Coefficient Estimate of Toeplitz Determinant for a Certain Class of Close to Convex Functions

Let S denote the class of analytic and univalent functions in D, where D is defined as unit disk and having the Taylor representation form of S. We will determine the estimation for the Toeplitz determinants where the elements are the Taylor coefficients of the class close-to-convex functions in S.

Fourth Toeplitz Determinants for Starlike Functions Defined by Using the Sine Function

In this article, we aim to study the upper bounds of the fourth Toeplitz determinant T 4 2 for the function class S s ∗ , which are connected with the sine function.

The Third-Order Hermitian Toeplitz Determinant for Alpha-Convex Functions

Sharp lower and upper bounds of the second- and third-order Hermitian Toeplitz determinants for the class of α-convex functions were found. The symmetry properties of the arithmetic mean underlying the definition of α-convexity and the symmetry properties of Hermitian matrices were used.

Applications in random matrix theory of a PIII’ τ-function sequence from Okamoto’s Hamiltonian formulation

We consider the singular linear statistic of the Laguerre unitary ensemble (LUE) consisting of the sum of the reciprocal of the eigenvalues. It is observed that the exponential generating function for this statistic can be written as a Toeplitz determinant with entries given in terms of particular [Formula: see text] Bessel functions. Earlier studies have identified the same determinant, but with the [Formula: see text] Bessel functions replaced by [Formula: see text] Bessel functions, as relating to the hard edge scaling limit of a generalized gap probability for the LUE, in the case of non-negative integer Laguerre parameter. We show that the Toeplitz determinant formed from an arbitrary linear combination of these two Bessel functions occurs as a [Formula: see text]-function sequence in Okamoto’s Hamiltonian formulation of Painlevé III[Formula: see text], and consequently the logarithmic derivative of both Toeplitz determinants satisfies the same [Formula: see text]-form Painlevé III[Formula: see text] differential equation, giving an explanation of a fact which can be observed from earlier results. In addition, some insights into the relationship between this characterization of the generating function, and its characterization in the [Formula: see text] limit, both with the Laguerre parameter [Formula: see text] fixed, and with [Formula: see text] (this latter circumstance being relevant to an application to the distribution of the Wigner time delay statistic), are given.

The fourth-order Hermitian Toeplitz determinant for convex functions

The sharp bounds for the fourth-order Hermitian Toeplitz determinant over the class of convex functions are computed.