Revisiting the Schwarz Lemma for the Pentablock: Counterexamples and Sharp Interpolation Criteria

Abstract

Alshehri and Lykova (\emph{J. Geom. Anal.} \textbf{33} (2023)) formulated a Schwarz lemma for the pentablock $P$, giving a criterion for the existence of an analytic map from the unit disc $\mathbb D$ into the closed pentablock $\overline P$ that sends the origin to the origin and a prescribed point $λ_0\in\mathbb D$ to a given point $x_0\in\overline P$. We show that the sufficiency direction of their criterion is false. We identify the underlying error as the extension to the open unit disc of a Fejér--Riesz identity that is valid only on the unit circle. We establish a corrected necessary and sufficient interpolation criterion when the last two components are prescribed by a fixed rational $Γ$-inner function and resolve the special case $s_0=0$. Using the Poisson integral representation of the associated outer spectral factor, we obtain a sharp comparison between the corrected bound and the bound proposed by Alshehri and Lykova, with a precise characterization of equality. We further express the logarithmic difference between these bounds as a nonnegative Green potential, thereby quantifying the discrepancy in terms of the derivative of the first coordinate of the prescribed $Γ$-inner function. Finally, we formulate a variational problem over rational $Γ$-inner interpolants satisfying the prescribed two-point conditions. We show that the supremum of the associated outer-factor quantities determines the maximal admissible modulus of the first coordinate among pentablock-valued interpolants whose last two components belong to this class. Equivalently, this extremal problem reduces to minimizing the corresponding Green potential. We conclude by discussing the remaining problem of characterizing unrestricted Schwarz interpolation for the pentablock and its connection with the Lempert function.