Convexity and matrix means

In this article we present some mean inequalities for convex functions that lead to some generalized inequalities treating the arithmetic, geometric and harmonic means for numbers and matrices. Our first main inequality will be(ντ)λ≤((1−ν)f(0)+νf(1))λ−fλ(ν)((1−τ)f(0)+τf(1))λ−fλ(τ)≤(1−ν1−τ)λ, for the convex function f, when λ≥1 and 0<ν≤τ<1. Moreover, when λ=1, the inequality will be valid for operator convex functions. Then by selecting an appropriate convex function, we obtain certain matrix inequalities. In particular, we obtain several mixed mean inequalities for operators using real and operator convexity. Our discussion will lead to new multiplicative refinements and reverses of the Heinz and Hölder inequalities for matrices, new and refined trace and determinant inequalities. The significance of this work is its general treatment, where convexity is the only needed property.