Optimal Lehmer mean bounds for the $n$th power-type Toader means of n=-1,1,3

In the article, we prove that λ1 = 0 , μ1 = 5/8 , λ2 = −1/8 , μ2 = 0 , λ3 = −1 and μ3 = −7/8 are the best possible parameters such that the double inequalities Lλ1 (a,b) < T3(a,b) < Lμ1 (a,b), Lλ2 (a,b) < T1(a,b) < Lμ2 (a,b), Lλ3 (a,b) < T−1(a,b) < Lμ3 (a,b) hold for a,b > 0 with a = b , and provide new bounds for the complete elliptic integral of the second kind E (r) = ∫ π/2 0 (1− r2 sin2 θ )1/2dθ on the interval (0,1) , where Lp(a,b) = (ap+1 + bp+1)/(ap +bp) is the p -th Lehmer mean and Tn(a,b) = ( 2 π ∫ π/2 0 √ an cos2 θ +bn sin2 θdθ )2/n is the n th power-type Toader mean. Mathematics subject classification (2020): 26E60, 33E05.