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

This thesis consists of four chapters.In chapterⅠ, the author introduces the definitions, main properties, some known resultsand related problems for the Gauss hypergeometric functions, elliptic integrals and somemean value functions.In chapterⅡ, 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 chapterⅢ, the author obtain some monotonicity, convexity and concavity propertiesfor the generalized elliptic integrals and their combinations. As applications, some newanalytic inequalities are found.In chapterⅣ, 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.