A new class of operator monotone functions via operator means

Abstract

Abstract. In this paper, we obtain a new class of functions, which is developed via the Hermite-Hadamard inequality for convex functions. The well-known one-one correspondence between the class of operator monotone functions and operator connections declares that the obtained class represents the weighted logarithmic means. We shall also consider weighted identric mean and some relationships between various operator means. Among many things, we extended the weighted arithmetic-geometric operator mean inequality as≤ A∇ t B and A# t B ≤ AI t B ≤ A∇ t B involving the considered operator means.

More results on weighted means

“…As given in (8), the distance between the geometric mean and the arithmetic mean is not so tight. Therefore the proof for the following relations are not so difficult.…”

Further mean inequalities via some integral inequalities

“…Remark 2.2. It was also possible to show (21) for scalar numbers a, b > 0 and then deduce (21) for operator arguments by using (15) and the standard techniques of Functional Calculus, which consist to multiply at left and at right by A 1/2 the related formulas obtained via (15). The proof of the following lemma explains more this latter point.…”

“…hold for any a, b > 0 and λ ∈ [0, 1]. For the construction of some weighted logarithmic means as well as some weighted identric means we can consult [9,21,26].…”

On a class of new means including the generalized Schwab-Borchardt mean

“…with ( 6.3 ) as weighted logarithmic mean, according to Definition 2.1 . This weighted logarithmic mean is simpler than those introduced in [ 16 ] and [ 17 ]. Another L -weighted mean will be introduced by analogy with those of T and M .…”