Extremal plurisubharmonic functions in $C^N$

Józef Siciak

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The logarithmic capacity in $C^{n}$

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Extremal sections of complex $l_{p}$ -balls, 0 < p ≤ 2

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We study the extremal volume of central hyperplane sections of complex n-dimensional $l_{p}$ -balls with 0 < p ≤ 2. We show that the minimum corresponds to hyperplanes orthogonal to vectors ξ = (ξ¹,...,ξⁿ) ∈ ℂⁿ with |ξ¹| = ... = |ξⁿ|, and the maximum corresponds to hyperplanes orthogonal to vectors with only one non-zero coordinate.

Siciak’s extremal function via Bernstein and Markov constants for compact sets in $ℂ^{N}$

Leokadia Bialas-Ciez (2012)

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The paper is concerned with the best constants in the Bernstein and Markov inequalities on a compact set $E \subset ℂ^{N}$ . We give some basic properties of these constants and we prove that two extremal-like functions defined in terms of the Bernstein constants are plurisubharmonic and very close to the Siciak extremal function $Φ_{E}$ . Moreover, we show that one of these extremal-like functions is equal to $Φ_{E}$ if E is a nonpluripolar set with $l i m_{n \to \infty} M ₙ {(E)}^{1 / n} = 1$ where $M ₙ (E) : = s u p {| | | g r a d P | | |}_{E} / {| | P | |}_{E}$ , the supremum is taken over all polynomials P of N variables...

On some extremal problems in classes of holomorphic functions $S_{p}^{} (ρ)$ , $S_{p}^{)} (ρ)$ , $Σ_{p}^{} (ρ)$ , $M C (ρ)$ at additional condition

B. Platynowicz (1980)

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Universally divergent Fourier series via Landau's extremal functions

Gerd Herzog, Peer Chr. Kunstmann (2015)

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We prove the existence of functions $f \in A (𝔻)$ , the Fourier series of which being universally divergent on countable subsets of $𝕋 = \partial 𝔻$ . The proof is based on a uniform estimate of the Taylor polynomials of Landau’s extremal functions on $𝕋 ∖ {1}$ .

Robin functions and extremal functions

T. Bloom, N. Levenberg, S. Ma'u (2003)

Annales Polonici Mathematici

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Given a compact set $K \subset ℂ^{N}$ , for each positive integer n, let $V^{(n)} (z)$ = $V_{K}^{(n)} (z)$ := sup $1 / (d e g p) V_{p (K)} (p (z))$ : p holomorphic polynomial, 1 ≤ deg p ≤ n. These “extremal-like” functions $V_{K}^{(n)}$ are essentially one-variable in nature and always increase to the “true” several-variable (Siciak) extremal function, $V_{K} (z)$ := max[0, sup1/(deg p) log|p(z)|: p holomorphic polynomial, ${| | p | |}_{K} \leq 1$ ]. Our main result is that if K is regular, then all of the functions $V_{K}^{(n)}$ are continuous; and their associated Robin functions $ϱ_{V_{K}^{(n)}} (z) : = l i m s u p_{| λ | \to \infty} [V_{K}^{(n)} (λ z) - l o g (| λ |)]$ increase to $ϱ_{K} : = ϱ_{V_{K}}$ for all z outside a pluripolar...

The postage stamp problem and arithmetic in base $r$

Amitabha Tripathi (2008)

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Let $h, k$ be fixed positive integers, and let $A$ be any set of positive integers. Let $h A : = {a_{1} + a_{2} + \dots + a_{r} : a_{i} \in A, r \leq h}$ denote the set of all integers representable as a sum of no more than $h$ elements of $A$ , and let $n (h, A)$ denote the largest integer $n$ such that ${1, 2, ..., n} \subseteq h A$ . Let $n (h, k) : = {max}_{A} : n (h, A)$ , where the maximum is taken over all sets $A$ with $k$ elements. We determine $n (h, A)$ when the elements of $A$ are in geometric progression. In particular, this results in the evaluation of $n (h, 2)$ and yields surprisingly sharp lower bounds for $n (h, k)$ , particularly for $k = 3$ .

Sharp estimation of the coefficients of bounded univalent functions close to identity

Lucjan Siewierski

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CONTENTSIntroduction...............................................................................................................................................................................5Definitions and notation.........................................................................................................................................................7The main result........................................................................................................................................................................91....

The gap theorems for some extremal submanifolds in a unit sphere

Xi Guo and Lan Wu (2015)

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Let $M$ be an $n$ -dimensional submanifold in the unit sphere $S^{n + p}$ , we call $M$ a $k$ -extremal submanifold if it is a critical point of the functional $\int_{M} ρ^{2 k} d v$ . In this paper, we can study gap phenomenon for these submanifolds.

Displaying similar documents to “Extremal plurisubharmonic functions in C N ”

Displaying similar documents to “Extremal plurisubharmonic functions in $C^{N}$ ”