Starlikeness Of Order Research Articles

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.

Radius of starlikeness through subordination

"A normalized function $f$ on the open unit disc is starlike (or convex) univalent if the associated function $zf'(z)/f(z)$ (or $1+zf''(z)/f'(z)$) is a function with positive real part. The radius of starlikeness or convexity is usually obtained by using the estimates for functions with positive real part. Using subordination, we examine the radius of various starlikeness, in particular, radii of Janowski starlikeness and starlikeness of order $\beta$, for the function $f$ when the function $f$ is either convex or $(zf'(z)+\alpha z^2f''(z))/f(z)$ lies in the right-half plane. Radii of starlikeness associated with lemniscate of Bernoulli and exponential functions are also considered."

On Starlike Functions of Negative Order Defined by q-Fractional Derivative

In this paper, two new classes of q-starlike functions in an open unit disc are defined and studied by using the q-fractional derivative. The class Sq*˜(α), α∈(−3,1], q∈(0,1) generalizes the class Sq* of q-starlike functions and the class Tq*˜(α), α∈[−1,1], q∈(0,1) comprises the q-starlike univalent functions with negative coefficients. Some basic properties and the behavior of the functions in these classes are examined. The order of starlikeness in the class of convex function is investigated. It provides some interesting connections of newly defined classes with known classes. The mapping property of these classes under the family of q-Bernardi integral operator and its radius of univalence are studied. Additionally, certain coefficient inequalities, the radius of q-convexity, growth and distortion theorem, the covering theorem and some applications of fractional q-calculus for these new classes are investigated, and some interesting special cases are also included.

Generalized Roper-Suffridge Operator for ϵ Starlike and Boundary Starlike Mappings

This article is devoted to a deep study of the Roper-Suffridge extension operator with special geometric properties. First, we prove that the Roper-Suffridge extension operator preserves ϵ starlikeness on the open unit ball of a complex Banach space ℂ × X, where X is a complex Banach space. This result includes many known results. Secondly, by introducing a new class of almost boundary starlike mappings of order α on the unit ball Bn of ℂn, we prove that the Roper-Suffridge extension operator preserves almost boundary starlikeness of order α on Bn. Finally, we propose some problems.

Some geometrical properties of free boundaries in the Hele-Shaw flows

In this paper, we are concerned with certain geometric properties of the moving boundary in the case of two-dimensional viscous fluid flows in Hele-Shaw cells under injection. We study the invariance in time of free boundary for such a bounded flow domain under the assumption of zero surface tension. By applying various results in the theory of univalent functions, we consider the invariance in time of starlikeness of a complex order, almost starlikeness of order α ∈ [0, 1), and almost spirallikeness of type γ∈(−π/2,π/2) and order α ∈ (0, cos γ). This work complements recent work on planar Hele-Shaw flow problems in the case of zero surface tension.

The order of starlikeness of some classes of strongly starlike functions

.In this paper we consider the order of strongly starlikeness for functions in Silverman’s class 𝒢b and in the generalized Mocanu’s class M (α, β1, β2). Some previous results are improved.

Geometric Properties of the Shifted Hypergeometric Functions

We will provide sufficient conditions for the shifted hypergeometric function $$z_2F_1(a,b;c;z)$$ to be a member of a specific subclass of starlike functions in terms of the complex parameters a, b and c. For example, we study starlikeness of order $$\alpha ,$$ $$\lambda $$ -spirallikeness of order $$\alpha $$ and strong starlikeness of order $$\alpha .$$ In particular, those properties lead to univalence of the shifted hypergeometric functions on the unit disk.

The Roles Played by Order of Starlikeness and the Bloch Condition in the Roper–Suffridge Extension Operator

In this article, we obtain that the relation \(\Phi _{G,p}(f)({\mathcal {S}}_1^{*}(\alpha ))\subseteq {\mathcal {S}}_n^{*}\) implies G is a homogeneous polynomial of degree at most p for all \(\alpha \in [0, 1)\) on the bounded convex circular domain \(\Omega _{2,p}\). We shall also show that if f is a spirallike function of type \(\beta \) and is also a Bloch function on the unit disk, then \(\Phi _{G,p}(f)\) is a spirallike mapping of type \(\beta \) on the domain \(\Omega _{2,p}\) for certain G having terms of degree greater than p, where \(p\in {\mathbb {Z}}^+\) and \(p\ge 2\). As corollaries, some well known results can be got.

The order of starlikeness of uniformly convex functions

AbstractWe investigate classes of

Properties of the modified roper-suffridge extension operators on reinhardt domains

In this paper, we mainly discuss the properties of the modified Roper-Suffridge operators on Reinhardt domains. By the analytical characteristics and distortion results of subclasses of biholomorphic mappings, we conclude that the modified Roper-Suffridge operators preserve the properties of S*Ω (β, A, B), almost starlike mapping of complex order λ on Ωn,p2, …,pn. Sequentially, we get that the modified Roper-Suffridge operators preserve spirallikeness of type β and order a, strongly pirallikeness of type β and order α, almost starlikeness of order α on Ωn,p2, …, Pn. The conclusions provide a new approach to construct these biholomorphic mappings which have special geometric properties in several complex variables.

RADII OF HARMONIC MAPPINGS IN THE PLANE

In this paper, for the convolution and convex combination of harmonic mappings, the radii of univalence, full convexity and starlikeness of order $\unicode[STIX]{x1D6FC}$ are explored. All results are sharp. By way of application, the univalent radius and the Bloch constant of the convolution of two bounded harmonic mappings are obtained.

Invariant Geometric Properties in Hele-Shaw Flows

In this paper, we investigate some geometric properties of the moving frontier of a viscous fluid for planar flows in Hele-Shaw cells under injection. We study invariance in time properties of the free boundary for bounded and unbounded domains (with bounded complement) under the assumption of zero surface tension. By applying certain results in the theory of univalent functions we partially solve an open problem of Vasil’ev concerning the invariance in time of starlikeness of order \(\alpha , \alpha \in [0,1)\), for bounded domains. In the case of unbounded domains with bounded complement we analyze the invariance in time of convexity of order \(\alpha , \alpha \in [0,1)\).

The invariance of strong and almost spirallike mappings of type β and order α

The invariance of strong and almost spirallike mappings of type β and order a is discussed in this paper. From the maximum modulus principle of holomorphic functions, we obtain that the generalized Roper-Suffridge operators preserve strong and almost spirallikeness of type β and order α on the unit ball Bn in ℂn and on bounded and complete Reinhardt domains. Therefore we obtain that the generalized Roper-Suffridge operators preserve strong spirallikeness of type β, strong and almost starlikeness of order α, strong starlikeness on the corresponding domains. Thus we can construct more subclasses of spirallike mappings in several complex variables.

On some class of convex functions

In this paper, we consider the order of starlikeness and strong starlikeness in the class of functions f(z)=z+a2z2+… analytic in |z|<1 in the complex plane and satisfyingRe{1+zf″(z)f′(z)}>|zf′(z)f(z)−1|,|z|<1.

Roper-Suffridge extension operator on a reinhardt domain

Let pj∈ℕ and pj≥1,j=2,…,k,k≥2, be a fixed positive integer. We introduce a Roper-Suffridge extension operator on the following Reinhardt domain ΩN={z=(z1,z′2,…,z′k)′∈ℂ×ℂn2×…×ℂnk:|z1|2+||z2||2p2+…+||zk||kpk<1} given by F(z)=(f(z1)+f′(z1)∑j=2kPj(zj),(f′(z1))1p2z′2,…,(f′(z1)1pkz′k)′,, where f is a normalized biholomorphic function on the unit disc D, and for 2≤j≤k,Pj:ℂnj→ℂ is a homogeneous polynomial of degree pj and zj=(zj1,…,zjnj)′∈ℂnj,nj≥1,pj≥1,||zj||j=(∑l=1nj|zjl|pj)1pj.. In this paper, some conditions for pj are found under which the operator preserves the properties of almost starlikeness of order α, starlikeness of order α and strongly starlikeness of order α on ΩN, respectively.

On circularly symmetric functions

Let D C and 02 D. A set D is circularly symmetric if for each % 2 R + a set D\f 2 C : j j = %g is one of three forms: an empty set, a whole circle, a curve symmetric with respect to the real axis containing %. A function f 2 A is circularly symmetric if f() is a circularly symmetric set. The class of all such functions we denote by X. The above denitions were given by Jenkins in (2). In this paper besides X we also consider some of its subclasses: X( ) and Y \S consisting of functions in X with the second coecient xed and univalent starlike functions respectively. According to the suggestion, in Abstract we add one more paragraph at the end of the section: For X( ) we nd the radii of starlikeness, starlikeness of order , univalence and local univalence. We also obtain some distortion results. For Y\S we discuss some coecient problems, among others the Fekete- Szego ineqalities.

A class of analytic functions involving in the Dziok-Srivastava operator

Let A be a class of functions f (z )o f the form f (z )= z + ∞ � n=2 anz n (.) which are analytic in the open unit disk U. By means of the Dziok-Srivastava operator, we introduce a new subclass S l m (α1, α, μ) � l ≤ m +1 ,l, m ∈ N ∪{ 0} ,– π 2 – cosα � of A .I n particular,S 1 (2, 0, 0) coincides with the class of uniformly convex functions introduced by Goodman. The order of starlikeness and the radius of α-spirallikeness of order β (β < 1) are computed. Inclusion relations and convolution properties for the class S l m (α1, α, μ) are obtained. A special member of S l m (α1, α, μ )i s also given. The results presented here not only generalize the corresponding known results, but also give rise to several other new results. MSC: Primary 30C45

Linear invariance of locally biholomorphic mappings in the unit ball of a [formula omitted]-triple

We study the notion of linear invariance on the unit ball of a J B ⁎ -triple X, and we obtain some connection between the norm-order of a linear invariant family and the starlikeness of order 1/2. Also, we give some result concerning the radius of univalence of some linear invariant families. Finally, if the dimension of X is finite and if the norm-order of a linear invariant family is finite, then we prove the normality of the linear invariant family and we also obtain upper bounds on the distortion and the growth of mappings in a linear invariant family with specified norm-order. In particular, our results are valid for the classical Cartan domains and the unit polydisc.

A certain class of starlike functions

This paper presents a new class of functions analytic in the open unit disc, and closely related to the class of starlike functions. Besides being an introduction to this field, it provides an interesting connections defined class with well known classes. The paper deals with several ideas and techniques used in geometric function theory. The order of starlikeness in the class of convex functions of negative order is also considered here.

On the order of starlikeness and convexity of complex harmonic functions with a two-parameter coefficient condition

We’ve built this digital research platform to provide academics everywhere with fast, stable and secure digital access to our library of over 110,000 scholarly books and 800,000 journal articles. We’ve tested the platform in close cooperation with leading academic institutions. If you have any questions or you notice something doesn’t quite work as it should, please visit our <a href="/publishing/faq?lang=en" class="text-primary">Help/FAQ</a> page and let us know.