} f (t) dt − (b − s) f (s)

for any s ∈ [a, b] .

Utilising the relation (3.6), we obtain

I ≤ \frac{1}{b − a} [∫_{a}^{b} \{(s − a) f (s) − ∫_{a}^{s} f (t) dt (3.10)

+ ∫_{s}^{b} f (t) dt − (b − s) f (s)} d (⋁_{a}^{s} (u))]

= \frac{1}{b − a} [∫_{a}^{b} (2s − a − b) f (s) d (⋁_{a}^{s} (u))

+ ∫_{a}^{b} (∫_{s}^{b} f (t) dt) d (⋁_{a}^{s} (u)) − ∫_{a}^{b} (∫_{a}^{s} f (t) dt) d (⋁_{a}^{s} (u))]

=: J.

However, integrating by parts in the Riemann-Stieltjes integral, we have:

∫_{a}^{b} (∫_{s}^{b} f (t) dt) d (⋁_{a}^{s} (u)) = (∫_{s}^{b} f (t) dt) · ⋁_{a}^{s} (u)|_{a}^{b} − ∫_{a}^{b} (⋁_{a}^{s} (u)) d (∫_{s}^{b} f (t) dt)

= ∫_{a}^{b} (⋁_{a}^{s} (u)) f (s) ds

and

∫_{a}^{b} (∫_{a}^{s} f (t) dt) d (⋁_{a}^{s} (u)) = (∫_{a}^{s} f (t) dt) · ⋁_{a}^{s} (u)|_{a}^{b} − ∫_{a}^{b} (⋁_{a}^{s} (u)) d (∫_{a}^{s} f (t) dt)

= ∫_{a}^{b} f (t) dt · ⋁_{a}^{b} (u) − ∫_{a}^{b} (⋁_{a}^{s} (u)) f (s) ds.

Therefore,

J = \frac{1}{b − a} [∫_{a}^{b} (2s − a − b) f (s) d (⋁_{a}^{s} (u))

+ ∫_{a}^{b} (⋁_{a}^{s} (u)) f (s) ds − ∫_{a}^{b} f (t) dt · ⋁_{a}^{b} (u) + ∫_{a}^{b} (⋁_{a}^{s} (u)) f (s) ds]

= \frac{1}{b − a} [∫_{a}^{b} (2s − a − b) f (s) d (⋁_{a}^{s} (u))

+ 2 ∫_{a}^{b} (⋁_{a}^{s} (u)) f (s) ds − ∫_{a}^{b} f (s) ds · ⋁_{a}^{b} (u)] .

This together with the inequalities (3.6) and (3.10) produces the desired result (3.9). □

4. Bounds in the Case when f is of Bounded Variation

We can state the following result as well.

Theorem 4. Assume that f : [a, b] → ℝ is of bounded variation on [a, b] . If u : [a, b] → ℝ is continuous and such that there exists the constants L_{a}, L_{b} > 0 and α, β > 0 with the properties that:

(4.1)  |u (t) − u (a)| ≤ L_{a} (t − a)^{α} ,  |u (t) − u (b)| ≤ L_{b} (b − t)^{β}

for any t ∈ [a, b] , then

(4.2) |D (f; u)|

≤ \frac{1}{b − a}L_{a} [∫_{a}^{b} (⋁_{a}^{t} (f)) (t − a)^{α} dt − α ∫_{a}^{b} (⋁_{a}^{t} (f)) (b − t) (t − a)^{α−1} dt]

+ \frac{1}{b − a}L_{b} [β ∫_{a}^{b} (⋁_{a}^{t} (f)) (t − a) (b − t)^{β−1} dt − ∫_{a}^{b} (⋁_{a}^{t} (f)) (b − t)^{β} dt] .

Proof. Utilising the identity (2.6) and the first inequality in (3.1), we have successively,

|D (f; u)| ≤ \frac{1}{b − a} ∫_{a}^{b} |∫_{a}^{b} Q (t, s) du (s)| d (⋁_{a}^{t} (f)) (4.3)

= \frac{1}{b − a} ∫_{a}^{b} |∫_{a}^{t} Q (t, s) du (s) + ∫_{t}^{b} Q (t, s) du (s)| d (⋁_{a}^{t} (f))

≤ \frac{1}{b − a} ∫_{a}^{b} [|∫_{a}^{t} Q (t, s) du (s)| + |∫_{t}^{b} Q (t, s) du (s)|] d (⋁_{a}^{t} (f))

= \frac{1}{b − a} ∫_{a}^{b} [(b − t) |u (t) − u (a)| + (t − a) |u (b) − u (t)|] d (⋁_{a}^{t} (f))

=: P.

Now, on making use of the condition (4.1), we can state that

(4.4)  P ≤ \frac{1}{b − a} ∫_{a}^{b} [L_{a} (b − t) (t − a)^{α} + L_{b} (t − a) (b − t)^{β}] d (⋁_{a}^{t} (f))

= \frac{1}{b − a} [L_{a} ∫_{a}^{b} (b − t) (t − a)^{α} d (⋁_{a}^{t} (f))

+ L_{b} ∫_{a}^{b} (t − a) (b − t)^{β} d (⋁_{a}^{t} (f))] .

However,

∫_{a}^{b} (b − t) (t − a)^{α} d (⋁_{a}^{t} (f))

= (b − t) (t − a)^{α} ⋁_{a}^{t} (f)|_{a}^{b} − ∫_{a}^{b} (⋁_{a}^{t} (f)) d [(b − t) (t − a)^{α}]

= − ∫_{a}^{b} (⋁_{a}^{t} (f)) [− (t − a)^{α} + α (b − t) (t − a)^{α−1}] dt

= ∫_{a}^{b} (⋁_{a}^{t} (f)) (t − a)^{α} dt − α ∫_{a}^{b} (⋁_{a}^{t} (f)) (b − t) (t − a)^{α−1} dt

12 SEVER S. DRAGOMIR

and

∫_{a}^{b} (t − a) (b − t)^{β} d (⋁_{a}^{t} (f))

= (t − a) (b − t)^{β} ⋁_{a}^{t} (f)|_{a}^{b} − ∫_{a}^{b} (⋁_{a}^{t} (f)) d [(t − a) (b − t)^{β}]

= − ∫_{a}^{b} (⋁_{a}^{t} (f)) [(b − t)^{β} − β (t − a) (b − t)^{β−1}] dt

= β ∫_{a}^{b} (⋁_{a}^{t} (f)) (t − a) (b − t)^{β−1} dt − ∫_{a}^{b} (⋁_{a}^{t} (f)) (b − t)^{β} dt

and from (4.4) we deduce the desired inequality (4.2). □ Corollary 3. If f is as in Theorem 4 and u is of r − H−Hölder type, i.e.,

(4.5)  |u (t) − u (s)| ≤ H |u − t|^{r} for any t, s ∈ [a, b] ,

where H > 0 and r ∈ (0, 1) are given, then

(4.6) |D (f; u)| ≤ \frac{1}{b − a}H ∫_{a}^{b} (⋁_{a}^{t} (f)) \{ (t − a)^{r} − (b − t)^{r}

+ r (b − t)^{r−1} (t − a)^{r−1} [(t − a)^{1−r} − (b − t)^{1−r}]} dt.

Remark 2. If r = \frac{1}{2} in Corollary 3, then we obtain the inequality:

(4.7) |D (f; u)| ≤ \frac{1}{b − a}H

× ∫_{a}^{b} (⋁_{a}^{t} (f)) (\sqrt{t − a} − \sqrt{b − t}) (1 + \frac{1}{2\sqrt{(b − t) (t − a)}}) dt.

The following particular result may be useful for applications. Corollary 4. If f is as in Theorem 4 and u : [a, b] → ℝ is Lipschitzian with the constant K > 0, then

(4.8)  |D (f; u)| ≤ \frac{4}{b − a} · K ∫_{a}^{b} (t − \frac{a + b}{2}) · ⋁_{a}^{t} (f) dt

≤ \{ \begin{matrix}K (b − a) ⋁_{a}^{b} (f) ; \\ \frac{2(b−a)^{\frac{1}{q}}}{(q+1)^{\frac{1}{q}}} K (∫_{a}^{b} [⋁_{a}^{t} (f)]^{p} dt)^{\frac{1}{p}} , p > 1, \frac{1}{p} + \frac{1}{q} = 1; \\ 2K ∫_{a}^{b} (⋁_{a}^{t} (f)) dt. \end{matrix}

The multiplication constant 4 is best possible.

Proof. The first inequality follows by Theorem 4 on choosing L_{a} = L_{b} = K and α = β = 1.

Now, on utilising Hölder’s inequality, we have

(4.9) ∫_{a}^{b} (t − \frac{a + b}{2}) · (⋁_{a}^{t} (f)) dt

≤ \{ \begin{matrix}sup_{t∈[a,b]} (⋁_{a}^{t} (f)) ∫_{a}^{b} |t − \frac{a+b}{2} | dt \\ (∫_{a}^{b} [⋁_{a}^{t} (f)]^{p} dt)^{\frac{1}{p}} (∫_{a}^{b} |t − \frac{a+b}{2} |^{q} dt)^{\frac{1}{q}} , p > 1, \frac{1}{p} + \frac{1}{q} = 1; \\ ∫_{a}^{b} (⋁_{a}^{t} (f)) dt sup_{t∈[a,b]} |t − \frac{a+b}{2} | . \end{matrix}

However, sup_{t∈[a,b]} |t − \frac{a+b}{2} | = \frac{b−a}{2} and

∫_{a}^{b} |t − \frac{a + b}{2}|^{q} dt = 2 ∫ b_{\frac{a+b}{2}} (t − \frac{a + b}{2})^{q} dt

= \frac{(b − a)^{q+1}}{2^{q} (q + 1)} , q ≥ 1

and by (4.9) we deduce:

(4.10) ∫_{a}^{b} (t − \frac{a + b}{2}) · ⋁_{a}^{t} (f) dt

≤ \{ \begin{matrix}\frac{(b−a)^{2}}{4} ⋁_{a}^{t} (f) ; \\ \frac{(b−a)^{1+ \frac{1}{q}}}{2(q+1)^{\frac{1}{q}}} (∫_{a}^{b} [⋁_{a}^{t} (f)]^{p} dt)^{\frac{1}{p}} , p > 1, \frac{1}{p} + \frac{1}{q} = 1; \\ \frac{b−a}{2} ∫_{a}^{b} (⋁_{a}^{t} (f)) dt \end{matrix}

and the second part is proved.

To prove the sharpness of the constant 4 in the first inequality in (4.8) assume that there exists A > 0 such that

(4.11)  |D (f; u)| ≤ \frac{A}{b − a} · K ∫_{a}^{b} (t − \frac{a + b}{2}) · ⋁_{a}^{t} (f) dt,

provided that f is of bounded variation and u is K−Lipschitzian.

Let f : [a, b] → ℝ,

f (t) = \{ \begin{matrix}0 & if t ∈ [a, \frac{a+b}{2}] , \\ k & if t ∈ (\frac{a+b}{2}, b] , \end{matrix}

with k > 0. Then

⋁_{a}^{t} (f) = \{ \begin{matrix}0 & if t ∈ [a, \frac{a+b}{2}] , \\ k & if t ∈ (\frac{a+b}{2}, b] . \end{matrix}

Also, we have

∫_{a}^{b} (t − \frac{a + b}{2}) · ⋁_{a}^{t} (f) dt = ∫ b_{\frac{a+b}{2}} (t − \frac{a + b}{2}) kdt

= \frac{k (b − a)^{2}}{8} .

Consider u : [a, b] → ℝ, u (t) = |t − \frac{a+b}{2} | . Then u is K−Lipschitzian with K = 1. Also,

D (f; u) = ∫_{a}^{b} f (t) du (t) − \frac{u (b) − u (a)}{b − a} ∫_{a}^{b} f (t) dt

= k ∫ b_{\frac{a+b}{2}} du (t) = k [u (b) − u (\frac{a + b}{2})]

= \frac{(b − a) k}{2} .

Substituting these values into (4.11) produces the inequality

\frac{(b − a) k}{2} ≤ \frac{A}{b − a} · \frac{k (b − a)^{2}}{8} ,

which implies that A ≥ 4. □

5. Inequalities for (l, L) −Lipschitzian Functions

The following simple lemma holds.

Lemma 4. Let u : [a, b] → ℝ and l, L ∈ ℝ with L > l. The following statements are equivalent:

(i) The function u− \frac{l+L}{2} ·e, where e (t) = t, t ∈ [a, b] is \frac{1}{2} (L − l) −Lipschitzian; (ii) We have the inequalities

l ≤ \frac{u (t) − u (s)}{t − s} ≤ L for each t, s ∈ [a, b] , t ≠ s;

(iii) We have the inequalities

l (t − s) ≤ u (t) − u (s) ≤ L (t − s) for each t, s ∈ [a, b] with t > s.

The proof is obvious and we omit the details.

Definition 1 (see also [10]). The function u : [a, b] → ℝ which satisfies one of the equivalent conditions (i) – (iii) from Lemma 4 is said to be (l, L) −Lipschitzian on [a, b] . If L > 0 and l = −L, then (−L, L) −Lipschitzian means L−Lipschitzian in the classical sense.

The following result can be stated.

Theorem 5. Let f : [a, b] → ℝ be a function of bounded variation and u : [a, b] → ℝ an (l, L) −Lipschitzian function. Then

(5.1) |D (f; u)| ≤ \frac{2}{b − a} (L − l) ∫_{a}^{b} (t − \frac{a + b}{2}) · ⋁_{a}^{t} (f) dt

≤ \{ \begin{matrix}\frac{1}{2} (L − l) (b − a) ⋁_{a}^{b} (f) ; \\ \begin{matrix}(b−a) & \frac{1}{q} \\ (q+1) & \frac{1}{q} \end{matrix} (L − l) (∫_{a}^{b} [⋁_{a}^{t} (f)]^{p} dt)^{\frac{1}{p}} , p > 1, \frac{1}{p} + \frac{1}{q} = 1; \\ (L − l) ∫_{a}^{b} (⋁_{a}^{t} (f)) dt. \end{matrix}

The constant 2 in the first inequality is sharp.

Proof. Observe that:

D (f; u − \frac{l + L}{2} · e)

= ∫_{a}^{b} (f (t) − \frac{1}{b − a} ∫_{a}^{b} f (s) ds) d [u (t) − \frac{l + L}{2} · t]

= ∫_{a}^{b} (f (t) − \frac{1}{b − a} ∫_{a}^{b} f (s) ds) du (t)

− \frac{l + L}{2} ∫_{a}^{b} (f (t) − \frac{1}{b − a} ∫_{a}^{b} f (s) ds) dt

= D (f; u) .

Now, applying Corollary 4 for the function u−\frac{l+L}{2}e, which is \frac{1}{2} (L − l) −Lipschitzian, we get:

|D (f; u − \frac{l + L}{2}e)| ≤ \frac{4}{b − a} · \frac{1}{2} (L − l) ∫_{a}^{b} (t − \frac{a + b}{2}) · ⋁_{a}^{t} (f) dt

= \frac{2}{b − a} (L − l) ∫_{a}^{b} (t − \frac{a + b}{2}) · ⋁_{a}^{t} (f) dt

and the theorem is proved. □

The second result may be stated as:

Theorem 6. Let u : [a, b] → ℝ be a function of bounded variation. If f : [a, b] → ℝ is (ϕ, Φ) −Lipschitzian with Φ > ϕ, then

|D (f; u) − \frac{ϕ + Φ}{2} [\frac{u (b) + u (a)}{2} (b − a) − ∫_{a}^{b} u (t) dt]| (5.2)

≤ \frac{1}{2} (Φ − ϕ) · [\frac{1}{2} (b − a) ⋁_{a}^{b} (u) − \frac{2}{b − a} ∫_{a}^{b} (⋁_{a}^{s} (u)) (s − \frac{a + b}{2}) ds]

≤ \frac{1}{4} · (Φ − ϕ) (b − a) ⋁_{a}^{b} (u) .

The constant \frac{1}{2} in front of (Φ − ϕ) and \frac{1}{4} are best possible.

Proof. Observe that

D (f − \frac{ϕ + Φ}{2} · e; u)

= ∫_{a}^{b} [f (t) − \frac{ϕ + Φ}{2} · t − \frac{1}{b − a} ∫_{a}^{b} (f (s) − \frac{ϕ + Φ}{2} · s) ds] du (t)

= ∫_{a}^{b} [f (t) − \frac{1}{b − a} ∫_{a}^{b} f (s) ds − (\frac{ϕ + Φ}{2}t − \frac{1}{b − a} ∫_{a}^{b} \frac{ϕ + Φ}{2} · sds)] du (t)

= D (f; u) − \frac{ϕ + Φ}{2} ∫_{a}^{b} (t − \frac{1}{b − a} ∫_{a}^{b} sds) du (t)

= D (f; u) − \frac{ϕ + Φ}{2} ∫_{a}^{b} (t − \frac{a + b}{2}) du (t) .

Since, integrating by parts in the Riemann-Stieltjes integral we have:

∫_{a}^{b} (t − \frac{a + b}{2}) du (t) = \frac{u (b) + u (a)}{2} (b − a) − ∫_{a}^{b} u (t) dt,

then

D (f − \frac{ϕ + Φ}{2}e; u) = D (f; u) − \frac{ϕ + Φ}{2} [\frac{u (b) + u (a)}{2} (b − a) − ∫_{a}^{b} u (t) dt] .

Now, on applying Theorem 2 for the function f−\frac{ϕ+Φ}{2}e which is \frac{1}{2} (L − l) −Lipschitzian, we deduce the desired result (5.2). □

6. Applications for the Čebyšev Functional

If we choose u (t) := ∫_{a}^{t} g (τ) dτ, t ∈ [a, b] , where g : [a, b] → ℝ is Lebesgue integrable on [a, b] , then we have the equality

C (f; g) = \frac{1}{b − a}D (f; u) .

Also, u is of bounded variation on any subinterval [a, s] , s ∈ [a, b] and if g is continuous on [a, b] , then

⋁_{a}^{s} (u) = ∫_{a}^{s} |g (τ)| dτ, s ∈ [a, b] .

RIEMANN-STIELTJES INTEGRAL VIA A ČEBYŠEV TYPE FUNCTIONAL 17

If f is of bounded variation on [a, b] , then on utilising the inequality (3.5) we have |C (f; g)| (6.1)

≤ \frac{1}{(b − a)^{2}} [∫_{a}^{b} (2s − a − b) |g (s)| ⋁_{a}^{s} (f) ds

+ 2 ∫_{a}^{b} (∫_{a}^{s} |g (τ)| dτ) ⋁_{a}^{s} (f) ds − ∫_{a}^{b} |g (τ)| dτ · ∫_{a}^{b} (⋁_{a}^{s} (f)) ds]

≤ \frac{1}{(b − a)^{2}} ∫_{a}^{b} (2s − a − b) |g (s)| ⋁_{a}^{s} (f) ds

+ \frac{1}{(b − a)^{2}} ∫_{a}^{b} (∫_{a}^{s} |g (τ)| dτ) ⋁_{a}^{s} (f) ds

≤ \frac{1}{(b − a)^{2}} ∫_{a}^{b} (2s − a − b) |g (s)| ⋁_{a}^{s} (f) ds + \frac{1}{b − a} ∫_{a}^{b} |g (τ)| dτ · ⋁_{a}^{b} (f) .

Now, if f is monotonic nondecreasing, then by (3.9) we have

(6.2) |C (f; g)| ≤ \frac{1}{(b − a)^{2}} [∫_{a}^{b} (2s − a − b) f (s) |g (s)| ds

+ 2 ∫_{a}^{b} (∫_{a}^{s} |g (τ)| dτ) f (s) ds − ∫_{a}^{b} f (s) ds · ∫_{a}^{b} |g (τ)| dτ] .

The case where f is L−Lipschitzian provides via (3.8) a much simpler inequality: |C (f; g)| (6.3)

≤ L [\frac{1}{2} ∫_{a}^{b} |g (τ)| dτ − \frac{2}{(b − a)^{2}} ∫_{a}^{b} (∫_{a}^{s} |g (τ)| dτ) (s − \frac{a + b}{2}) ds]

≤ \frac{1}{2}L ∫_{a}^{b} |g (s)| ds.

Now, if f is of bounded variation and |g| is bounded above by M, i.e., |g (t)| ≤ M for a.e. t ∈ [a, b] , then by (4.8) we have:

(6.4) |C (f; g)| ≤ \frac{4}{(b − a)^{2}} M ∫_{a}^{b} (t − \frac{a + b}{2}) ⋁_{a}^{t} (f) dt

≤ \{ \begin{matrix}M ⋁_{a}^{b} (f) ; \\ \frac{2(b−a)^{\frac{1}{q} −1}}{(q+1)^{\frac{1}{q}}} M (∫_{a}^{b} [⋁_{a}^{t} (f)]^{p} dt)^{\frac{1}{p}} , p > 1, \frac{1}{p} + \frac{1}{q} = 1; \\ \frac{2M}{b−a} ∫_{a}^{b} (⋁_{a}^{t} (f)) dt. \end{matrix}

The constant 4 in (6.4) is best possible.

Finally, if −∞ < ϕ ≤ g (t) ≤ Φ for a.e. t ∈ [a, b] , then |g (t) − \frac{ϕ+Φ}{2} | ≤ \frac{1}{2} (Φ − ϕ) and since

C (f; g − \frac{ϕ + Φ}{2}) = C (f; g) ,

hence, by (6.4) we deduce the inequalities

(6.5) |C (f; g)| ≤ \frac{2}{(b − a)^{2}} (Φ − ϕ) ∫_{a}^{b} (t − \frac{a + b}{2}) ⋁_{a}^{t} (f) dt

≤ \{ \begin{matrix}\frac{1}{2} (Φ − ϕ) ⋁_{a}^{b} (f) ; \\ \begin{matrix}(b−a) & ^{\frac{1}{q} −1} \\ (q+1) & \frac{1}{q} \end{matrix} (Φ − ϕ) (∫_{a}^{b} [⋁_{a}^{t} (f)]^{p} dt)^{\frac{1}{p}} , p > 1, \frac{1}{p} + \frac{1}{q} = 1; \\ \frac{Φ−ϕ}{b−a} ∫_{a}^{b} (⋁_{a}^{t} (f)) dt. \end{matrix}

The constant 2 in the first inequality is best possible.

References

[1] G.A. ANASTASSIOU, Univariate Ostrowski inequalities, revisited. Monatsh. Math. 135 (2002), no. 3, 175–189.

[2] G.A. ANASTASSIOU, Chebyshev-Grüss type and comparison of integral means inequalities for the Stieltjes integral. Panamer. Math. J. 17 (2007), no. 3, 91–109.

[3] T. APOSTOL, Mathematical Analysis, Addison Wesley.

[4] S.S. DRAGOMIR, Inequalities of Grüss type for the Stieltjes integral, Kragujevac J. Math., 26 (2004), 89-122.

[5] S.S. DRAGOMIR, Inequalities for Stieltjes integrals with convex integrators and applications, Appl. Math. Lett., 20 (2007), 123-130.

[6] S.S. DRAGOMIR, A Sharp bound of the Cebysev functional for the Riemann-Stieltjes integral and applications, Preprint, RGMIA Res. Rep. Coll. 10(2007), Supplement, Article 16.

[ONLINE http://rgmia.vu.edu.au/v10(E).html].

[7] S.S. DRAGOMIR, Approximation of the Riemann-Stieltjes integral with (l, L)-Lipschitzian integrators, Preprint, RGMIA Res. Rep. Coll. 10(2007), No. 3, Article 5. [ONLINE

http://rgmia.vu.edu.au/v10n3.html].

[8] S.S. DRAGOMIR and I. FEDOTOV, An inequality of Grüss type for the Riemann-Stieltjes integral and applications for special means, Tamkang J. Math., 29(4) (1998), 287-292.

[9] S.S. DRAGOMIR and I. FEDOTOV, A Grüss type inequality for mappings of bounded variation and applications for numerical analysis, Nonlinear Funct. Anal. Appl., 6(3) (2001), 425-433.

[10] Z. LIU, Refinement of an inequality of Grüss type for Riemann-Stieltjes integral, Soochow J. Math., 30(4) (2004), 483-489.

School of Computer Science & Mathematics, Victoria University, PO Box 14428, Melbourne, VIC 8001, Australia.

E-mail address: Sever.Dragomir@vu.edu.au
URL: http://rgmia.vu.edu.au/dragomir