Sharp L p Bound for Holomorphic Functions on the Unit Disc

Abstract

For any 1 < p < ∞ and any \({X, Y\in \mathbb{R}}\) satisfying \({|X|\leq Y}\) , we determine the optimal constant Cp(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 $$||F||_{L^p(\mathbb{T})}\geq C_p(X,Y).$$ This can be regarded as a reverse version of the classical estimates of Riesz and Essen. The proof rests on the exploitation of certain families of special subharmonic functions on the plane.