Complete Elliptic Integrals Of The Nature And Mean Inequality And Its Application

This thesis consists of four chapters.In chapterâ… , the author introduces the definitions, main properties, some known resultsand related problems for the Gauss hypergeometric functions, elliptic integrals and somemean value functions.In chapterâ…¡, the author give a positive answer to a conjecture which involving the ellipticintegral of the first kind, and present the best possible generalized Seiffert's mean boundsfor the elliptic integral of the second kind.In chapterâ…¢, the author obtain some monotonicity, convexity and concavity propertiesfor the generalized elliptic integrals and their combinations. As applications, some newanalytic inequalities are found.In chapterâ…£, the author give the best possible geometric combinations bounds of arith-metic and logarithmic means for the first Seiffert's mean, and establish a sharp doubleinequality between the identric and generalized Heronian power means.