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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 $- 1 \leq α < \infty$ . 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...

Harmonic majorization of $| x_{1} |$ in subsets of $ℝ^{n}$ , $n \geq 2$ .

Eiderman, Vladimir; Essén, Matts — 1996

Annales Academiae Scientiarum Fennicae. Series A I. Mathematica

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Sharp $L \log^{α} L$ inequalities for conjugate functions

Harmonic majorization of $| x_{1} |$ in subsets of $ℝ^{n}$ , $n \geq 2$ .