Inequalities For Gaussian Hypergeometric Function、Generalized Trigonometric Functions And Complete Elliptic Integrals Of The Second Kind

It is well known that F (a, b; c; x) has many important applications in geometryfunction theory, number theory and several other contexts, and many classes of specialfunction in mathematical physics are particular or limiting cases of this function. Inthe recent past, the Gaussian hypergeometric function F (a, b; c; x) and its special caseshave been the subject of intensive research. The frst result in this thesis is that a classof quadratic transformation inequalities for zero-balanced hypergeometric functions isproved.The convex function is a important class of functions, it is fundamental to the re-search in the felds of mathematical economics, optimization theory and method, andthe optimal control theory, which can be used to prove many important inequalities. Inthe latest several years, the Hp,qconcavity and convexity is one of the hottest researchsubject in the theory of convexity, and there exist many important properties and in-equalities. The second main result in this thesis is that we established the necessaryand sufcient conditions for the convexity (concavity) of the generalized trigonomet-ric sine function sinp(x) and the generalized inverse hyperbolic sine function sinhp(x)with respected to H¨older mean. Since it is hard to study the necessary and sufcientconditions for the convexity (concavity) of the generalized trigonometric sine functionsinp(x) and the generalized inverse hyperbolic sine function sinhp(x) with respected toH¨older mean, we turned to study the necessary and sufcient conditions for the con-vexity (concavity) of the generalized inverse trigonometric sine function arcsinp(x) andthe generalized inverse hyperbolic sine function arcsinhp(x) with respected to H¨oldermean, then we got the necessary and sufcient conditions for the convexity (concavi-ty) of the generalized trigonometric sine function sinp(x) and the generalized inversehyperbolic sine function sinhp(x) with respected to H¨older mean.It is well known that complete elliptic integrals have many important applica-tions in the felds of engineering, physics and mathematics containing the subject ofquasiconformal mapping, mean value theory and geometric functional theory. In thelatest years there are many mathematicians studied the complete elliptic integrals fromdiferent views and obtained many properties and their generalizations. The last mainresult is that we presented two upper and lower bounds for Toader mean in terms of arithmetic and contraharmonic means. As applications, several inequalities for thecomplete elliptic integrals of the second kind are derived.