Optimal inequalities for bounding Toader mean by arithmetic and quadratic means

摘要

In this paper, we present the best possible parameters α(r) and β(r) such that the double inequality [α(r)Ar(a,b)+(1−α(r))Qr(a,b)]1/r<TD[A(a,b),Q(a,b)]<[β(r)Ar(a,b)+(1−β(r))Qr(a,b)]1/r holds for all r ≤ 1 and a, b  0 with a ≠ b, and we provide new bounds for the complete elliptic integral E(r)=∫0π/2(1−r2sin2θ)1/2dθ (r∈(0,2/2)) of the second kind, where TD(a,b)=2π∫0π/2a2cos2θ+b2sin2θdθ, A(a, b) = (a + b)/2 and Q(a,b)=(a2+b2)/2 are the Toader, arithmetic, and quadratic means of a and b, respectively.