A proof of Khavinson's conjecture in R4

Abstract

The paper deals with an extremal problem for bounded harmonic functions in the unit ball of R4. We solve the generalized Khavinson problem in R4. This precise problem was formulated by Kresin and Maz'ya for harmonic functions in the unit ball and in the half‐space of Rn. We find the optimal pointwise estimates for the norm of the gradient of bounded real‐valued harmonic functions.

Khavinson conjecture for hyperbolic harmonic functions on the unit ball

“…See [16,Chapter 6] for various Khavinson-type extremal problems for harmonic functions. Kalaj [8] showed that the conjecture for n = 4 and Melentijević [19] confirmed the conjecture in R 3 . Marković [18] solved the Khavinson problem for points near the boundary of the unit ball.…”

A proof of the Khavinson conjecture

“…Just like in [6], [11] and [12], we shall prove the following equivalent formulation of Theorem 1. For fixed ρ ∈ [0, 1), the function α −→ C(ρe 1 , ℓ α ) attains its maximum on [0, π/2] at α = 0.…”

“…Just like that in [6], [11] and [12], our proof is based on an observation of M. Marković in [11] that the generalized Khavinson conjecture is equivalent to the statement that the optimization problem…”

Schwarz Lemmas for mappings satisfying Poisson’s equation

“…The definition of the classical hypergeometric function F (a, b; c; z) is given in Section 2. Next, we state a Schwarz-Pick type lemma, which is a generalization of[20, Theorem 1.3], [28, Theorem 2.12] and [19, Corollary 2.2].…”