In the contemporary paper, we introduce new subclasses of analytic and bi-univalent functions involving integral operator based upon Lucas polynomial. Furthermore, find estimates on first two Taylor-Maclaurin coefficients a2 a3 for in these obtain Fekete-Szegö problem subclasses.

We consider a compact Riemannian manifold with a Hermitian line bundle whose curvature is non-degenerate. The Laplacian acting on high tensor powers (the semiclassical regime) of the bundle exhibits a cluster of low-energy states. We demonstrate that the orthogonal projectors onto these states are the Fourier components of an operator with the structure of the Szegö projector, i.e. a Fourier in...

We study the strong asymptotics of orthogonal polynomials with respect to a measure of the type dμ/2π + ∑∞ j=1Ajδ(z− zk), where μ is a positive measure on the unit circle Γ satisfying the Szegö condition and {zj}j=1 are fixed points outside Γ. The masses {Aj}j=1 are positive numbers such that ∑∞ j=1Aj < +∞. Our main result is the explicit strong asymptotic formulas for the corresponding orthogo...