Inequalities and bounds for generalized complete elliptic integrals
The generalized complete elliptic integrals of the first and second kind are defined by \[ {\mathcal K}_{a,b,c}=\frac{B(a,b)}{2} F(a,b;c;k^2), \] and \[ {\mathcal E}_{a,b,c}=\frac{B(a,b)}{2} F(a-1,b;c;k^2), \] where \(B(a,b)=\int_0^1 u^{a-1} (1-u)^{b-1}\,du\) and \(F(a,b;c;z)\) is the Gauss hypergeometric function. The author gives computable bounds for these integrals as well as for their differences and products. Besides other results, he shows that \({\mathcal K}_{a,b,c}\) and \({\mathcal E}_{a,b,c}\) are strictly log-convex functions in \(a\).
generalized complete elliptic integrals of the first and second kind
scientific article; zbMATH DE number 5804452
scientific article