A Companion of Ostrowski inequality for the Stieltjes integral of monotonic functions

: Some companions of Ostrowski’s integral inequality for the Riemann-Stieltjes integral (cid:82) ba f ( t ) du ( t ), where f is assumed to be of r - H -H¨older type on [ a,b ] and u is of bounded variation on [ a,b ], are proved. Applications to the approximation problem of the Riemann-Stieltjes integral in terms of Riemann-Stieltjes sums are also pointed out.


Introduction
In [10], Dragomir has proved an Ostrowski inequality for the Riemann-Stieltjes integral, as follows: Theorem 1.1.
Let f : [a, b] → R be a r-H-Hölder type mapping, that is, it satisfies the condition where, H > 0 and r ∈ (0, 1] are given, and u : [a, b] → R is a mapping of bounded variation on [a, b]. Then we have the inequality for all x ∈ [a, b], where, b a (u) denotes the total variation of u on [a, b]. Furthermore, the constant 1 2 is the best possible in the sense that it cannot be replaced by a smaller one, for all r ∈ (0, 1].
Motivated by [17], Dragomir in [13], established the following companion of the Ostrowski inequality for mappings of bounded variation.
for any x ∈ [a, a+b 2 ] , where b a (f ) denotes the total variation of f on [a, b]. The constant 1/4 is best possible.
In this paper, we establish a companion of Ostrowski's integral inequality for the Riemann-Stieltjes integral

The Results
The following companion of Ostrowski's inequality for Riemann-Stieltjes integral holds.
for all x ∈ a, a+b 2 .
Proof. Using the integration by parts formula for Riemann-Stieltjes integral, we have Adding the above equalities, we have It is well known that if p : [c, d] → R is continuous and ν : [c, d] → R is monotonic nondecreasing, then the Stieltjes integral d c p(t)dν(t) exists and the following inequality holds: Making use of this property and the fact that f is of r-H-Hölder type on [a, b], we can state that By the integration by parts formula for the Stieltjes integral, we have similarly, which together with (6) proves the first inequality in (4). The second inequality is obvious by the property of max function and we omit the details here.
The following inequalities are hold: Corollary 2.1.
Let f and u as in Theorem 2.1. In (4) choose 1. x = a, then we get the following trapezoid type inequality 2. x = a+b 2 , then we get the following mid-point type inequality We may state the following Ostrowski type inequality: Let f and u as in for all x ∈ [a, a+b 2 ].

Corollary 2.3.
Let u as in Theorem 2.1, and f : [a, b] → R be an L-Lipschitzian mapping on [a, b], that is, where, L > 0 is fixed. Then, for all x ∈ [a, a+b 2 ], we have the inequality The constant 1 4 is the best possible in the sense that it cannot be replaced by a smaller one.

Corollary 2.4.
In Theorem 2.1, if u is monotonic on [a, b], and f is of r-H-Hölder type. Then, for all x ∈ [a, a+b 2 ], we have the inequality for all x ∈ [a, a+b 2 ], where g 1 = b a |g (t)| dt.
Proof. Define the mapping u : [a, b] → R, u(t) = t a g(s)ds. Then u is differentiable on (a, b) and u (t) = g(t). Using the properties of the Riemann-Stieltjes integral, we have for all x ∈ [a, a+b 2 ]. For instance, choose x = a+b 2 , then we get for all x ∈ a, a+b Proof. As u is continuous and f is of bounded variation on [a, b], the following Riemann-Stieltjes integrals exist and, by the integration by parts formula, we can state that If we add the above three identities, we obtain for all x ∈ [a, a+b 2 ]. Now, using the properties of absolute value, we have: Now, using (17), we have for all x ∈ [a, a+b 2 ], and the first inequality in (15) is proved.
Summing the above inequality over i from 0 to n − 1 and using the generalized triangle inequality, we deduce |R (f, u, In, ξ)| which completely proves the first inequality in (17).