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1 J ournal of Mathematical I nequalities Volume 6, Number 1 (2012), 6568 RELATIVE CONVEXITY AND QUADRATURE RULES FOR THE RIEMANNSTIELTJES INTEGRAL P ETER R. M ERCER (Communicated by A. Cizmesija) Abstract. We develop Trapezoid, Midpoint, and Simpsons rules for the Riemann-Stieltjes in- tegral, the latter two being new. These rules are completely natural when the notion of relative convexity is used. 1. For f continuous (say) on [a, b], the classical Midpoint Rule is b f (x)dx = f ( a+b 2 )[b a] , (M) a and the classical Trapezoid Rule is b f (x)dx = f (a)+ f (b) 2 [b a] . (T) a A useful relationship between these is the well-known H ADAMARD S I NEQUALITY . If f is a convex function on [a, b] then b f (a)+ f (b) f ( a+b 2 )[b a] f (x) dx 2 [b a] . (H) a Moreover, if f is continuous then the errors can be expressed as follows [1]: There are 1 and 2 (a, b) such that b 1 3 f (a)+ f (b) 1 f ( a+b 2 )[b a]+ 24 f (1 )(b a) = f (x) dx = 2 [b a] 12 f (2 )(b a)3 . a (E) Mathematics subject classification (2010): 65D30. Keywords and phrases: trapezoid rule, midpoint rule, Simpsons rule, Riemann-Stieltjes integral, rel- ative convexity, Hadamards inequality. c , Zagreb 65 Paper JMI-06-06

2 66 P ETER R. M ERCER Now for f continuous (say), and g increasing (say) on [a, b], a possible Riemann- Stieltjes Midpoint Rule is b f dg = f ( a+b 2 )[g(b) g(a)] (RSM) a and a possible Riemann-Stieltjes Trapezoid Rule is b f dg = f (a)+ f (b) 2 [g(b) g(a)] . (RST) a We write possible to mean that each reduces to its corresponding classical quadrature rule when g(x) = x. These RS quadrature rules have been used and studied by a good number of au- thors (e.g. [27]). In [8] it is shown that for f convex, the obvious analog of (H) for RSM and RST does not hold. In that paper alternatives to RSM and RSM are offered, for which (H) does hold. Error terms are also obtained. In this investigation we take a different approach this time adjusting, in a natural way, the notion of convexity. The very nature of the adjustment ensures that the right hand side of the obvious analog for (H) holds and it forces us to rethink the left hand side, thus engendering an alternative to the RSM above. Error terms analogous to (E) are obtained as well as a RS Simpsons rule. 2. Let f be defined on [a, b] and get g : [a, b] [a, b], with g increasing. We say f is convex with respect to g if f g1 is convex [9]. As g is increasing we may write b g(b) f dg(x) = f g1(x)dx. a g(a) Then by applying (M) and (T) and using (H), we immediately obtain the following. T HEOREM 1. Let f and g be defined on [a, b] and suppose that f is convex with respect to g there. Then b f (a)+ f (b) f g1( g(a)+g(b) 2 )[g(b) g(a)] f dg(x) 2 [g(b) g(a)] . a We can obtain error terms for these quadrature rules which are perfectly analogous to those in (E), as follows. T HEOREM 2. Suppose that f and g are defined on [a, b], that f is convex with respect to g there, and that f and g are twice continuously differentiable. Then there are 1 and 2 (a, b) such that b d2 f (g(b) g(a))3 f dg = f g1( g(a)+g(b) 2 )[g(b) g(a)] + ( 1 ) . dg2 24 a

3 R ELATIVE CONVEXITY AND QUADRATURE RULES FOR THE R IEMANN -S TIELTJES INTEGRAL 67 and b f (a)+ f (b) d2 f (g(b) g(a))3 f dg = 2 [g(b) g(a)] ( 2 ) . dg2 12 a Proof. It is a routine matter to verify that g (g1 (x)) f (g1 (x)) f (g1 (x))g (g1 (x)) ( f g1) (x) = , ( g (g1 (x)) )3 so we may apply (E) to obtain b g (1 ) f (1 ) f (1 )g (1 ) (g(b)g(a))3 f dg f g1 ( g(a)+g(b) 2 )(g(b)g(a)) = (g (1 ))3 24 a and b f (a)+ f (b) g (2 ) f (2 ) f (2 )g (2 ) (g(b) g(a))3 f dg 2 (g(b) g(a)) = , ( g (2 ) )3 12 a where 1 = g1 (1 ) and 2 = g1 (2 ). Now f and g are each defined for t [a, b], so df f d 2 f dg g f f g = and = . dg g dg2 dt (g)2 Therefore d2 f g f f g = , dg2 (g)3 and the proof is complete. We close with two observations. It is a useful fact that Simpsons Rule = 13 (Trape- zoid Rule) + 23 (Midpoint Rule). Using this in the RS context, and following the same idea as in Theorem 2, we obtain the following. (Another possible RS Simpsons Rule for g only of bounded variation can be found in [5].) C OROLLARY 3. Let f and g be defined on [a, b], that f is convex with respect to g there, and that f and g are four times continuously differentiable. Then there is 3 (a, b) such that b f (a)+ f (b) d4 f (g(b) g(a))5 f dg = 2 3 f g1( g(a)+g(b) 2 )+ 6 [g(b) g(a)] ( 3 ) . dg4 2880 a Also, it is a fact that ([9]) f g f is convex with respect to g , f g and this ensures that each of the terms [ ] in the proof of Theorem 2 is nonnegative. Therefore, we have

4 68 P ETER R. M ERCER C OROLLARY 4. Let f and g be defined on [a, b], and let f and g be twice continuously differentiable. Then d2 f f is convex with respect to g 0. dg2 REFERENCES [1] S. D. C ONTE & C. DE B OOR, Elementary Numerical Analysis: An Algorithmic Approach, 3rd edition, McGraw-Hill, 1980. [2] T. K. B OEHME , W. P REUSS & V. VAN DER WALL, On a simple numerical method for computing Stieltjes integrals in reliability theory, Prob. in Eng. & Inf. Sci. 5 (1991), 113128. [3] K. D IETHELM, A note on the midpoint rectangle formula for Riemann-Stieltjes integrals, J. Stat. & Com. Simul. 74 (2004), 920922. [4] S. S. D RAGOMIR, Some inequalities of midpoint and trapezoid type for the Riemann-Stieltjes integral, Nonlinear. Anal. 47 (2001), 23332340. [5] M. T ORTORELLA, Closed Newton-Cotes quadrature rules for Stieltjes integrals and numerical con- volution of life distributions, SIAM J. Sci. Stat. Comp. 11 (1990), 732748. [6] M. T ORTORELLA, Numerical solutions of renewal-type integral equations, INFORMS J. Comp. 17 (2005), 6674. [7] M. X IE , W. P REUSS & L. R. C UI , Error analysis of some integrations procedures for renewal equa- tions and convolution integrals, J. Stat. & Comp. Simul. 73 (2003), 5970. [8] P. R. M ERCER, Hadamards inequality and trapezoid rules for the Riemann-Stieltjes integral, J. Math. Anal. Appl. 344 (2008), 921926. [9] C. N ICULESCU & L. E. P ERSSON, Convex Functions and their Applications, CMS Books in Mathe- matics, Springer, 2006. (Received August 31, 2010) Peter R. Mercer Department of. Mathematics Buffalo State College 1300 Elmwood Avenue Buffalo NY 14222 U.S.A. e-mail: [email protected] Journal of Mathematical Inequalities www.ele-math.com [email protected]

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