Ramanujan Constant Gauss Hypergeometric Function Of The Nature Of Its Application

It is well known that the Gaussian hypergeometric function F (a, b; c; z) plays an im-portant role in the theory of special functions. Many other important special functions arespecial cases of F (a, b; c; z). On the other hand, the Ramanujan constant R(a) plays animportant role in the study of properties of these special functions. In this thesis, we studythe analytic properties of the Gaussian hypergeometric function F (a, b; c; z), R(a) and theirrelated special functions. From these results some inequalities follow. By these results, newkinds of estimates for the bounds of the explicit quasiconformal Schwarz lemma and thesolutions to Ramanujan modular equations are presented. Moreover, several inequalities forthe generalized elliptic integrals with respect to H¨older means are obtained.This thesis is divided into five chapters.In Chapter1, we introduce some concepts, notation and the developing history of theGaussian hypergeometric function F (a, b; c; z) and its related special functions, and Ramanu-jan modular equations.In Chapter2, we show some monotonicity and concavity properties of certain functionsdefined in terms of the Ramanujan constant R(a) and certain elementary functions, fromwhich new properties and inequalities of R(a) follow.In Chapter3, we study the dependence of the Gaussian hypergeometric function onits parameters, and show some properties of certain combinations of the generalized ellipticintegrals Ka, Eaand elementary functions. Some known properties of complete ellipticintegrals are extended to the generalized elliptic integrals Ka, Eaand even to the zero-balanced hypergeometric functions.In Chapter4, we obtain several lower and upper bounds of the complete elliptic inte-gral of the first kind K in terms of trigonometric functions sine and cosines, and those ofthe Hu¨bner upper bound function. By these results, new estimates for the Hersch-Pflugerdistortion function K(r) are presented.In Chapter5, we study the inequalities for the generalized elliptic integrals with respectto Ho¨lder means, and obtain the generalized elliptic integrals of the first and the secondkinds with respect to H¨older mean of order p.