Automorphisms and generalized projections on spaces of analytic functions

Hadamard type operators on spaces of real analytic functions in several variables

By using the most generalized gamma function (parametric gamma function, or p-gamma function), we present the most generalized Rabotnov function, called the p-Rabotnov function. Consequently, new fractal–fractional differential and integral operators of a complex variable in an open unit disk are defined and investigated analytically and geometrically. We address some inequalities involving the generalized fractal–fractional integral operator in some spaces of analytic functions. A novel complex fractal–fractional integral transform (CFFIT) is presented. A normalization of the proposed CFFIT is observed in the open unit disk. Examples are illustrated for power series of analytic functions.

We characterize continuity and compactness of the Volterra integral operator $T_g$ with the non-constant analytic symbol $g$ between certain weighted Fréchet or (LB)-spaces of analytic functions on the open unit disc, which arise as projective (resp. inductive) limits of intersections (resp. unions) of Bergman spaces of order $1<p<\infty$ induced by the standard radial weight $(1-|z|^2)^\alpha$ for $0<\alpha<\infty$. Motivated from the earlier results obtained for weighted Bergman spaces of standard weight, we also establish several results concerning the spectrum of the Volterra operators acting on the weighted Bergman Fréchet space $A^p_{\alpha+}$, and acting on the weighted Bergman (LB)-space $A^p_{\alpha-}$.

The spectrum of the Cesaro operator $$\mathsf {C}$$ , which is always continuous (but never compact) when acting on the classical Korenblum space and other related weighted Frechet or (LB) spaces of analytic functions on the open unit disc, is completely determined. It turns out that such spaces are always Schwartz but, with the exception of the Korenblum space, never nuclear. Some consequences concerning the mean ergodicity of $$\mathsf {C}$$ are deduced.

A study of a special family of analytic functions at infinity

Two spaces of analytic functions are considered, each comprised of functions analytic on the open disk NR(0) of radius R (0 < R < +∞ ) centred at the origin. The first space consists of all analytic functions on NR(0) topologized according to the metric of uniform convergence on compact sets. As the second space we allow any Fréchet space of analytic functions on NR(0) for which the topology is stronger than that induced by . Our objective is then to present a scheme for constructing simultaneous automorphisms on and .

Let {mathbb {D}} be the unit disc in {mathbb {C}}. If mu is a finite positive Borel measure on the interval [0, 1) and f is an analytic function in {mathbb {D}}, f(z)=sum _{n=0}^infty a_nz^n (zin {mathbb {D}}), we define Cμ(f)(z)=∑n=0∞μn∑k=0nakzn,z∈D,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} {\\mathcal {C}}_\\mu (f)(z)= \\sum _{n=0}^\\infty \\mu _n\\left( \\sum _{k=0}^na_k\\right) z^n,\\quad z\\in {\\mathbb {D}}, \\end{aligned}$$\\end{document}where, for nge 0, mu _n denotes the n-th moment of the measure mu , that is, mu _n=int _{[0, 1)}t^ndmu (t). In this way, {mathcal {C}}_mu becomes a linear operator defined on the space {mathrm{Hol}}({mathbb {D}}) of all analytic functions in {mathbb {D}}. We study the action of the operators {mathcal {C}}_mu on distinct spaces of analytic functions in {mathbb {D}}, such as the Hardy spaces H^p, the weighted Bergman spaces A^p_alpha , BMOA, and the Bloch space {mathcal {B}}.

Let the differential system where D is the diagonal matrix and the elements αij (Z) of the matrix A(z) are analytic functions in some neighborhood of the closed unit disc. In this paper under several assumptions with respect to the constant matrices and the diagonal D, it is proved that the conjugate system of (1) has exactly k(p − l) linearly independent solutions in the product space: where is the usual Hilbert space of analytic functions in the open unit disc.

The Geometric Theory of Analytic Functions was initially developed for the space of functions that are analytic in the unit disk. The convexity and starlikeness of functions are the first geometric ideas considered in this theory. We can notice a symmetry between the subjects considered in the space of analytic functions and those in the space of harmonic functions. In the presented paper, we consider the starlikeness of functions in the space of meromorphic harmonic functions.

We provide new sharp decomposition theorems for multifunctional Bergman spaces in the unit ball and bounded pseudoconvex domains with smooth boundary expanding known results from the unit ball. Namely we prove that mΠ j=1 jjfj jjXj ≍ jjf1 : : : fmjj Ap for various (Xj) spaces of analytic functions in bounded pseudoconvex domains with smooth boundary where f; fj ; j = 1; : : : ;m are analytic functions and where Ap ; 0 < p < 1; > �����1 is a Bergman space. This in particular also extend in various directions a known theorem on atomic decomposition of Bergman Ap spaces.

A quantitative majorization-subordination result of Goluzin and Tao Shah for univalent functions is generalized to H", the linear invariant family of locally univalent functions of finite order or.If f(z) is subordinate to F(z) in the open unit disc, /'(0) > 0, and F(z) is in U" 1.65 < a < oo, then f'(z) is majorized by F'(z) in \z\ < (a + 1) -(a2 + 2a)vi.The result is sharp.I. Introduction.Let <S denote the set of all normalized analytic univalent functions in the open unit disc D. Let/(z), F(z) and <p(z) be analytic in |z| < r.We say that/(z) is majorized byLet tla be the set of all locally univalent (f'(z) = 0) analytic functions in D with order < a which are of the form f(z) = z A-.The family U" is known as the universal linear invariant family of order a [4].A concise summary and introduction to properties of linear invariant families which relate to the following material is contained in [1].The present paper concludes the proof of results announced in [1].Majorization-subordination theory begins with Biernacki who showed in 1936 that if/(z) is subordinate in D to F(z) (F(z) G <S), then/(z) is majorized by F(z) in |z| < 1/4.In the succeeding years Goluzin, Tao Shah, Lewandowski and MacGregor examined various related problems but always under the stipulation that the dominant function F(z) is in <S (for greater detail see [1]).In 1951 Goluzin showed that if f(z) is majorized by a univalent function F(z), then f'(z) would be majorized by F'(z) in |z| < 0.12.He conjectured that majorization would always occur for jz| < 3 -y and this was proved by Tao Shah in 1958.In this paper we show that the result is actually true for functions in 11. and obtain the sharp radius of majorization as a + 1 -(a2 + 2a)^2 for 1.65 < o < oo.This yields 3 -y for the case a = 2.Our investigation shows that the important datum for majorization-subordination theory is not univalence, but the order of a linear invariant family.In particular, many classically derived estimates for univalent functions are true for functions of infinite valence.

LetVbe an arbitrary system of weights on an open connected subsetGofℂN(N≥1)and letB(E)be the Banach algebra of all bounded linear operators on a Banach spaceE. LetHVb(G,E)andHV0(G,E)be the weighted locally convex spaces of vector-valued analytic functions. In this survey, we present a development of the theory of multiplication operators and composition operators from classical spaces of analytic functionsH(G)to the weighted spaces of analytic functionsHVb(G,E)andHV0(G,E).

On Newton's interpolation polynomials

Let f be an analytic function in the unit disc 𝔻. The Volterra integral operator If is defined as follows: If(h)(z)=∫0zf(w)h'(w)dw,h∈H(𝔻),z∈𝔻. In this paper, we compute the norm of If on some analytic function spaces.

We correct a mistake found in our the article “Cesàro-like operators acting on spaces of analytic functions”, published in Anal. Math. Phys. 12 (2022), no. 2, Paper No. 51, 29 pp.