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We prove that for any given upper semicontinuous function $varphi$ on an open subset $E$ of $mathbb C^nsetminus{0}$, such that the complex cone generated by $E$ minus the origin is connected, the homogeneous Siciak-Zaharyuta function with the weight $varphi$ on $E$, can be represented as an envelope of a disc functional.
In this paper, we present an alternative and elementary proof of a sharp version of the classical boundary Schwarz lemma by Frolova et al. with initial proof via analytic semigroup approach and Julia-Caratheodory theorem for univalent holomorphic sel
We study the smoothness of the Siciak-Zaharjuta extremal function associated to a convex body in $mathbb{R}^2$. We also prove a formula relating the complex equilibrium measure of a convex body in $mathbb{R}^n$ to that of its Robin indicatrix. The main tool we use are extremal ellipses.
Let $mathsf M$ and $mathsf M _{mathsf S}$ respectively denote the Hardy-Littlewood maximal operator with respect to cubes and the strong maximal operator on $mathbb{R}^n$, and let $w$ be a nonnegative locally integrable function on $mathbb{R}^n$. We
This is an auxiliary note to [12]. To be precise, here we have gathered the proofs of all the statements in [12, Section 5] that happen to have points of contact with techniques recently developed in Chousionis-Pratt [5] and Chunaev [6].
We give an in-depth analysis of a 1-parameter family of electrified droplets first described in D. Khavinson et. al. (2005). We also investigate a technique for searching for new solutions to the droplet equation, and rederive via this technique a 1-