GENERAL THREE POINTS INEQUALITIES FOR WEIGHTED RIEMANN-STIELTJES INTEGRAL

In this paper we provide amongst others some simple error bounds in approximating the weighted Riemann-Stieltjes integral $\int_{a}^{b}f\left(t\right) g\left(t\right) dv\left(t\right) $ by the use of three points formula \begin{equation*} f\left(b\right) \int_{c}^{b}g\left(s\right) dv\left(s\right) +f\left(a\right) \int_{a}^{d}g\left( s\right) dv\left( s\right) -f\left(x\right) \int_{c}^{d}g\left(t\right) dv\left(t\right) \end{equation*} where $x,$ $c,$ $d\in \left[a,b\right],$ $g,$ $v:\left[a,b\right] \rightarrow \mathbb{C}$ under bounded variation and Lipschitzian assumptions for the function $f$ and such that the involved Riemann-Stieltjes integrals exist.

where 2 [0; 1] and x 2 [a; b] ; under bounded variation assumptions for the functions u and f and such that the involved Riemann-Stieltjes integral exists, in the recent paper [26] we have obtained the following result: In [27] we also obtained the following result in the case of Lipschitzian integrands: where f and u belong to di¤erent classes of function for which the Riemann-Stieltjes integral exists, see [22], [21], [20], and [8] and the references therein.Bounds for the functional can be found in [15], [16] and [8], while for the functional they may be found in [28], [8], [3] and [2].The details are omitted.
In this paper we provide some simple error bounds in approximating the weighted Riemann-Stieltjes integral R b a f (t) g (t) dv (t) by the use of various general three points formulae out of which we mention the following one

Some Preliminary Facts
The following properties of Riemann-Stieltjes integral are well know, [1, p. 158-159]: which also gives (2.1).We start with the following simple fact: In particular, for = we have Proof.Assume that x; c; d 2 [a; b] : Using the integration by parts formula for the Riemann-Stieltjes integral and Lemma 1, we have (2.4) In a similar way, (2.5) which is equivalent to the desired result (2.2) If we take d = c above, we get: In particular, for = we have 2) and (2.3), then we get If we take = v (d) + and = + v (c) ; then by (2.8) we get In particular, for = we get by (2.9) that In particular, for = we obtain (2.12) If we take c = a and d = b in Lemma 2, then we get In particular, for = we obtain If this equality we take = ; then we get  which is equivalent to 2), then we get

Inequalities for Integrands of Bounded Variation
We have:  In particular, for = we have By using the identity (2.2) and the property (3.3) we get which proves the …rst inequality in (3.1).Observe that and With the assumptions of Theorem 3, and if d = c; then In particular, for = ; we get In particular, for = ; we get  In particular, for = we have With the assumptions of Theorem 3 and if the Riemann-Stieltjes integrals below exist, then In particular, for c = a and d = b; we have for c = b and d = a; we have , then under the assumptions of Corollary 4, we have Using the equalities (2.9) and (2.10) one can obtain various inequalities as in the recent paper [26].The details are omitted.
In particular, for = we have By using the identity (2.2) and the property (4.4) we get which proves the …rst inequality in (4.2).The rest is obvious.
In particular, for = we have In particular, for = ; we get Remark 10.If we take c = x in Corollary 5, then we get (4.9)In particular, for = ; we get (4.10) !
In particular, for = we have (4.12) In particular, for c = a and d = b; we have  Remark 11.If we take x = a+b 2 ; then under the assumptions of Corollary 7, we have provided f; v are of bounded variation and g is continuous and such that the integral f ) ; then from the …rst inequality in (5.1) we get ; then from the inequality (5.1) we get provided v is of bounded variation, f is Lipschitzian with the constant L > 0 and g is continuous on [a; b] : In particular, for x = a+b 2 we get from the …rst inequality in (5.4) that ; then from the inequality (5.4) we get  Using the equalities (2.9) and (2.10) one can obtain various inequalities as in the recent paper [27].The details are omitted.

1 .
Introduction Assume that u; f : [a; b] !C are bounded.If the Riemann-Stieltjes integral R b a f (t) du (t) exists, we write for simplicity, like in [1, p. 142] that f 2 R C (u; [a; b]) ; or R C (u) when the interval is implicitly known.If the functions u; f are real valued, then we write f 2 R (u; [a; b]) ; or R (u) : In order to approximate the Riemann-Stieltjes integral R b a f (t) du (t) by the use of a three points formula, namely to establish bounds for the error functional T (f; u; a; b; x; ) := (1 ) f[u (b) u (x)] f (b) + [u (x) u (a)] f (a)g + [u (b) u (a)] f (x) Z b a f (t) du (t) ;

Theorem 1 .
Let f; u : [a; b] !C and x 2 [a; b] are such that f 2 R C (u; [a; b]) : If f and u are of bounded variation, then (1.1) jT (f; u; a; b; x;

Theorem 2 .
Let f; u : [a; b] !C and x 2 [a; b].If f is Lipschitzian with the constant L > 0; namely jf (t) f (s)j L jt sj for all t; s 2 [a; b] and u is of bounded variation, then f 2 R C (u; [a; b]) and (1.2) jT (f; u; a; b; x; (f; u; a; b; x) := Z b a f (t) du (t) f (x) [u (b) u (a)]

g
(t) dv (t) ; where x; c; d 2 [a; b] ; g; v : [a; b] !C under bounded variation and Lipschitzian assumptions for the function f and such that the involved Riemann-Stieltjes integral exist.

gRemark 7 .
(s) dv (s) ; max t2[x;b]Z t c g (s) dv (s) If we take c = x in Corollary 2, then we get

Corollary 3 .
s) dv (s) With the assumptions of Theorem 3, and if c = b and d = a; then (3.10)

Corollary 6 .
s) dv (s) (b a) : With the assumptions of Theorem 4, and if c = b and d = a; then

Corollary 7 .
With the assumptions of Theorem 4 and if the Riemann-Stieltjes integrals below exist, then (4.13) f (b)

g
for c = b and d = a; we have(4.15)f (x) Z b a g (t) dv (t) Z b a f (t) g (t) dv (t) (s) dv (s) (x a) + max t2[x;b] Z b t g (s) dv (s) (b x) dv (s) (b x) L max t2[a;b]

((b a+b 2 gg
max s2[a;x] jg (s)j x _ a (v) ; max s2[x;b] jg (s)j max s2[a;x] jg (s)j x _ a (f ) ; max s2[x;b] jg (s)jprovided f; v are of bounded variation and g is continuous and such that the integralR b a f (t) g (t) dv (t) exists.If m 2 (a; b) is such that m ; max s2[x;b] jg (s)j max max s2[a;x] jg (s)j (x a) ; max s2[x;b] jg (s)j (b x) b _ a (v)provided that v is of bounded variation, f is Lipschitzian with the constant L > 0 and g is continuous on [a; b] :In particular, for x = a+b 2 we get from the …rst inequality in (5.10) that (5.11) f (b)Z (s) dv (s) + f (a) ; then from the inequality (5.10) we get (5.12)f (b)Z b p g (s) dv (s) + f (a) Z p a g (t) dv (t) Z b a f(t) g (t) dv (t) (s) dv (s) (p a) + max t2[p;b] Z t p g (s) dv (s) (b p)