Study Of Holomorphic Functions Research Articles

Boundary values of Hankel and Toeplitz determinants for [formula omitted]-convex functions

The study of holomorphic functions has been recently extended through the application of diverse techniques, among which quantum calculus stands out due to its wide-ranging applications across various scientific disciplines. In this context, we introduce a novel q-differential operator defined via the generalized binomial series, which leads to the derivation of new classes of quantum-convex (q-convex) functions. Several specific instances within these classes were explored in detail. Consequently, the boundary values of the Hankel determinants associated with these functions were analyzed. All graphical representations and computational analyses were performed using Mathematica 12.0.•These classes are defined by utilizing a new q-differential operator.•The coefficient values |ai|(i=2,3,4) are investigated.•Toeplitz determinants, such as the second T2(2) and the third T3(1) order inequalities, are calculated.

Classification of Holomorphic Functions as Pólya Vector Fields via Differential Geometry

We review Pólya vector fields associated to holomorphic functions as an important pedagogical tool for making the complex integral understandable to the students, briefly mentioning its use in other dimensions. Techniques of differential geometry are then used to refine the study of holomorphic functions from a metric (Riemannian), affine differential or differential viewpoint. We prove that the only nontrivial holomorphic functions, whose Pólya vector field is torse-forming in the cannonical geometry of the plane, are the special Möbius transformations of the form f(z)=b(z+d)−1. We define and characterize several types of affine connections, related to the parallelism of Pólya vector fields. We suggest a program for the classification of holomorphic functions, via these connections, based on the various indices of nullity of their curvature and torsion tensor fields.

Variation of Neumann and Green functions under homotopies of the boundary

We develop variational formulas for certain Neumann and Green functions of multiply connected planar domains, valid for any smooth homotopy of the boundary. The modified Neumann and Green functions under consideration arise in the study of holomorphic functions with single-valued primitives.

Polynomial extreme elements of the closed unit ball ofH∞(B)

Properties of homogeneous polynomials which are extreme points of the closed unit ball of the space H ∞(B) of bounded analytic functions on the open unit ball in C n are studied. A complete characterization is given in the case of degree 2 and applied to the study of holomorphic functions on B having positive real part. A sufficient condition for extremality is also given.

Cauchy and poisson intergral of ultredistributions

Let C be an open convex cone in such that [Cbar] does not contain any entire straight line and let be the corresponding rube domain. Generalized Cauchy and Poisson integrals of ultradistributions of Beurling type and of Roumieu type both of which generalize the Schwartz distributions are defined and studied for appropriate values of s The Caucthy integral is shown to be holomorphic in T to satisfy a growth property, and to obtain an ultradistribution boundary value. Which leads to a holomorphic decomposition dresult for the ultradistributions. The Poission integral is shown to have an ultradistribution boundary value. The analysis of this paper lays the foundation and is necessary for future research concerning a study of holomorphic functions in tubes which are characterixed by either pointwise or norm growths, their boundary values, their recovery in terms of generalized integrals, and other associated properties.