We prove sharp inequalities for determinants of Toeplitz operators and twisted Laplace operators on the two-sphere, generalizing the Moser-Trudinger-Onofri inequality. In particular a sharp version of conjectures of Gillet-Soule and Fang motivated by Arakelov geometry is obtained; applications to SU(2)-invariant determinantal random point processes on the two-sphere are also discussed. The inequalities are obtained as corollaries of a general theorem about the maximizers of a certain non-local functional defined on the space of all positively curved Hermitian metrics on an ample line bundle over a compact complex manifold. This functional is an adjoint version, introduced by Berndtsson, of Donaldson's L-functional and generalizes the Ding-Tian functional whose critical points are Kahler-Einstein metrics. In particular, new proofs of some results in Kahler geometry are also obtained, including a lower bound on Mabuchi's K-energy and the uniqueness result for Kahler-Einstein metrics on Fano manifolds of Bando-Mabuchi.
We consider the $\dbar-$ equation in $\C^1$ in classes of functions with Gaussian decay at infinity. We prove that if the right-hand side of the equation is majorated by $\exp(-q|z|^2)$, with some positive $q$, together with derivatives up to some order, and is orthogonal, as a distribution, to all analytical polynomials, then there exists a solution with decays, together with derivatives, as $\exp(-q'|z|^2)$, for any $q'<q/e$. This result carries over to the $\dbar$-equation in classes of distributions, again, with Gaussian decay at infinity, in some precisely defined sense. The properties of…
On a smooth domain in complex n space of finite D'Angelo q-type at a point, an effective upper bound for the vanishing order of the Levi determinant $\text{coeff}\{\partial r \wedge \dbar r \wedge (\partial \dbar r)^{n-q}\}$ at that point is given in terms of the D'Angelo q-type, the dimension of the space n, and q itself. The argument uses Catlin's notion of a boundary system as well as techniques pioneered by John D'Angelo.
In this paper, we derive the sharp bounds of Toeplitz determinants for a class of holomorphic mappings on the bounded starlike circular domain $Ω$ in $\mathbb{C}^n$, which extend certain known bounds for various subclasses of normalized analytic univalent functions in the unit disk to higher dimensions.
In this study, we derive the sharp bounds of certain Toeplitz determinants whose entries are the coefficients of holomorphic functions belonging to a class defined on the unit disk $\mathbb{U}$. Further, these results are extended to a class of holomorphic functions on the unit ball in a complex Banach space and on the unit polydisc in $\mathbb{C}^n$. The obtained results provide the bounds of Toeplitz determinants for various subclasses of normalized univalent functions in higher dimensions.
Sharp upper and lower bounds for the second and third order Hermitian-Toepilitz determinants are obtained for some generalized subclasses of starlike and convex functions. Applications of these results are also discussed for several widely known classes.
In this study, we deal with the sharp bounds of certain Toeplitz determinants whose entries are the logarithmic coefficients of analytic univalent functions $f$ such that the quantity $z f'(z)/f(z)$ takes values in a specific domain lying in the right half plane. The established results provide the bounds for the classes of starlike and convex functions, as well as various of their subclasses.
A starlike function $f$ is characterized by the quantity $zf'(z)/f(z)$ lying in the right half-plane. This paper deals with sharp bounds for certain symmetric Toeplitz determinants whose entries are the coefficients of the functions $f$ for which the quantity $zf'(z)/f(z)$ takes values in certain specific subset in the right half-plane. The results obtained include several new special cases and some known results.
Let $\mathcal{S}$ denote the class of analytic and univalent functions in $\mathbb{D}:=\{z\in\mathbb{C}:\, |z|<1\}$ of the form $f(z)= z+\sum_{n=2}^{\infty}a_n z^n$. In this paper, we determine sharp estimates for the Toeplitz determinants whose elements are the Taylor coefficients of functions in $\mathcal{S}$ and its certain subclasses. We also discuss similar problems for typically real functions.
For the classes of analytic functions $f$ defined on the unit disk satisfying $$\frac{z {f}'(z)}{f(z) - f(-z)} \prec φ(z) \quad \text{and} \quad \frac{(2 z {f}'(z))'}{(f(z) - f(-z))'} \prec φ(z),$$ denoted by $\mathcal{S}^*_s(φ)$ and $\mathcal{C}_s(φ)$ respectively, the sharp bound of the $n^{th}$ Taylor coefficients are known for $n=2,$ $3$ and $4$. In this paper, we obtain the sharp bound of the fifth coefficient. Additionally, the sharp lower and upper estimates of the third order Hermitian Toeplitz determinant for the functions belonging to these classes are determined. The applications of…
We study the determinants of Toeplitz matrices as the size of the matrices tends to infinity, in the particular case where the symbol has two jump discontinuities and tends to zero on an arc of the unit circle at a sufficiently fast rate. We generalize an asymptotic expansion by Widom [22], which was known for symbols supported on an arc. We highlight applications of our results in the Circular Unitary Ensemble and in the study of Fredholm determinants associated to the sine kernel.
We obtain an asymptotic formula for $n\times n$ Toeplitz determinants as $n\to \infty$, for real valued symbols with any fixed number of Fisher-Hartwig singularities, which is uniform with respect to the location of the singularities. As an application, we prove a conjecture by Fyodorov and Keating regarding moments of averages of the characteristic polynomial of the Circular Unitary Ensemble. In addition, we obtain an asymptotic formula regarding the momentum of impenetrable bosons in one dimension with periodic boundary conditions.
We consider canonical determinantal random point processes with N particles on a compact Riemann surface X defined with respect to the constant curvature metric. In the higher genus (hyperbolic) cases these point processes may be defined in terms of automorphic forms. We establish strong exponential concentration of measure type properties involving Dirichlet norms of linear statistics. This gives an optimal Central Limit Theorem (CLT), saying that the fluctuations of the corresponding empirical measures converge, in the large N limt, towards the Laplacian of the Gaussian free field on X in th…
We study asymptotic behavior for determinants of $n\times n$ Toeplitz matrices corresponding to symbols with two Fisher-Hartwig singularities at the distance $2t\ge0$ from each other on the unit circle. We obtain large $n$ asymptotics which are uniform for $0<t<t_0$ where $t_0$ is fixed. They describe the transition as $t\to 0$ between the asymptotic regimes of 2 singularities and 1 singularity. The asymptotics involve a particular solution to the Painlevé V equation. We obtain small and large argument expansions of this solution. As applications of our results we prove a conjecture of Dyson o…
This paper examines the coefficient problems for the class of semigroup generators, a topic in complex dynamics that has recently been studied in context of geometric function theory. Further, sharp bounds of coefficient functional such as second order Hankel determinant, third order Toeplitz and Hermitian-Toeplitz determinants are derived. Additionally, the sharp growth estimates and the bounds of difference of successive coefficients are determined, which are used to prove the Bohr and the Bohr-Rogosinski phenomenon for the class of semigroup generators.
