Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.31043

Comparison of the Bergman and Szegő kernels on arbitrary bounded pseudoconvex domains with $C^2$ boundary

Xu Xing
View original source ↗Read source document ↗Export metadata ↓

Abstract

On every bounded pseudoconvex domain $Ω\Subset\mathbb C^n$, $n\ge2$, with $C^2$ boundary, we prove $K_Ω/S_Ω\le C_{Ω,\varepsilon} δ_Ω^{-1}|\logδ_Ω|^{1+\varepsilon}$ near the boundary for every $\varepsilon>0$, where $δ_Ω$ denotes the boundary distance. For each positive nondecreasing $H(s)=o(s)$, a smoothly bounded pseudoconvex complete Reinhardt domain in $\mathbb C^2$ shows that the logarithmic factor cannot be replaced by $H(|\logδ_Ω|)$.

Explore related subjects

math.CV

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Xu Xing. 2026-10-22. Comparison of the Bergman and Szegő kernels on arbitrary bounded pseudoconvex domains with $C^2$ boundary. https://arxiv.org/abs/2610.31043

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Fekete Points as Norming Sets for Random Holomorphic Sections

We prove that Fekete configurations associated with a positive Hermitian line bundle over a compact Kähler manifold are asymptotically exact norming sets for random holomorphic sections. For every fixed $a>0$, the ratio of the supremum norm of a degree-$k$ section to its maximum on the degree-$\lceil ak\rceil$ Fekete configuration converges to one in probability. This holds for any background Kähler form: Fekete equidistribution provides enough sampling points near the maximum of the limiting pointwise variance. We obtain polynomial upper bounds for the tail of the ratio and almost-sure norm comparisons. In contrast, when the limiting variance is nonconstant, we construct separated arrays of cardinality comparable to the section-space dimension for which the ratio converges to a constant strictly greater than one. General separated arrays still satisfy probabilistic norming inequalities with degree-independent constants, and every such array is asymptotically exact if and only if the volume form is proportional to $c_1(L,h)^n$. We also prove a discrete $L^2$ norm comparison with exponentially small exceptional probability. The norm comparisons and convergence statements hold in both the Gaussian and spherical ensembles.

math.CV↗
arXiv2026

Sharp coefficient, logarithmic and Schwarzian estimates for $α$-convex functions on the real line

The class $M_α$ of $α$-convex functions, introduced by Mocanu in 1969, interpolates between starlike ($α=0$) and convex ($α=1$) mappings and has traditionally been studied only for $α\geq0$. We show that several of its sharp invariant estimates in fact extend to the entire real line. Our main result characterizes $M_α$ for every real $α\neq0$ through an explicit pointwise inequality, generalizing a theorem of Chuaqui, Duren and Osgood from the convex case, and we determine the sharp value of $β$ for which $M_α\subset C_β$ when $α\geq1$; we also show that the reciprocal inclusion $C_β\subset M_α$ fails for every $α>1$. Building on this, we obtain a sharp Fekete--Szegő inequality, valid for every $α\notin\{0,-1,-1/2\}$, and sharp bounds for the first two logarithmic coefficients of $M_α$, which recover the classical values for starlike and convex functions as special cases. We also determine the order of $M_α$, establish a piecewise sharp bound for its Schwarzian norm together with an explicit quasiconformal extension criterion via the Ahlfors--Weill theorem, and compute the Schwarzian norm of the $α$-Koebe function explicitly for $α=1/n$, $n\in\mathbb N$, verified for $n\leq9$ and numerically supported for $n<50$, leading to a conjecture for the general case. Finally, we propose a conjectural sharp bound for the second Hankel determinant of $M_α$, consistent with the known values $1$ and $1/8$ for the starlike and convex subclasses.

math.CV↗
arXiv2026

Asymptotic Root Dynamics under Repeated and Fractional Integration

Fix a polynomial $P(x)=\sum_{j=0}^m a_jx^j$ of degree $m$, and let $I_kP$ be its $k$-fold antiderivative with all integration constants zero. Borwein, Chen, and Dilcher proved that, after $x=ky$, the nonzero zeros of $I_kP$ converge to the nonzero zeros of $L[P](y)=\sum_j a_j j! y^j$. That convergence determines the leading linear motion but neither the constant offset nor successive differences. We compute the next two terms at a simple nonzero zero $λ$ of $L[P]$. Integer-order and Riemann-Liouville integration reduce, after scaling, to one rational family $c_j(h)=\prod_{\ell\le j}(1+\ell h)^{-1}$ sampled at $h=1/k$ and $h=1/α$. We prove $r_k=kλ+C_P(λ)+D_P(λ)/k+O(k^{-2})$ with explicit $C_P$ and $D_P$. Hence $λ$ is at once the scaled zero, the limit of successive root differences, and the continuous-order velocity, each with an explicit correction; $C_P(λ)$ is the offset of an asymptotic line. For binomials the limiting spacing is a multiple of a gamma-ratio quantity, the Factorial Fraction, and the nonzero reduced zeros form an exact regular polygon at every positive order. For real quadratics the collision order of two branches takes a simple discriminant form. In every degree the collision discriminant carries a universal factor, the one familiar from Laguerre polynomials and supported at negative orders. In the coordinates $(α,x)$ all trajectories lie on one plane curve of degree $m$ with leading form $L[P]$. At a nonzero zero of $L[P]$ of multiplicity $q$, the nearby zeros split on the scale $\sqrt{k}$ like the zeros of the Hermite polynomial $He_q$, and for real $P$ the nonzero zeros stay real at every positive order exactly when $L[P]$ is real-rooted.

math.CV↗