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

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