A new sharp double inequality for generalized Heronian, harmonic and power means

For a real number pp, let Mp(a,b)Mp(a,b) denote the usual power mean of order pp of positive real numbers aa and bb. Further, let H=M−1H=M−1 and Heα=αM0+(1−α)M1Heα=αM0+(1−α)M1 for α∈[0,1]α∈[0,1]. We prove that the double mixed-means inequality M−α2(a,b)≤12[H(a,b)+Heα(a,b)]≤Mln2ln4−ln(1−α)(a,b) holds for all α∈[0,1]α∈[0,1] and positive real numbers aa and bb, with equality only for a=ba=b, and that the orders of power means involved in its left-hand and right-hand sides are optimal.