Some Inequalities Involving Interpolations Between Arithmetic and Geometric Mean

In this article, we mainly study the interpolations between arithmetic mean and geometric mean—power mean, Heron mean and Heinz mean. First, we obtain the improvement and reverse improvement of arithmetic-power mean inequalities by the convexity of the function. We show that the proof of Heron mean inequality due to Yang and Ren: [Some results of Heron mean and Young’s inequalities, J. Inequal. Appl. 2018 (2018), paper no, 172], is not substantial. In addition, we also obtain Heron-Heinz mean inequalities for t ∈ ℝ. Further corresponding operator versions and generalizations are also established.