Special Functions: Approximations and Bounds
نویسنده
چکیده
There are a number of results that provide bounds for integrals of products of functions. The main techniques that shall be employed in the current article involve the Steffensen inequality and a variety of bounds related to the Čebyšev functional. There have been some developments in both of these in the recent past with which the current author has been involved. These have been put to fruitful use in a variety of areas of applied mathematics including quadrature rules, in the approximation of integral transforms, as well as in applied probability problems (see [31], [22] and [11]. This article is a review of these developments and some new results are also presented. It is intended that in the current article the techniques will be utilised to obtain useful bounds for special functions. The methodologies will be demonstrated through obtaining bounds for the Bessel function of the first kind, the Beta function and the Zeta function. It is instructive to introduce some techniques for approximating and bounding integrals of the product of functions. We first present inequalities due to Steffensen and then review bounds for the Čebyšev functional.
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