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.

We show that functions whose derivatives lie in a half-plane are preserved under the Pommerenke, Chandra-Singh, Libera, Alexander and Bernardi integral transforms. We determine precisely how these transforms act on such functions. We prove that if the derivative of a function lies in a convex region then the derivative of its Pommerenke, Chandra-Singh, Libera, Alexander and Bernardi transforms lie in a strictly smaller convex region which can be determined. We also consider iterates of these transforms....

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...

In this work we consider the Dunkl operator on the complex plane, defined by $${𝒟}_{k}f\left(z\right)=\frac{d}{dz}f\left(z\right)+k\frac{f\left(z\right)-f(-z)}{z},k\ge 0.$$ We define a convolution product associated with ${𝒟}_k$ denoted ${*}_k$ and we study the integro-differential-difference equations of the type $\mu {*}_{k}f={\sum }_{n=0}^{\infty }{a}_{n,k}{𝒟}_k^{n}f$, where $\left(a_{n,k}\right)$ is a sequence of complex numbers and $\mu$ is a measure over the real line. We show that many of these equations provide representations for particular classes of entire functions of exponential type.

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.