Young-type inequalities and their matrix analogues

Convexity and matrix means

“…x b are convex functions on [0, 1]. Applying Theorem 2.1 using these functions implies the results in [1] and [11]. Now if f is convex on the interval [α, β], we have the following.…”

“…Our results will be stated in a quotient form, comparing between the convex function f = f (x) and its secant (1). Notice that when f is a linear function, f coincides with its secant.…”

“…Then by selecting an appropriate function, we get the results in [1] and [11]. We encourage the reader to look at the operator versions presented in the references [1] and [11], to be able to compare them with our results.…”

“…Let f (x) = x s , 0 ≤ s ≤ 1. Then f is operator concave, hence is operator log-concave [2]. Therefore, for 0 ≤ ν ≤ 1, we have f (I∇ ν X) ≥ f (I)# ν f (X), for X = A − 1 2 BA − 1 2 .…”

“…proved in [1] as a refinement and a reverse of the numerical Young's inequality a# t b ≤ a∇ t b for a, b > 0, where a# t b = a 1−t b t and a∇ t b = (1 − t)a + tb are the weighted geometric and arithmetic means. We refer the reader to [7][8][9]13,14,16] for some fresh refinements and discussion of Young's and related inequalities.…”