Best constant inequalities for conjugate functions
Let \(F=f+i\widetilde f\) be analytic in the unit disc (in the complex plane) with \(\widetilde f(0)=0\). The authors give a survey on inequalities involving the functions \(F\), \(f\) and \(\widetilde f\) starting with work from the 1920s of M. Riesz and A. Zygmund. The emphasis is on their work during the last 15 years on different kinds of best constant inequalities. In all cases, key points are to construct subharmonic minorants to certain real-valued functions in the complex plane. They explain a general method which will give the old results as well as new inequalities. One of the new results is as follows: There exist absolute constants \(B_0\) and \(B_1\) such that \(\|\widetilde f\|_1\leq 2/\pi\int|f|\log(e+|f|)+B_0\int|f|\log\log(e+|f|)+B_1\|f\|_1\). The first constant \(2/\pi\) is sharp.
scientific article; zbMATH DE number 1334867
scientific article