In this paper we survey several recent results that highlight an interplay between a relatively new class of quasiseparable matrices and polynomials. Quasiseparable matrices generalize two classical matrix classes, Jacobi (tridiagonal) matrices and unitary Hessenberg matrices that are known to correspond to real orthogonal polynomials and Szegö polynomials, resp. The latter two polynomial famil...

The Eneström-Kakeya theorem (the absolute value of any zero of a0+a1x+· · ·+akx, ai ∈ R is at most max( a0 a1 , a1 a2 , . . . , ak−1 ak ) ) implies |νj| ≤ k . It was shown by Szegö [8] that the numbers νj k cluster around the simple closed curve Γ = {z : |ze1−z| = 1, |z| ≤ 1} as k → ∞ and conversely each point of the Szegö curve is a limit point of the normalized zeros. Moreover Szegö [8] and a...

Several different subclasses of the bi-univalent function class Σ were introduced and studied by many authors using distribution series like Pascal distribution, Poisson Borel Mittag-Leffler-type Miller–Ross-Type Distribution. In present paper, making use Bell we introduce investigate a new family GΣt(x,p,q,λ,β,γ) normalized functions in open unit disk U, which are associated with Horadam polyn...

Abstract The concentration-compactness principle of Lions type in Euclidean space relies on the Pólya-Szegö inequality, which is not available non-Euclidean settings. first proof setting, such as Heisenberg group, was given by Li et al. using a symmetrization-free nonsmooth truncation argument. In this article, we study second-order Adams’ inequality Lorentz-Sobolev <m:math xmlns:m="http://www....

in this work, we obtain the fekete-szegö inequalities for the class $p_{sigma }left( lambda ,phi right) $ of bi-univalent functions. the results presented in this paper improve the recent work of prema and keerthi [11].

A change to the Szegö matrix recurrence relation, satisfied by orthonormal polynomials on the unit circle, gives rise to a linear map by the action of matrices belonging to the group SU(1; 1). The companion factorization of such matrices, via 2-order linear homogeneous difference equations, provides a compact representation of the orthogonal polynomial on the circle. Moreover, an isomorphism SU...

We apply Forelli-Rudin construction and Nakazawa’s hodograph transformation to prove a graph theoretic closed formula for invariant theoretic coefficients in the asymptotic expansion of the Szegö kernel on strictly pseudoconvex complete Reinhardt domains. The formula provides a structural analogy between the asymptotic expansion of the Bergman and Szegö kernels. It can be used to effectively co...

A change to the Szegö matrix recurrence relation, satisfied by orthonormal polynomials on the unit circle, gives rise to a linear map by the action of matrices belonging to the group SU(1; 1). The companion factorization of such matrices, via 2-order linear homogeneous difference equations, provides a compact representation of the orthogonal polynomial on the circle. Moreover, an isomorphism SU...