Sharp approximations for the complete elliptic integrals of the second kind by one-parameter means

Let E(r) be the complete elliptic integrals of the second kind and Jp(a,b) be the one-parameter mean of two distinct positive numbers a and b defined byJp(a,b)=pp+1ap+1−bp+1ap−bp for p≠−1,0. In this study using the recurrence method with the sign rule of a special power series, we prove that the double inequality7π22J2/(π−2)(1,r′)<2πE(r)