A Majorant Problem
نویسنده
چکیده
Let f(z) akzk a 0 be analytlc in the unlt disc. Any k=O o Inflnlte complex vector e (eo,et,e2 ) such that lekl 1, k 0,1,2 induces a function re(Z) akekZk whlch is still analytic k=O In the unit disc. In this paper we study the problem of maximizing the p-means: over all possible vectors e and for values of r close to 0 and for all p<2. k It is proved that a maxlmlzlng function Is f,{z} -laoi + . laklZ k=l and that r could be taken to be any positive number which Is smaller than the radius of the largest disc centered at the orlgln which can be Inscrlbed in the zero sets of f Thls problem is originated by a well known maJorant problem for Fourier coefficients that was studied by Hardy and Llttlewood. One consequence of our paper is that for p < 2 the extremal function for the Hardy-Llttlewood problem should be -[ao[ + r. laklz k=1 We also give some appllcatlons to derive some sharp Inequalltles for the classes of Schllcht functions and of functions of positive real part.
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