We give a method for constructing functions and for which has a specified subharmonic minorant . 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 , for . In particular, the case yields an improvement of Pichorides’ form of Zygmund’s classical inequality for the conjugates of functions in . We also apply the method to produce a new proof of the M. Riesz’s inequality for functions in , , also with sharp constant. All these inequalities are special cases of a general sharp inequality for conjugate functions (cf. Theorem 6).
Nous donnons une méthode pour la construction des fonctions et telles que aî t une minorante sousharmonique spécifiée. D’après un théorème de B. Cole, ce procédé établit des inégalités d’intégrales pour les fonctions conjuguées. Nous appliquons cette méthode pour déduire des inégalités optimales pour les conjuguées des fonctions de la classe , pour . En particulier, le cas procure une amélioration de la version de Pichorides de l’inégalité classique de Zygmund pour les conjuguées des fonctions de . Nous appliquons aussi cette méthode pour obtenir une nouvelle preuve de l’inégalité de M. Riesz pour les fonctions de (), avec meilleure constante. Toutes ces inégalités sont des cas spéciaux d’une inégalité générale et optimale pour les fonctions conjuguées (cf. Théorème 6).
Keywords: conjugate functions, norm estimates, minimal thinness
Mot clés : fonctions conjuguées, estimation des normes, effilement minimal
@article{AIF_2002__52_2_623_0, author = {Ess\'en, Matts and Shea, Daniel F. and Stanton, Charles S.}, title = {Sharp $L\;{\rm log}^\alpha L$ inequalities for conjugate functions}, journal = {Annales de l'Institut Fourier}, pages = {623--659}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {52}, number = {2}, year = {2002}, doi = {10.5802/aif.1896}, zbl = {1053.42012}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.1896/} }
TY - JOUR AU - Essén, Matts AU - Shea, Daniel F. AU - Stanton, Charles S. TI - Sharp $L\;{\rm log}^\alpha L$ inequalities for conjugate functions JO - Annales de l'Institut Fourier PY - 2002 SP - 623 EP - 659 VL - 52 IS - 2 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.1896/ DO - 10.5802/aif.1896 LA - en ID - AIF_2002__52_2_623_0 ER -
%0 Journal Article %A Essén, Matts %A Shea, Daniel F. %A Stanton, Charles S. %T Sharp $L\;{\rm log}^\alpha L$ inequalities for conjugate functions %J Annales de l'Institut Fourier %D 2002 %P 623-659 %V 52 %N 2 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.1896/ %R 10.5802/aif.1896 %G en %F AIF_2002__52_2_623_0
Essén, Matts; Shea, Daniel F.; Stanton, Charles S. Sharp $L\;{\rm log}^\alpha L$ inequalities for conjugate functions. Annales de l'Institut Fourier, Volume 52 (2002) no. 2, pp. 623-659. doi : 10.5802/aif.1896. http://archive.numdam.org/articles/10.5802/aif.1896/
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