Univalent Functions Research Articles

On possible limit functions on a Fatou component in non-autonomous iteration

Abstract The possibilities for limit functions on a Fatou component for the iteration of a single polynomial or rational function are well understood and quite restricted. In non-autonomous iteration, where one considers compositions of arbitrary polynomials with suitably bounded degrees and coefficients, one should observe a far greater range of behavior. We show this is indeed the case and we exhibit a bounded sequence of quadratic polynomials which has a bounded Fatou component on which one obtains as limit functions every member of the classical Schlicht family of normalized univalent functions on the unit disc. The proof is based on quasiconformal surgery and the use of high iterates of a quadratic polynomial with a Siegel disc which closely approximate the identity on compact subsets. Careful bookkeeping using the hyperbolic metric is required to control the errors in approximating the desired limit functions and ensure that these errors ultimately tend to zero.

Chemical and Acoustical Mixed-Mapping of Geological Materials from Laser-Induced Plasmas: A Comprehensive Approach to Differentiate Mineral Phases.

The acoustic wave produced alongside laser-induced plasmas can be used in conjunction with the recorded atomic spectra of plasma emission to expand the physicochemical information acquired from a single inspection event. Among the most interesting uses of acoustic information is the differentiation of mineral phases with similar optical responses coexisting in geological targets. In addition, laser-induced plasma acoustics (LIPAc) can provide data related to the inspected material's hardness, density, and compactness. In this paper, we present a dual acoustic-optic laser-based strategy for the generation of high-resolution surface images of mineral samples. By combining simultaneous multimodal LIBS (laser-induced breakdown spectroscopy) and LIPAc spectral data from laser-induced plasmas, we explore the mineralogical composition of rocks embedded in resin matrixes to distinguish their chemical composition as well as their crystal phases based on physical changes caused by the different spatial arrangements of the constituent atoms. The multispectral polyhedron created by merging singular optical maps, one per detected elements, and the coincidental acoustic map enhance the distinction between regions present within the matrix of a host rock as compared to the differentiation yielded by each technique when used separately. The chemical information guides the composition of the mineral phases in the host rock. Then, the physical information obtained from acoustics may reinforce the identification of the detected mineral phase, draw the geological history of the inspected section, and showcase possible transformations, mainly of polymorphic nature. To test the combination proposed herein, we also inspected a septarian nodule featuring an ensemble of mineral phases with different origins. Mixed optical and acoustic responses from laser-produced plasmas of this complex sample allowed us to obtain more specific information. This approach constitutes a reliable and high-throughput tool for studying the surface of geological samples, which can substantially supplement well-established techniques for mineralogical analysis such as Raman spectroscopy and X-ray diffraction.

ON THE GENERALIZATION OF SOME CLASSES OF CONVEX IN DIRECTION AND TYPICALLY REAL FUNCTIONS

In the article by M.O. Reade (Duke Math. Journal, 1956) using the condition |arg (f'(z)/ g'(z)) | ≤ γπ/2, where g(z) is a convex function,0≤γ≤1 , a class of functions close-to-convex (almost convex) of order γ is introduced. In our paper, we introduce a subclass of the class of close-to-convex (almost convex) order γ functions satisfying the condition |arg [(1-λzn ) f'(z)]| ≤ γπ/2, which, for different parameter values, gives a number of well-known subclasses of univalent (schlicht) functions. Based on this subclass, a class of close-to-starlike (almost star-shaped) functions is constructed, containing a number of subclasses that have been actively studied by many authors in recent years, as well as a classical class of typically real functions. For this classes exact theorems of distortion (growth) and radii of convexity (starlikeness) are obtained, generalizing previously known results. The case is also considered when the functions of the introduced classes have missing members in the power series expansion. The results obtained are accurate and not only generalize previously known results, but also reveal the properties of a number of new subclasses of univalent (schlicht) functions.

A generalized Fekete-Szegö functional and initial successive coefficients of univalent functions

The paper deals with sharp lower and upper estimations of the functional |c3(f)−λc2(f)2|−μ|c2(f)| in the class S, where λ∈C, μ∈R and cn(f) denotes the n-th coefficient of f. This functional can be regarded as a generalized Fekete-Szegö functional as it coincides with the classical one for μ:=0. The obtained results are applicable to estimate successive coefficients of holomorphic functions relative to univalent functions.

Bounds for the Second Hankel Determinant of a General Subclass of Bi-Univalent Functions

The Hankel determinant, which plays a significant role in the theory of univalent functions, is investigated here in the context of bi-univalent analytic functions. Specifically, this paper is dedicated to deriving an upper-bound estimate for the second-order Hankel determinant for a general subclass of bi-univalent analytic functions that incorporate Gegenbauer polynomials within the unit disk. Through the careful specialization of parameters in our main result, we unveil several novel findings.

Some new facts about linear maps that preserve the orthogonal group

In this article, Albert Wei's result which described the forms of the linear maps that preserve the set of orthogonal matrices is treated and its new proof is provided. Wei's result is simplified by introducing a new description of the singular linear maps acting into the space n by n real matrices that preserve orthogonal matrices. From this description, one can immediately conclude that the range of such singular maps is isometrically isomorphic with the space R n . The proof of the Wei's result that is presented here is based on many new facts about linear maps that preserve the orthogonal group of matrices that are of interest independently of issues considered here.

Differential Subordination and Coefficient Functionals of Univalent Functions Related to $\cos z$

Differential subordination in the complex plane is the generalization of a differential inequality on the real line. In this paper, we consider two subclasses of univalent functions associated with the trigonometric function $\cos z$. Using some properties of the hypergeometric functions, we determine the sharp estimate on the parameter $\beta$ such that the analytic function $p(z)$ satisfying $p(0)=1$, is subordinate to $\cos z$ when the differential expression $p(z)+\beta z (dp(z)/dz)$ is subordinate to the Janowski function. We compute sharp bounds on coefficient functional Hermitian--Toeplitz determinants of the third and the fourth order with an invariance property for such functions. In addition, we estimate bound on Hankel determinants of the second and the third order.

On the Fekete–Szegö Problem for Certain Classes of (γ,δ)-Starlike and (γ,δ)-Convex Functions Related to Quasi-Subordinations

In the present paper, we propose new generalized classes of (p,q)-starlike and (p,q)-convex functions. These classes are introduced by making use of a (p,q)-derivative operator. There are established Fekete–Szegö estimates |a3−μa22| for functions belonging to the newly introduced subclasses. Certain subclasses of analytic univalent functions associated with quasi-subordination are defined.

A General and Comprehensive Subclass of Univalent Functions Associated with Certain Geometric Functions

A General and Comprehensive Subclass of Univalent Functions Associated with Certain Geometric Functions

Hankel and Toeplitz Determinants of Logarithmic Coefficients of Inverse Functions for the Subclass of Starlike Functions with Respect to Symmetric Conjugate Points

This paper focuses on finding the upper bounds of the second Hankel and Toeplitz determinants, whose entries are logarithmic coefficients of inverse functions for a new subclass of starlike functions with respect to symmetric conjugate points associated with the exponential function defined by subordination. Results on initial Taylor coefficients and logarithmic coefficients of inverse functions for a new subclass are also presented. This study may inspire others to focus further to the coefficient functional problems associated with the inverse functions of various classes of univalent functions.

A New Class of Univalent Functions Defined by Differential Operator

In the present work, we submit and study a new class $\mathcal{S}_{\lambda, \alpha, b}^{z, m, t}(\beta, \gamma, \omega, \mu)$ containing analytic univalent functions defined by new differential operator $D_{\lambda, \alpha}^{z, m, t}$ in the open unit disk $E=\{s \in \mathbb{C}:|s|<1\}$. We get some geometric properties, such as, coefficient estimate, growth and distortion theorems, convex set, radii of convexity and starlikeness, weighted mean, arithmetic mean and partial sums for functions belonging to the class $\mathcal{S}_{\lambda, \alpha, b}^{z, m, t}(\beta, \gamma, \omega, \mu)$.

Sharp Coefficient Estimates for Analytic Functions Associated with Lemniscate of Bernoulli

The main purpose of this work is to study the third Hankel determinant for classes of Bernoulli lemniscate-related functions by introducing new subclasses of star-like functions represented by SLλ* and RLλ. In many geometric and physical applications of complex analysis, estimating sharp bounds for problems involving the coefficients of univalent functions is very important because these coefficients describe the fundamental properties of conformal maps. In the present study, we defined sharp bounds for function-coefficient problems belonging to the family of SLλ* and RLλ. Most of the computed bounds are sharp. This study will encourage further research on the sharp bounds of analytical functions related to new image domains.

Coefficient Functionals of Sakaguchi-Type Starlike Functions Involving Caputo-Type Fractional Derivatives Subordinated to the Three-Leaf Function

A challenging part of studying geometric function theory is figuring out the sharp boundaries for coefficient-related problems that crop up in the Taylor–Maclaurin series of univalent functions. Using Caputo-type fractional derivatives to define the families of Sakaguchi-type starlike functions with respect to symmetric points, this article aims to investigate the first three initial coefficient estimates, the bounds for various problems such as Fekete–Szego inequality, and the Zalcman inequalities, by subordinating to the function of the three leaves domain. Fekete–Szego-type inequalities and initial coefficients for functions of the form H−1 and ζH(ζ) and 12logHζζ connected to the three leaves functions are also discussed.

A study On Quasi- Hadamard Product of A class of Analytic Functions Defined by Sălăgean Operator

The purpose of this paper is to study some known properties for certain class of uniformly functions defined in the open unite disk Especially, is to obtain quasi-Hadamard products for certain class of univalent functions which are defined by means of the Sălăgean operator. Relevant connections of the results presented here with those obtained in earlier works are also pointed out.

Simple proofs of certain inequalities with logarithmic coefficients of univalent functions

In this paper, we give simple proofs for the bounds (some of them sharp) of the difference of the moduli of the second and the first logarithmic coefficient for the general class of univalent functions and for the class of convex univalent functions.

Some Classes of Bazilevič-Type Close-to-Convex Functions Involving a New Derivative Operator

In the present paper, we are merging two interesting and well-known classes, namely those of Bazilevič and close-to-convex functions associated with a new derivative operator. We derive coefficient estimates for this broad category of analytic, univalent and bi-univalent functions and draw attention to the Fekete–Szegö inequalities relevant to functions defined within the open unit disk. Additionally, we identify several specific special cases of our results by specializing the parameters.

On a Class of Analytic Univalent Functions Associated with (H-R) Fractional Derivative

In this paper, we introduced a class of analytic functions defined by (H–R) fractional derivative defined in the unit disk, we obtained coefficient bounded, so we obtained some theorems of this class. 2000Mathematics Subject classification: 30C45.

On a subclass of close-to-convex functions

In this paper, we define a new subclass of close-to-convex functions with respect to certain univalent functions studied by [Z. Peng and G. Zhong, Some properties for certain classes of univalent functions defined by differential inequalities, Acta Math. Sci. Ser. B (Engl. Ed.) 37(1) (2017) 69–78]. We find certain examples of functions belong to the class. Further, we give coefficient estimates, distortion theorem and some more results for this new subclass. Finally, we obtain radius of convexity for functions belong to the class.

Certain Properties on Univalent Functions Related to New Linear and Integral Operators

Certain Properties on Univalent Functions Related to New Linear and Integral Operators

Generalized Fractional Integral Operator in a Complex Domain

A new fractional integral operator is used to present a generalized class of analytic functions in a complex domain. The method of definition is based on a Hadamard product of analytic function, which is called convolution product. Then we formulate a convolution integral operator acting on the sub-class of normalized analytic functions. Consequently, we investigate the suggested convolution operator geometrically. Differential subordination inequalities, taking the starlike formula are given. Some consequences of well-known results are illustrated. Keywords: Analytic function, subordination and superordination, univalent function, open unit disk, fractional integral operator, convolution operator, fractional calculus.