Some inequalities for operator (p, h)-convex functions

Abstract

Abstract. Let p be a positive number and h a function on R + satisfying h(xy) ≥ h(x)h(y) for any x, y ∈ R + . A non-negative continuous function f onholds for all positive semidefinite matrices A, B of order n with spectra in K, and for any α ∈ (0, 1).In this paper, we study properties of operator (p, h)-convex functions and prove the Jensen, Hansen-Pedersen type inequalities for them. We also give some equivalent conditions for a function to become an operator (p, h)-convex. In applications, we obtain Choi-Davis-Jensen type inequality for operator (p, h)-convex functions and a relation between operator (p, h)-convex functions with operator monotone functions.

MϕA-h-Convexity and Hermite-Hadamard Type Inequalities

“…If M ϕ = H, then the Jensen type inequality for HA-h-convex function is given in [2]. If M ϕ = M p and h(t) = t, then the Jensen inequality for M p A-convex was proved in [4]. If M ϕ ∈ {A, G, H}, then results from this section are given in [1].…”

“…HG-h-convexity investigated in [10] and AG-hconvexity or log-h-convexity in [9]. AM p -h-convexity or (h, p)-convexity is described in [6] while some properties of M p A-h-convex functions are given in [4]. Also, we have to mention article [1] devoted to the MN-h-convexity where M, N ∈ {A, G, H}.…”

Some Inequalities for Operator (p, h)‐Convex Function

“…Lemma 6 (see [22]). Let φ be a unital positive linear map on B(H), A a positive operator in H, and f an operator (p, h)-convex function on R + such that f(0) � 0.…”

“…en, f is the operator monotone if and only if it is operator concave. Definition 1 (see [22]). Let p > 0 and K, J ⊆ R + , (0, 1) ⊂ J, and h: J ⟶ R + be a nonnegative function nonidentical to 0.…”

“…for any positive operators A, B with spectra in K and λ ∈ (0, 1). is class contains several well-known classes of nonnegative operator convex function, operator P-class function, operator Q-class function, operator s-convex function, and operator h-convex functions on K. We can see some results operator (p, h)-convex functions by Dinh and Khue in [22].…”