Abstract
For any 1 < p < ∞ and any \({X, Y\in \mathbb{R}}\) satisfying \({|X|\leq Y}\) , we determine the optimal constant C p (X,Y) such that the following holds. If F is a holomorphic function on the unit disc satisfying ReF(0) = X and \({||{\rm Re}F||_{L^{p}(\mathbb{T})}=Y}\) , then
This can be regarded as a reverse version of the classical estimates of Riesz and Essén. The proof rests on the exploitation of certain families of special subharmonic functions on the plane.
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Research supported by the NCN Grant DEC-2012/05/B/ST1/00412.
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Osȩkowski, A. Sharp L p Bound for Holomorphic Functions on the Unit Disc. Mediterr. J. Math. 13, 127–139 (2016). https://doi.org/10.1007/s00009-014-0495-x
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DOI: https://doi.org/10.1007/s00009-014-0495-x