A new insight into meromorphic functions: An exploration of negative r order of two convex and starlike dynamics

Numerous scholars have shown remarkable interest in geometric function theory, a thriving area of study with roots in complex analysis. The investigation of the geometric properties of analytic (in addition to univalent) functions, with numerous applications in numerous branches of mathematics, mathematical physics, and engineering, is largely responsible for this. Applications in engineering designs, fluid flows in physics, special functions, orthogonal polynomials, and conformal mappings are noteworthy. The article investigates a subclass of quasi-subordination-related analytic and univalent functions established in the domain of logistics sigmoid functions on a unit disk. Using quasi-subordination, the Fekete-Szego functional |e3 – μe2 2|can be obtained for functions of the classes Sq* (τ, γ, ℧), and Gq (ϑ, α, ℧). We also briefly discuss the subordination results and the improved results for the related classes involving majorization. Some renowned subclasses of analytic and univalent functions, including the classes of starlike functions and convex functions, are included in the definition of the novel class. The description of the class involves the application of two fundamental mathematical concepts: quasi-subordination and Taylor’s series.

Motivated by the Liu-Srivastava linear operator, we introduce here a modification of the operator of multivalent meromorphic functions in the punctured unit disk. A new subclass of analytic functions involving this operator is given. Some sufficient conditions for star-likeness, which generalize and refine some previous results were determined. Key words: Hypergeometric function, Liu-Srivastava linear operator, convolution, meromorphic function, univalent functions, starlike functions, convex functions, -convex function, Carlson-Shaffer linear operator, Ruscheweyh derivative operator.

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.

In the present paper, we introduce and investigate three interesting superclasses SD, SD* and KD of analytic, normalized and univalent functions in the open unit disk D. For functions belonging to these classes SD, SD* and KD, we derive several properties including (for example) the coefficient bounds and growth theorems. The various results presented here would generalize many well known results. We also get a new univalent criterion and some interesting properties for univalent function,starlike function,convex function and close-to-convex function. Many superclasses which are already studied by various researchers are obtained as special cases of our two new superclasses.

The theory of geometric functions was first introduced by Bernard Riemann in 1851. In 1916, with the concept of normalized function revealed by Bieberbach, univalent function concept has found application area. Assume f(z)=z+??, n?(anzn) converges for all complex numbers z with |z|<1 and f(z)is one-to-one on the set of such z. Convex and starlike functions f(z) and g(z) are discussed with the help of subordination. The f(z) and g(z) are analytic in unit disc and f(0)=f'(0)=1, and g(0)=0, g'(0)-1=0. A single valued function f(z) is said to be univalent (or schlict or one-to-one) in domain D?C never gets the same value twice; that is, if f(z1)-f(z2)?0 for all z1 and z2 with z1 ? z2. Let A be the class of analytic functions in the unit disk U={z:|z|<1} that are normalized with f(0)=F'(0)=1. In this paper we give the some necessary conditions for f(z) ? S* [a, a2] and 0?a2?a?1 f'(z)(2r-1)[1-f'(z)]+zf?(z / 2r[f'(z)]2. This condition means that convexity and starlikeness of the function f of order 2-r.

The function $ \xi(z)$ is obtained from the logarithmic derivative function $\sigma(z)$. The elliptic function $ \wp(z) $ is also derived from the $ \xi(z) $ function. The function $ \wp(z) $ is a function of double periodic and meromorphic function on lattices region. The function $ \wp(z) $ is also double function. The function $ \varphi(z) $ meromorphic and univalent function was obtained by the serial expansion of the function $ \wp(z)$. The function $ \varphi(z) $ obtained here is shown to be a convex function.

This article presents an over view of Geometric Function Theory (GFT) and utilized the analytic properties of meromorphic functions. Geometric Function Theory is a branch of mathematics that focuses on the geometric interpretations and implications of analytic functions defined in the complex plane. Our exploration begins with an itemized discussion of key concepts within GFT, emphasizing their relevance and theoretical underpinnings. Central to our study is the investigation of meromorphic functions, which are functions that are analytic except for isolated singularities where they may have poles. We examined various classes of meromorphic functions and elucidate their properties, including their behavior near singularities and their broader geometric implications. A significant portion of our inquiry involves the Hadamard product of functions. This operation allows us to explore the combined effect of two analytic functions, considering their series expansions and how their product transforms under this operation. By studying the Hadamard transformation, we uncover analogues and interesting results that shed light on the interplay between analytic functions and their geometric representations. We also provide detailed diagrammatic descriptions of fundamental geometric shapes such as circles, open unit disks, and closed unit disks. These diagrams serve to visually illustrate key concepts and relationships within GFT, aiding in the understanding of how analytic functions behave in different spatial configurations. Our article offers a comprehensive exploration of Geometric Function Theory, emphasizing its foundational concepts and their applications in analyzing analytic and meromorphic functions.

Let $$\Delta (r,f)$$ denote the area of the image of the subdisk $$|z|<r,\,0<r\le 1,$$ under an analytic function f in the unit disk $$|z|<1$$ . Without loss of generality, in this context, we consider only the analytic functions f in the unit disk with the normalization $$f(0)=0=f'(0)-1$$ . We set $$F_f(z)=z/f(z)$$ . Our objective in this paper is to obtain a sharp upper bound of $$\Delta (r,F_f),$$ when f varies over the class of normalized analytic univalent functions in the unit disk with quasiconformal extension to the entire complex plane.

Let \(\Lambda = \{ \lambda n\}\) be a sequence of points on the complex plane, and let \(\Lambda (r)\) be the number of points of the sequence \(\Lambda\) in the disk \(\{ \left| z \right| < r\}\). We study the following problem in terms of the counting function \(\Lambda (r)\): what is the minimal possible growth of the characteristic \(M_f (r) = \max \{ \left| {f\left( z \right)} \right|:\left| z \right| = r\}\) in the class of all entire functions \(f\not \equiv 0\) vanishing on \(\Lambda\)? Let \(F\) be a meromorphic function in \(\mathbb{C}\). In terms of the Nevanlinna characteristic \(T_F (r)\) of the function \(F\), we estimate the minimal possible growth of the characteristics \(M_g (r)\) and \(M_h (r)\) in the class of all pairs of entire functions \(g\) and \(h\) such that \(F = g/h\). We present analogs of the obtained results for holomorphic and meromorphic functions in the unit disk in the complex plane.

Abdullah Alotaibi defined a starlike function connected with a cosine hyperbolic function in the year 2020. We establish some appropriate conditions for several features of multivalent analytic starlike function subordinated with cosine hyperbolic function in this article. We determine conditions on α are subordinated by Janowski function. We acquire some suitable conditions by selecting specific values for functions we get some adequate conditions for multivalent starlik function related with cosine hyperbolic. Over the last decade, starlike functions have grown in popularity in both literature and application. Our goal in this work is look at some practical challenges with q-starlike functions. Moreover, we will show that the class described in this research, as well as the results gained, generalizes numerous previously published papers. We need to add some fundamental Geometric function theory literature here to comprehend the notions employed in our work in a straightforward way. To do so, we'll start with the notation, which signifies the class of holomorphic or analytic functions in the holomorphic or analytic functions. Then the relationships must be stable. In addition, all univalent functions will belong to the subfamily. Furthermore, the possibility of subjections between analytic functions and, as shown by, as; the functions, are related by the connection of subjection, if there exists an analytic function with restrictions and such that in addition, if the function is in, we get The aim of this paper is to define a family of multivalent q-starlike functions associated with circular domains and to study some of its useful properties of multivalent analytic functions subordinated cosine hyperbolic function.

B. Friedman discovered in his 1946 paper that the set of analytic univalent functions on the unit disk in the complex plane with integral Taylor coefficients consists of nine functions. In the present paper, we prove that the similar set obtained by replacing “integral” by “half-integral” consists of another 12 functions in addition to the nine. We also observe geometric properties of the 12 functions.

Let be the unit disc in the complex plane and a class of holomorphic functions in distinguished by a restriction on their growth in a neighbourhood of the boundary of the disc which is stated in terms of weight functions of moderate growth. Some results which describe the sequences of zeros for holomorphic functions in classes of this type are obtained. The weight functions defining are not necessarily radial; however the results obtained are new even in the case of radial constraints. Conditions for meromorphic functions in ensuring that they can be represented as a ratio of two functions in sharing no zeros are investigated.Bibliography: 28 titles.

In the present paper, we will consider the class of meromorphic starlike functions with fixed residue&nbsp;.&nbsp;&nbsp; Silverman et al. (2008) has obtained sharp upper bounds for Fekete-Szeg&ouml; like functional&nbsp;&nbsp;for certain subclasses of meromorphic functions. In this paper, we will find sharp upper bounds for&nbsp;&nbsp;for the class meromorphic starlike functions with fixed residue&nbsp;. The aim of the present paper, is to completely solve Fekete Szeg&ouml; problem for a certain subclass of meromorphic starlike functions with fixed residue d. &nbsp; Key words:&nbsp;Fekete-Szeg&ouml; inequality, starlike function, analytic function, subordination, meromorphic function.

Let f be a meromorphic function in the unit disc and ( a ν ) ν = 1 k ${(a_\nu )_{\nu =1}^k}$ a set of distinct meromorphic functions small with respect to f. An analogue of the second main theorem for f and ( a ν ) ν = 1 k ${(a_\nu )_{\nu =1}^k}$ is given. Upper limits for the sum of defects of an admissible meromorphic function and an admissible holomorphic function follow. For meromorphic and holomorphic functions in the unit disc and their small functions the analogues of Ullrich's theorem are presented.

The main object of this paper is to investigate the problem of majorization of certain class of meromorphic p-valent functions of complex order involving certain integral operator. Moreover we point out some new or known consequences of our main result. Keywords: Meromorphic functions, Starlike functions, Convex functions, Majorization problems, Hadamard product (convolution), Integral operator.