Starlike Functions Research Articles

New Properties of Analytic Functions

In the present paper, we consider the class A¯ of functions f(z) of the form f(z)=z+∑k=1∞a1+k3z1+k3 that are analytic in the open unit disc U. If a1+k3=0 for k≠3n(n=1,2,3,⋯), then f(z) is given by f(z)=z+∑k=2∞akzk. For such functions f(z)∈A¯, some interesting properties for subordinations and strongly starlike functions are given. Also, some interesting examples for the results are shown.

On Uniformly Starlike Functions with Respect to Symmetrical Points Involving the Mittag-Leffler Function and the Lambert Series

The aim of this paper is to define the linear operator based on the generalized Mittag-Leffler function and the Lambert series. By using this operator, we introduce a new subclass of β-uniformly starlike functions ΤJ(αi). Further, we obtain coefficient estimates, convex linear combinations, and radii of close-to-convexity, starlikeness, and convexity for functions f∈ΤJ(αi). In addition, we investigate the inclusion conditions of the Hadamard product and the integral transform. Finally, we determine the second Hankel inequality for functions belonging to this subclass.

Geometric properties of an integral operator associated with Mittag-Leffler functions

The main object of this paper is to introduce a new integral operator associated with Mittag-Leffler function. Further, we obtain some sufficient condition for this integral operator belonging to certain classes of starlike functions.

Sharp coefficient inequalities of starlike functions connected with secant hyperbolic function

This article comprises the study of class SE∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{S}_{E}^{\\ast }$\\end{document} that represents the class of normalized analytic functions f satisfying ςf′(z)/f(ς)≺sech(ς)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}${\\varsigma \\mathsf{f}}^{\\prime }(z)/\\mathsf{f}( {\\varsigma })\\prec \\sec h ( \\varsigma ) $\\end{document}. The geometry of functions of class SE∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{S}_{E}^{\\ast }$\\end{document} is star-shaped, which is confined in the symmetric domain of a secant hyperbolic function. We find sharp coefficient results and sharp Hankel determinants of order two and three for functions in the class SE∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{S}_{E}^{\\ast }$\\end{document}. We also investigate the same sharp results for inverse coefficients.

Ozaki-Type Bi-Close-to-Convex and Bi-Concave Functions Involving a Modified Caputo’s Fractional Operator Linked with a Three-Leaf Function

The focus of the present work is on the establishment and investigation of the coefficient estimates of two new subclasses of bi-close-to-convex functions and bi-concave functions; these are of an Ozaki type and involve a modified Caputo’s fractional operator that is associated with three-leaf functions in the open unit disc. The classes are defined using the notion of subordination based on the previously established fractional integral operators and classes of starlike functions associated with a three-leaf function. For functions in these classes, the Fekete-Szegö inequalities and the initial coefficients, |a2| and |a3|, are discussed. Several new implications of the findings are also highlighted as corollaries.

Estimating the Second Order Hankel Determinant for the Subclass of Bi-Close-to-Convex Function of Complex Order

The article aims to estimate the coefficient bounds for the second Hankel determinant by using the class of bi-close-to-convex functions of a complex order in the open unit disk. Making the direct application of Carathéodory function along with the closely related properties of starlike functions, we obtain the upper bound for the second Hankel determinant via certain subclass of bi-close-to-convex functions of complex order. The study discusses the maximization of the second Hankel determinant in both conventional graph and analytic methods. Moreover, we explore and modify some results on the study of bi-close-to-convex functions and its second degree Hankel determinant. At the end of the article, we remark on improvement in the earlier work of some researchers and discover a better value than the one they obtained.

محدد هنكل الثاني لفئة معينة من الدوال المعرفة بواسطة مؤثر تفاضلي

The objective of this paper is to obtain an upper bound of second Hankel determinant for a class of functions M_(α,β,λ,δ)^k (φ) defined in the open unit disc U by using the differential operator〖 D〗_(α,β,λ,δ)^k f(z). In addition, we give particular values to the parameters A,B and k to study special cases of the results of this article. The class M_(α,β,λ,δ)^k (φ) and the differential operator〖 D〗_(α,β,λ,δ)^k f(z). Key words: differential operator, Second Hankel determinant, Starlike functions, Subordination property

Estimates for a class of analytic and univalent functions connected with quasi-subordination

Geometric Function Theory, an active field of studied that has its roots in complex analysis, has gained an impressive attention from many researchers. This occurs largely because it deals with the study of geometric properties of analytic (and univalent) functions where many of its applications spread across many fields of mathematics, mathematical physics and engineering. Notable areas of application include conformal mappings, special functions, orthogonal polynomials, fluid flows in physics and engineering designs. The investigations in this paper are on a subclass of analytic and univalent functions defined in the unit disk Ω and denoted by Qaq(m). The definition of the new class encompasses some well-known subclasses of analytic and univalent functions such as the classes of starlike functions, Yamaguchi functions, and Ma-Minda functions. Two key mathematical principles involved in the definition of the class are the principles of Taylor’s series and quasi-subordination. Some of the investigations carried out on functions f ∈ Qaq(m) are however, the upper estimates for some initial bounds, the solution to the well-known Fekete-Szego problem and the upper estimate for a Hankel determinant.

Differential subordination of certain class of starlike functions

This paper presents several results concerning second and third-order differential subordination for the class S e ∗ := { f ∈ A : z f ′ ( z ) / f ( z ) ≺ e z } , which represents the class of starlike functions associated with exponential function.

Certain radii problems for 𝓢∗(ψ) and special functions

Abstract In Geometric function theory, the Ma-Minda class of starlike functions has a unique place as it unifies various subclasses of starlike functions. There has been an vivid interplay between special functions and their geometric properties, like starlikeness. In this article, we establish certain special function’s radius of Ma-Minda starlikness. As an application, we obtain conditions on parameters for these special functions to be in the Ma-Minda class. Further, we focus on certain convolution properties for the Ma-Minda class that are not done so far, and study their applications in radius problem. Finally, we prove a variational problem of Goluzin, namely, the region of variability for the Ma-Minda class. Our results simplify and generalize the already-known ones.

Properties of a Class of Analytic Functions Influenced by Multiplicative Calculus

Motivated by the notion of multiplicative calculus, more precisely multiplicative derivatives, we used the concept of subordination to create a new class of starlike functions. Because we attempted to operate within the existing framework of the design of analytic functions, a number of restrictions, which are in fact strong constraints, have been placed. We redefined our new class of functions using the three-parameter Mittag–Leffler function (Srivastava–Tomovski generalization of the Mittag–Leffler function), in order to increase the study’s adaptability. Coefficient estimates and their Fekete-Szegő inequalities are our main results. We have included a couple of examples to show the closure and inclusion properties of our defined class. Further, interesting bounds of logarithmic coefficients and their corresponding Fekete–Szegő functionals have also been obtained.

Geometric Properties of Normalized Galué Type Struve Function

The field of geometric function theory has thoroughly investigated starlike functions concerning symmetric points. The main objective of this work is to derive certain geometric properties, such as the starlikeness of order δ, convexity of order δ, k-starlikeness, k-uniform convexity, lemniscate starlikeness and convexity, exponential starlikeness and convexity, and pre-starlikeness for the Galué type Struve function (GTSF). Furthermore, the conditions for GTSF belonging to the Hardy space are also derived. The results obtained in this work generalize several results available in the literature.

Inverse logarithmic coefficient bounds for starlike functions subordinated to the exponential functions

In recent years, many subclasses of univalent functions, directly or not directly related to the exponential functions, have been introduced and studied. In this paper, we consider the class of Se∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{S}^{\\ast}_{e}$\\end{document} for which zf′(z)/f(z)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$zf^{\\prime}(z)/f(z)$\\end{document} is subordinate to ez\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$e^{z}$\\end{document} in the open unit disk. The classic concept of Hankel determinant is generalized by replacing the inverse logarithmic coefficient of functions belonging to certain subclasses of univalent functions. In particular, we obtain the best possible bounds for the second Hankel determinant of logarithmic coefficients of inverse starlike functions subordinated to exponential functions. This work may inspire to pay more attention to the coefficient properties with respect to the inverse functions of various classes of univalent functions.

Coefficient Bounds of Bi-Univalent Function Involving Pseudo-Starlikeness Associatedwith Sigmoid Function Defined by Salagean Operator via Chebyshev Polynomial

Some special classes of univalent functions play an important role in geometric function theory because of their geometric properties. Many of such classes have been introduced and studied; some became well known, for example, the classes of convex, starlike, close to convex, strongly convex and strongly starlike functions. Previous studies by Awolere and Oladipupo (2018) now served as motivation and background to investigate certain classes of analytic, univalent and bi-univalent functions in terms of their coefficient bounds involving salagean and sigmoid functions via Chebyshev poynomial. The classes 퐻푛(휆,훽,훾(푠),휙(푧,푡))are newly established classes for which coefficient bounds will be determined. The aim of the present work is to investigate coefficient bound for class 퐻푛(휆,훽,훾(푠),휙(푧,푡))of pseudo-starlikeness associated with sigmoid functions defined by Salagean operator via Chebyshev polynomial, Fekete-szego problem will also be established and the Hankel of the function will be determined

A New Special Class of Ma–Minda Type Starlike Functions

A New Special Class of Ma–Minda Type Starlike Functions

Subclass of bi-starlike function associated with touchard polynomials

The current paper aims to study bi- starlike functions. It presents a new class using Touchard polynomial, which fulfills the subordination conditions. In addition, an estimation of the Taylor-Maclaurin coefficients |a2| and |a3| was found.

Sharp inequalities for $ q $-starlike functions associated with differential subordination and $ q $-calculus

<p>This paper employs differential subordination and quantum calculus to investigate a new class of $ q $-starlike functions associated with an eight-like image domain. Our study laid a foundational understanding of the behavior of these $ q $-starlike functions. We derived the results in first-order differential subordination. We established sharp inequalities for the initial Taylor coefficients and provided optimal estimates for solving the Fekete-Szegö problem and a second-order Hankel determinant applicable to all $ q $-starlike functions in this class. Furthermore, we presented a series of corollaries that demonstrate the broader implications of our findings in geometric function theory.</p>

Applications of fuzzy differential subordination theory on analytic $ p $ -valent functions connected with $ \mathfrak{q} $-calculus operator

In recent years, the concept of fuzzy set has been incorporated into the field of geometric function theory, leading to the evolution of the classical concept of differential subordination into that of fuzzy differential subordination. In this study, certain generalized classes of $ p $ -valent analytic functions are defined in the context of fuzzy subordination. It is highlighted that for particular functions used in the definitions of those classes, the classes of fuzzy $ p $-valent convex and starlike functions are obtained, respectively. The new classes are introduced by using a $ \mathfrak{q} $-calculus operator defined in this investigation using the concept of convolution. Some inclusion results are discussed concerning the newly introduced classes based on the means given by the fuzzy differential subordination theory. Furthermore, connections are shown between the important results of this investigation and earlier ones. The second part of the investigation concerns a new generalized $ \mathfrak{q} $-calculus operator, defined here and having the $ (p, \mathfrak{q)} $-Bernardi operator as particular case, applied to the functions belonging to the new classes introduced in this study. Connections between the classes are established through this operator.

Study of quantum calculus for a new subclass of $ q $-starlike bi-univalent functions connected with vertical strip domain

<abstract><p>In this study, using the ideas of subordination and the quantum-difference operator, we established a new subclass $ \mathcal{S} ^{\ast }\left(\delta, \sigma, q\right) $ of $ q $-starlike functions and the subclass $ \mathcal{S}_{\Sigma }^{\ast }\left(\delta, \sigma, q\right) $ of $ q $-starlike bi-univalent functions associated with the vertical strip domain. We examined sharp bounds for the first two Taylor-Maclaurin coefficients, sharp Fekete-Szegö type problems, and coefficient inequalities for the function $ h $ that belong to $ \mathcal{S}^{\ast }\left(\delta, \sigma, q\right) $, as well as sharp bounds for the inverse function $ h $ that belong to $ \mathcal{S}^{\ast }\left(\delta, \sigma, q\right) $. We also investigated some results for the class of bi-univalent functions $ \mathcal{S}_{\Sigma }^{\ast }\left(\delta, \sigma, q\right) $ and well-known corollaries were also highlighted to show connections between previous results and the findings of this paper.</p></abstract>

On H3(1) Hankel determinant for starlike and convex functions associated with exponential function

This research aims to present and study the convex subclasses and starlike functions of the analytic functions defined in the open unit disk Ư and to investigate the third-order Hankel determinant H3,1 for these subclasses related to the exponential function. As well as the upper bounds of H2,2 (g) and H3,1 (g) for the convex and starlike functions, and the estimates of the coefficients |d3|,|d4|, and |d5| were obtained.