Sharp bounds for Toader mean by arithmetic and centroidal means in exponential type with applications

We establish the optimal lower and upper bounds for the Toader mean by arithmetic and centroidal means in exponential type. That is, for any fixed p ? R, we find the best possible parameters ? = ?(p) and ? = ?(p) such that ?p(a, b; ?) ? T(a, b) ? ?p(a, b; ?) are valid for all a, b > 0 with a ? b, where T(a, b) is the Toader mean of a and b, and ?p(a, b; ?) is the exponential type mean of a and b with ? ? (0, 1). As applications, several new sharp inequalities for the complete elliptic integral of the second kind will be derived.