Converse growth estimates for ODEs with slowly growing solutions

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.

Recently J. Miao proved that if is a holomorphic function with Hadamard gaps on the open unit disc D then f ∈ Xp if and only if f ∈ Bp if and only if if and if only if where Xp, Bp and denote respectively the class of holomorphic functions on D which satisfy |f′(z)|p(1 − |z|2)p − 1dxdy is a finite measure, a Carleson measure and a little Carleson measure on D. In this paper we give a higher-dimensional version of Miao's result.

This study focuses on deriving Fekete-Szegö inequalities for a specific subclass of univalent functions with complex order, related to quasi-subordination. It investigates the geometric properties of these functions and the conditions for subordination between analytic functions within the open unit disk. The research also discusses generalized starlikeness and convexity as examples of subordination and explores the role of analytic Schwarz functions in this context. New results are provided for superior functions of various orders, contributing to the field of geometric function theory. The findings offer insights into quasi-subordination and its practical applications in pure mathematics, advancing the understanding of function behavior under subordination constraints.

In this article, we introduce a general family of analytic and bi-univalent functions in the open unit disk, which is defined by applying the principle of differential subordination between analytic functions and the Tremblay fractional derivative operator. The upper bounds for the general coefficients | a n | of functions in this subclass are found by using the Faber polynomial expansion. We have thereby generalized and improved some of the previously published results.

Moments of Riesz measures on Poincaré disk and homogeneous tree–A comparative study

The goal of this paper is to investigate some of the features of differential subordination of analytic univalent functions in an open unit disc. In addition, it has shed light on geometric features such as coefficient inequality, Hadamard product qualities, and the Komatu integral operator. Some intriguing results for third-order differential subordination and superordination of analytic univalent functions have been installed. Then, using the convolution of two linear operators, certain results of third order differential subordination involving linear operators were reported. As a result, we use features of the Komatu integral operator to analyze and study third-order subordinations and superordinations in relation to the convolution. Finally, several results for third order differential subordination in the open unit disk using generalized hypergeometric function have been addressed using the convolution operator.

Let Ω denote the class of functions f ( z ) = z + a 2 z 2 + a 3 z 3 + ⋯ belonging to the normalized analytic function class A in the open unit disk U = z : z &lt; 1 , which are bi-univalent in U , that is, both the function f and its inverse f − 1 are univalent in U . In this paper, we introduce and investigate two new subclasses of the function class Ω of bi-univalent functions defined in the open unit disc U , which are associated with a new differential operator of analytic functions involving binomial series. Furthermore, we find estimates on the Taylor–Maclaurin coefficients | a 2 | and | a 3 | for functions in these new subclasses. Several (known or new) consequences of the results are also pointed out.

A normalized analytic function f defined on the open unit disc D is called Ma-Minda convex if 1 + zf′′(z)/f′(z) is subordinate to the function φ. For 0 ⩽ α ⩽ β, the Kaplan class K(α, β) of type α and β consists of normalized analytic functions of the form p^{α}g defined on D where p with p(0) = 1 is an analytic function taking values in the right half-plane and g is an analytic function with g(0) = 1 satisfying Re(zg′(z)/g(z)) &gt; (α − β)/2. For functions f with f′ ∈ K(α, β), we obtain the radius of Ma-Minda convexity for various choices of φ. The radius of lemniscate convexity, lune convexity, nephroid convexity, exponential convexity and several other radius estimates are examined. The results obtained are sharp.

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.

Neighborhoods of certain analytic functions

Let X be a set of analytic functions on the open unit disk D, and let phi be an analytic function on D such that phi(D) is contained in D and f |-> f o phi takes X into itself. We present conditions on X ensuring that if f |-> f o phi is invertible on X, then phi is an automorphism of D, and we derive a similar result for mappings of the form f |-> psi.(f o phi), where psi is some analytic function on D. We obtain as corollaries of this purely function-theoretic work, new results concerning invertibility of composition operators and weighted composition operators on Banach spaces of analytic functions such as S^p and the weighted Hardy spaces H^2(beta).

The boundedness and the compactness of the two integral operators;, where is an analytic function on the open unit disk in the complex plane, on the Zygmund space are studied.

We study the following two integral operators equation image where g is an analytic function on the open unit disk in the complex plane. The boundedness and compactness of these two operators between the α ‐Bloch space Bα and the Besov space are discussed in this paper (© 2009 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Let f be an analytic function in the open unit disk that assumes the values 0 and 1 at least once and an unequal number of rimes in a disk of radius 1 10-5 entered at the origin. We prove that fmust assume the value 0 or 1 elsewhere in the unit disk. This result is motivated by a control engineering design problem which is also presented.

1. IntroductionLet D = fz : jzj< 1gbe the unit disk in the complex plane and letH(D) denote the space of all analytic functions on D. Recall that a positiveBorel measure on Dis called a Carleson measure if there is a positive niteconstant Ksuch that(1) (S(I)) KjIjfor all arcs Iˆ@D, where jIjdenotes the normalized arc length of I(so thatj@Dj= 1) and S(I) is the Carleson square de ned byS(I) = fz: 1 j Ij<jzj<1; z=jzj2Ig:Carleson measures are ubiquitous in the study of function-theoretic operatortheory. A fundamental property of Carleson measures due to L. Carlesonaddresses the issue of when the inclusion map is bounded from the Hardyspace H