OPTIMAL LEHMER MEAN BOUNDS FOR THE nTH POWER-TYPE TOADER MEANS OF n =-1,1,3

In the article, we prove that lambda(1) = 0, mu(1) = 5/8, lambda(2) = -1/8, mu(2) = 0, lambda(3) = -1 and mu(3) = -7/8 are the best possible parameters such that the double inequalities L-lambda 1(a,b) 0 with a not equal b, and provide new bounds for the complete elliptic integral of the second kind E (r) = integral(pi/2)(0) (1- r(2)sin(2) theta)(1/2)d theta on the interval (0,1), where L-p(a,b) = (a(p+1) + b(p+1))/(a(p)+b(p)) is the p-th Lehmer mean and T-n(a,b) = (2/pi integral(pi/2)(0) root a(n)cos(2)theta+b(n)sin(2)theta d theta)(2/n) is the nth power-type Toader mean.

Automated summary

In this article, we prove the optimality of specific parameters for a double inequality and provide new bounds for a complete elliptic integral.