Sharp Stolarsky mean bounds for the complete elliptic integral of the second kind

Abstract

In the article, we prove that the double inequality 25/16 < E(r)/S 5/2,2 (1, r ) < π/2, holds for all r ∈ (0, 1) with the best possible constants 25/16 and π/2, where r = (1 − r 2 ) 1/2 , E(r) = π/2 0 1 − r 2 sin 2 (t)dt, is the complete elliptic integral of the second kind and, is the Stolarsky mean of a and b.

Sharp inequalities for Toader mean in terms of other bivariate means

“…There are many bounds for the Toader mean in terms of various elementary means, see for example, [6,7,[9][10][11][12][13][14][15]17,[22][23][24][25][26][27][28], and recent papers [16,[29][30][31][32]. In particular, we mention here several interesting results.…”

Monotonicity rule for the quotient of two functions and its application

“…Very recently, the accurate bounds for in terms of the Stolarsky mean were given in [ 44 , 45 ]: where .…”