Sharp bounds for neuman-sándor mean in terms of the convex combination of quadratic and first Seiffert means

Abstract

In this article, we prove that the double inequality αP(a,b)+1(1−α)Q(a,b) < M(a,b) < βP(a,b)+(1−β)Q(a,b) holds for any a,b > 0 with a ≠ b if and only if α ≥ 1/2 and β ≤ [π(2log(1+2)−1)]/[(2π−2)log(1+2)]=03595…, where M(a,b), Q(a,b), and P(a,b) are the Neuman-Sándor, quadratic, and first Seiffert means of a and b, respectively.