MSC 2010: 33E12, 30A10, 30D15, 30E15We consider some families of 3-index generalizations of the classical Mittag-Le²er functions and study the behaviour of these functions in domains of the complex plane. First, some inequalities in the complex plane and on its compact subsets are obtained. We also prove an asymptotic formula for the case of "large" values of the indices of these functions. Similar results have also been obtained by the author for the classical Bessel functions and their Wright's...

In 1945 the first author introduced the classes $H^p\left(\alpha \right)$, 1 ≤ p<∞, α > -1, of holomorphic functions in the unit disk with finite integral (1) ∬ |f(ζ)|p (1-|ζ|²)α dξ dη < ∞ (ζ=ξ+iη) and established the following integral formula for $f\in H^p\left(\alpha \right)$: (2) f(z) = (α+1)/π ∬ f(ζ) ((1-|ζ|²)α)/((1-zζ̅)2+α) dξdη, z∈ . We have established that the analogues of the integral representation (2) hold for holomorphic functions in Ω from the classes $L^p(\Omega ;{\left[K\left(w\right)\right]}^{\alpha }dm\left(w\right))$, where: 1) $\Omega =w=(w₁,...,w_n)\in {ℂ}^n:Imw₁>{\sum }_{k=2}^n\left|w_k\right|²$, $K\left(w\right)=Imw₁-{\sum }_{k=2}^n\left|w_k\right|²$; 2) Ω is the matrix domain consisting of those complex m...

For α ≥ 0 let ${ℱ}_{\alpha }$ denote the class of functions defined for |z| < 1 by integrating $1/{(1-xz)}^{\alpha }$ if α > 0, and log(1/(1-xz)) if α = 0, against a complex measure on |x| = 1. We study families of starlike functions where zf’(z)/f(z) ranges over a parabola with given focus and vertex. We prove a number of properties of these functions, among others that they are bounded and that they belong to ${ℱ}_0$. In general, it is only known that bounded starlike functions belong to ${ℱ}_{\alpha }$ for α > 0.

An exploratory study is performed to investigate the use of a time-dependent discrete adjoint methodology for design optimization of a high-lift wing configuration augmented with an active flow control system. The location and blowing parameters associated with a series of jet actuation orifices are used as design variables. In addition, a geometric parameterization scheme is developed to provide a compact set of design variables describing the wing...

The article puts up the problem of finding harmonic functions on a domain D, which for simplicity is a disk with the origin as a boundary point, continuous on D, and with arbitrary asymptotic harmonic expansion. To solve it, in the space Ac(D) of harmonic functions on D, continuous on D and with aymptotic harmonic expansion at 0, we define the topology Tc for which it is a Fréchet space. There we define the linear functionals which map each function to the coefficients of its asymptotic harmonic...

The extremal solutions to the Nevanlinna Pick problem are studied. If there is more than one solution, Nevanlinna showed that all extremal solutions are inner functions. With some extra information on the interpolation data we find that the extremal solutions are Blaschke products whose zeroes form a finite union of interpolating sequences.

This paper deals with an interpolation problem in the open unit disc $𝔻$ of the complex plane. We characterize the sequences in a Stolz angle of $𝔻$, verifying that the bounded sequences are interpolated on them by a certain class of not bounded holomorphic functions on $𝔻$, but very close to the bounded ones. We prove that these interpolating sequences are also uniformly separated, as in the case of the interpolation by bounded holomorphic functions.

We give a pure complex variable proof of a theorem by Ismail and Stanton and apply this result in the field of integer-valued entire functions. Our proof rests on a very general interpolation result for entire functions.

The aim of the paper is to get an estimation of the error of the general interpolation rule for functions which are real valued on the interval $[-a,a]$, $a\in (0,1)$, have a holomorphic extension on the unit circle and are quadratic integrable on the boundary of it. The obtained estimate does not depend on the derivatives of the function to be interpolated. The optimal interpolation formula with mutually different nodes is constructed and an error estimate as well as the rate of convergence are obtained. The general...