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Sharp L log α L inequalities for conjugate functions

Matts Essén, Daniel F. Shea, Charles S. Stanton (2002)

Annales de l’institut Fourier

We give a method for constructing functions φ and ψ for which H ( x , y ) = φ ( x ) - ψ ( y ) has a specified subharmonic minorant h ( x , y ) . By a theorem of B. Cole, this procedure establishes integral mean inequalities for conjugate functions. We apply this method to deduce sharp inequalities for conjugates of functions in the class L log α L , for - 1 ≤ α < ∞ . In particular, the case α = 1 yields an improvement of Pichorides’ form of Zygmund’s classical inequality for the conjugates of functions in L log L . We also apply the method to produce a new proof of the...

Some results on thin sets in a half plane

Howard Lawrence Jackson (1970)

Annales de l'institut Fourier

When one is restricted to a Stolz domain in a half plane we prove that internal thinness of a set at the origin structly implies minimal thinness there. Furthermore this result extends to the half plane itself. We also work out some relations among the concepts of minimal thinness, semi-thinness and finite logarithmic length. Finally we show that a theorem of Ahlfors and Heins can be improved.

Some simple proofs in holomorphic spectral theory

Graham R. Allan (2007)

Banach Center Publications

This paper gives some very elementary proofs of results of Aupetit, Ransford and others on the variation of the spectral radius of a holomorphic family of elements in a Banach algebra. There is also some brief discussion of a notorious unsolved problem in automatic continuity theory.

Subharmonicity in von Neumann algebras

Thomas Ransford, Michel Valley (2005)

Studia Mathematica

Let ℳ be a von Neumann algebra with unit 1 ℳ . Let τ be a faithful, normal, semifinite trace on ℳ. Given x ∈ ℳ, denote by μ t ( x ) t ≥ 0 the generalized s-numbers of x, defined by μ t ( x ) = inf||xe||: e is a projection in ℳ i with τ ( 1 ℳ - e ) ≤ t (t ≥ 0). We prove that, if D is a complex domain and f:D → ℳ is a holomorphic function, then, for each t ≥ 0, λ ↦ ∫ 0 t l o g μ s ( f ( λ ) ) d s is a subharmonic function on D. This generalizes earlier subharmonicity results of White and Aupetit on the singular values of matrices.

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