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In this survey paper we present the natural application of cer- tain integral inequalities such as, Chebychev's inequality for synchronous and asynchronous mappings, Holder's inequality and Gruss' and Ostrowski's in- equalities for the celebrated Euler's Beta and Gamma functions. Natural applications dealing with some adaptive quadrature formulae which can be deduced from Ostrowski's inequality are also pointed out. This survey paper is an attempt to present the natural application of certain integral inequalities such as, Chebychev's inequality for synchronous and asynchro- nous mappings, Holder's inequality and Gruss' and Ostrowski's inequalities for the celebrated Euler's Beta and Gamma functions. In the rst section, following the well known book on special functions by Larry C. Andrews, we present some fundamental relations and identities for Gamma and Beta functions which will be used frequently in the sequel. The second section is devoted to the applications of some classical integral in- equalities for the particular cases of Beta and Gamma functions in their integral representations. The rst subsection of this is devoted to the applications of Chebychev's inequal- ity for synchronous and asynchronous mappings for Beta and Gamma functions whilst the second subsection is concerned with some functional properties of these functions which can be easily derived by the use of Holder's inequality. Applica- tions of Gruss' integral inequality, which provides a more general approach than Chebychev's inequality, are considered in the last subsection. The third and fourth sections are entirely based on some very recent results on Ostrowski type inequalities developed by Dragomir et al. in (10) - (16). It is shown that Ostrowski's type inequalities can provide general quadrature formulae of the Riemann type for the Beta function. The remainders of the approximation are analyzed and upper bounded using dierent techniques developed for general classes of real mappings. Those sections can be also seen themselves as new and powerful tools in Numerical Analysis and the interested reader can use them for other applications besides those considered here. For a dierent

Inequalities for Beta and Gamma functions via some classical and new integral inequalities