COSAM » Events » 2016 » February » Algebra/Linear Algebra Seminar

Linear Algebra/Algebra Seminar
Time: Feb 16, 2016 (04:00 PM)
Location: Parker Hall 244


Speaker: Trung Hoa Dinh

Title: Some inequalities for operator \((p,h)\)-convex functions.

Abstract: Let \(p\) be a positive number and \(h\) a function on \(\mathbb{R}^+\) satisfying \(h(xy) \ge h(x) h(y)\) for any \(x, y \in \mathbb{R}\). A non-negative function \(f\) on \(K (\subset \mathbb{R}^+)\) is said to be \(\it operator\) \((p,h)\)-convex if 

\(f ([\alpha A^p + (1-\alpha)B^p]^{1/p}) \leq h(\alpha)f(A) +h(1-\alpha)f(B)\)

holds for all self-adjoint matrices \(A, B\) of order \(n\) with spectra in \(K\), and for any \(\alpha \in (0,1)\). 

In this talk, we study properties of \((p,h)\)-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.

Last updated: 02/15/2016