منابع مشابه
Convexity Theorems
M(a, fi) S M'(ah ft)Jf^(a,, ft). This is M. Riesz's fundamental theorem. (See M. Riesz [5] and a different proof in Paley [4] ; see also a generalization of the theorem in L. C. Young [ i l ] . ) M. Riesz's argument proved the convexity only in the triangle O ^ a ^ l , 0 ^ / 3 ^ 1 , a+f3*zl. The extension to the whole quadrant is due to Thorin [9]. We shall not give the proof of the theorem her...
متن کاملSome Operator and Trace Function Convexity Theorems
We consider trace functions (A,B) 7→ Tr[(ABA)] where A and B are positive n × n matrices and ask when these functions are convex or concave. We also consider operator convexity/concavity of ABA and convexity/concavity of the closely related trace functional Tr[ABAC]. The concavity questions are completely resolved, thereby settling cases left open by Hiai; the convexity questions are settled in...
متن کاملConvexity of quasi-entropy type functions: Lieb’s and Ando’s convexity theorems revisited
Given a positive function f on (0,∞) and a non-zero real parameter θ, we consider a function Iθ f (A,B,X) = TrX (f(LAR −1 B )RB) −θ(X) in three matrices A,B > 0 and X. This generalizes the notion of monotone metrics on positive definite matrices, and in the literature θ = ±1 has been typical. We investigate how operator monotony of f is sufficient and/or necessary for joint convexity/concavity ...
متن کاملTurán theorems and convexity invariants for directed graphs
This paper is motivated by the desire to evaluate certain classical convexity invariants (specifically, the Helly and Radon numbers) in the context of transitive closure of arcs in the complete digraph. To do so, it is necessary to establish several new Turán type results for digraphs and characterize the associated extremal digraphs. In the case of the Radon number, we establish the following ...
متن کاملConvexity of Coverings of Projective Varieties and Vanishing Theorems
Let X be a projective manifold, ρ : X̃ → X its universal covering and ρ∗ : V ect(X) → V ect(X̃) the pullback map for isomorphism classes of vector bundles. This article makes the connection between the properties of the pullback map ρ∗ and the properties of the function theory on X̃. Our approach motivates a weakened version of the Shafarevich conjecture: the universal covering X̃ of a projective m...
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ژورنال
عنوان ژورنال: Bulletin of the American Mathematical Society
سال: 1949
ISSN: 0002-9904
DOI: 10.1090/s0002-9904-1949-09277-7