Song-Liang Qiu

Sharp bounds for generalized elliptic integrals of the first kind

In this paper, we prove that the double inequality begin{equation*} 1+alpha r^2<frac{mathcal{K}_{a}(r)}{sin(pi a)log(e^{R(a)/2}/r)}<1+beta r^2 end{equation*} holds for all

Optimal combinations bounds of root-square and arithmetic means for Toader mean

ain (0, 1/2]

Concavity of the complete elliptic integrals of the second kind with respect to Hölder means

and

Generalized convexity and inequalities involving special functions

rin (0, 1)

Generalized Hersch–Pfluger distortion function and complete elliptic integrals

if and only if

The bounds of the solutions to generalized modular equations

alphaleq pi/[R(a)sin(pi a)]-1

Inequalities for the generalized elliptic integrals with respect to Hölder means

and

Some properties of the difference between the Ramanujan constant and beta function

betageq a(1-a)

Distortion theorems of plane quasiconformal mappings

, where

Monotonicity properties and inequalities for the generalized elliptic integral of the first kind

r=sqrt{1-r^2}