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. μ Keywords: Ostrowski inequality, Grüss inequality, Cebyš ev inequality, Numer- ical 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 math- ematics 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. Mitrinovi ́c, J. E. Peμcari ́c 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. Mitrinovi ́c, J. E. Peμcari ́c 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 μ functions namely, inequalities of Grüss [46] and Cebyš ev type (see [64], p. 297). iiiThese 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 μ applications of inequalities of Ostrowski-Grüss and Cebyš ev type were established. The main aim of this dissertation is to address the domains of establishing μ inequalities of Ostrowski-Grüss and Cebyš ev type and their applications in Statis- tics, Numerical integration and Non-linear analysis. The tools that are used are Peano kernel approach, the most classical and extensively used approach in devel- oping such integral inequalities, Lebesgue and Riemann-Stieltjes integrals, Lebesgue μ spaces, Korkine’s identity [52], the classical Cebyš ev functional, Pre-Grüss and Pre- μ Cebyš 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 vari- able 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 ex- tensively addressed by N. S. Barnett et al. and Zheng Liu in [9] and [59]. We have presented some perturbed inequalities of Ostrowski type in L p (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 n- di¤erentiable (l; L)- Lipschitzian functions, a generalizations of such inequalities for L-Lipschitzian func- ivtions and (l; L)-Lipschitzian functions. The ...rst inequality of Ostrowski-Grüss type was presented by S. S. Dragomir and S. Wang in [39]. In this dissertation, some improved and generalized Ostrowski- Grüss type inequalities are further generalized for the ...rst and twice di¤erentiable functions in L 2 (a; b). Some generalizations of Ostrowski-Grüss type inequality in terms of upper and lower bounds of the ...rst and twice di¤erentiable functions are also given. The inequalities are then applied to probability density functions, special means, generalized beta random variable and composite quadrature rules. μ In the recent past, many researchers have used Cebyš ev type functionals to μ obtain some new product inequalities of Ostrowski-, Cebyš ev-, and Grüss type. We, in here, have also taken into account this domain to present some generalizations and improvements of such inequalities. The generalizations are obtained for ...rst di¤erentiable absolutely continuous functions with ...rst derivatives in L p (a; b) ; p > 1 and for twice di¤erentiable functions in L 1 (a; b). A product inequality is also given for monotonic non-decreasing functions. The inequalities are then applied to the expectation of a random variable. μ In [3], G. A. Anastassiou has extended Cebyš ev-Grüss type inequalities on R N over spherical shells and balls. We have extended this inequality for n-dimensional Euclidean space over spherical shells and balls on L p [a; b] ; p > 1. Some weighted Ostrowski type inequalities for a random variable whose proba- bility density functions belong to fL p (a; b) ; p = 1; p > 1g are presented as weighted extensions of the results of [10] and [33]. Ostrowski type inequalities are also applied to obtain various tight bounds for the random variables de...ned on a ...nite intervals whose probability density functions belong to fL p (a; b) ; p = 1; p > 1g. This dissertation also describes the applications of specially derived Ostrowski type inequalities to obtain some two-step and three-step iterative methods for solv- ing non-linear equations. Some Ostrowski type inequalities for Newton-Cotes formulae are also presented in a generalized or optimal manner to obtain one-point, two-point and four-point Newton-Cotes formulae of open as well as closed type. The results presented here extend various inequalities of Ostrowski type upto their year of publication.