Some Generalizations of Ostrowski Inequalities and Their Applications to Numerical Integration and Special Means

Some Generalization of Ostrowski Inequalities with Applications in Numerical Integration and Special Means by Fiza Zafar Submitted to the Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan on March 05, 2009 in partial ful llment of the requirement for the degree of Doctor in Philosophy of Mathematics. Keywords: Ostrowski inequality, Gruss inequality, μ Cebyev inequality, Numerical Integration, Special Means, Random variable, Probability Density Functions, Cumulative Distribution Function, Nonlinear Equations, Iterative Methods. 2000 Mathematics Subject Classi cation: 26D10; 26D15; 26D20; 41A55; 60E15; 34A34; 26C10; 65H05. In the last few decades, the eld of mathematical inequalities has proved to be an extensively applicable eld. It is applicable in the following manner: Integral inequalities play an important role in several other branches of mathematics and statistics with reference to its applications. The elementary inequalities are proved to be helpful in the development of many other branches of mathematics. The development of inequalities has been established with the publication of the books by G. H. Hardy, J. E. Littlewood and G. Polya [47] in 1934, E. F. Beckenbach and R. Bellman [13] in 1961 and by D. S. Mitrinovic, J. E. Peμcaric and A. M. Fink [64] & [65] in 1991. The publication of later has resulted to bring forward some new integral inequalities involving functions with bounded derivatives that measure bounds on the deviation of functional value from its mean value namely, Ostrowski inequality [69]. The books by D. S. Mitrinovic, J. E. Peμcaric and A. M. Fink have also brought to focus integral inequalities which establish a connection between the integral of the product of two functions and the product of the integrals of the two functions namely, inequalities of Gruss [46] and μ Cebyev type (see [64], p. 297). iii These type of inequalities are of supreme importance because they have immediate applications in Numerical integration, Probability theory, Information theory and Integral operator theory. The monographs presented by S. S. Dragomir and Th. M. Rassias [36] in 2002 and by N. S. Barnett, P. Cerone and S. S. Dragomir [8] in 2004 can well justify this statement. In these monographs, separate aspects of applications of inequalities of Ostrowski-Gruss and μ Cebyev type were established. The main aim of this dissertation is to address the domains of establishing inequalities of Ostrowski-Gruss and μ Cebyev type and their applications in Statistics, Numerical integration and Non-linear analysis. The tools that are used are Peano kernel approach, the most classical and extensively used approach in developing such integral inequalities, Lebesgue and Riemann-Stieltjes integrals, Lebesgue spaces, Korkines identity [52], the classical μ Cebyev functional, Pre-Gruss and Preμ Cebyev inequalities proved in [60]. This dissertation presents some generalized Ostrowski type inequalities. These inequalities are being presented for nearly all types of functions i.e., for higher di¤erentiable functions, bounded functions, absolutely continuous functions, (l; L)Lipschitzian functions, monotonic functions and functions of bounded variations. The inequalities are then applied to composite quadrature rules, special means, probability density functions, expectation of a random variable, beta random variable and to construct iterative methods for solving non-linear equations. The generalizations to the inequalities are obtained by introducing arbitrary parameters in the Peano kernels involved. The parameters can be so adjusted to recapture the previous results as well as to obtain some new estimates of such inequalities. The Ostrowski type inequalities for twice di¤erentiable functions have been extensively addressed by N. S. Barnett et al. and Zheng Liu in [9] and [59]. We have presented some perturbed inequalities of Ostrowski type in Lp (a; b) ; p 1; p =1 which generalize and re ne the results of [9] and [59]. In the past few years, Ostrowski type inequalities are developed for functions in higher spaces i.e., for L-Lipschitzian functions and (l; L)-Lipschitzian functions. We, in here, have obtained Ostrowski type inequality for ndi¤erentiable (l; L)Lipschitzian functions, a generalizations of such inequalities for L-Lipschitzian func-