Starlike Functions Of Order Research Articles

Applications of q-Calculus Multiplier Operators and Subordination for the Study of Particular Analytic Function Subclasses

In this article, a new linear extended multiplier operator is defined utilizing the q-Choi–Saigo–Srivastava operator and the q-derivative. Two generalized subclasses of q—uniformly convex and starlike functions of order δ—are defined and studied using this new operator. Necessary conditions are derived for functions to belong in each of the two subclasses, and subordination theorems involving the Hadamard product of such particular functions are stated and proven. As applications of those findings using specific values for the parameters of the new subclasses, associated corollaries are provided. Additionally, examples are created to demonstrate the conclusions’ applicability in relation to the functions from the newly introduced subclasses.

An analysis of convexity and starlikeness attributes for breaz integro-differential operator

The Geometric Theory of Analytic Functions (GTAF) is the attractive part ofcomplex analysis, which correlates with the rest of the themes in mathematics. Itsessential purpose is to formulate numerous classes of geometric analytic functions and todiscuss their geometric attributes. In continuation, the association between operatortheory and the GTAF area started to take shape and has remained a topic of wide attentiontoday. In the previous century, operator theory was extended to the complex open unit diskand has been applied to propose diverse sorts of generalizations of normalized analyticfunctions. As a result, the operator theory appears to be a good way to look for thingsin the GTAF area. Since then, the acquisition of geometric attributes by employingoperators has become a significant theme of research studies. The current studycenters on and investigates, in the classes of $\ell$-uniformly convex and starlikefunctions of order $\beta$, the convexity attribute by utilizing a modified Breazintegro-differential operator in the unit disk. Furthermore, in the class of analyticfunctions, some conditions that make the Breaz operator look like a star are looked into.

The Sharp Bound of the Fifth Coefficient of Strongly Starlike Functions with Real Coefficients

In this paper, we found the sharp bound of the fifth coefficient of strongly starlike functions of order $$\alpha $$ with real coefficients for all $$\alpha \in (0,1]$$ .

Coefficient inequality for a subclass of biholomorphic mappings in several complex variables

Let be the familiar class of starlike functions of order in the unit disk. In Keogh and Merkes proved that for a function in the class , then In this paper, we investigate the corresponding problem for the subclass of starlike mappings of order defined on the unit ball in a complex Banach space, on the unit polydisk in and the bounded starlike circular domain in , respectively.

Some properties and results for certain subclasses of starlike and convex functions

In the present paper, we introduce and investigate some properties of two subclasses $ Lambda_{n}( lambda , beta ) $ and $ Lambda_{n}^{+}( lambda , beta ) $; meromorphic and starlike functions of order $ beta $. In particular, several inclusion relations, coefficient estimates, distortion theorems and covering theorems are proven here for each of these function classes.

COEFFICIENT ESTIMATES FOR SOME CLASSES OF FUNCTIONS ASSOCIATED WITH -FUNCTION THEORY

For every $q\in (0,1)$, we obtain the Herglotz representation theorem and discuss the Bieberbach problem for the class of $q$-convex functions of order $\unicode[STIX]{x1D6FC}$ with $0\leq \unicode[STIX]{x1D6FC}<1$. In addition, we consider the Fekete–Szegö problem and the Hankel determinant problem for the class of $q$-starlike functions, leading to two conjectures for the class of $q$-starlike functions of order $\unicode[STIX]{x1D6FC}$ with $0\leq \unicode[STIX]{x1D6FC}<1$.

A note on a class of p-valent starlike functions of order β

We obtain sharp coefficient bounds for some p-valent starlike functions of order β, 0 ≤ β < p. Initially this problem was handled by Aouf in [1]. We pointed out that the proof by Aouf is incorrect and present a correct proof.

On some m-fold symmetric functions with generalized bounded variation

In this paper we study a class Bk(�; b; c; m; ) which uni�es a number of classes studied previously by Mocanu and others. This class includes several known classes of analytic functions such as convex and starlike functions of order b, �-convex functions, functions with bounded boundary rotation, bounded radius rotation and bounded Mocanu variation. Several interesting properties like inclusion results, In- tegral representation, arclength problem, coe�cient bounds and distortion results of this class are discussed.

Coefficient inequality for parabolic starlike functions of order alpha

The aim of this paper is to obtain a sharp upper bound to the second Hankel determinant \(|a_{2}a_{4}-a_{3}^{2}|\) for the function \(f\) when it belongs to the class of parabolic starlike functions of order \(\alpha (0\le \alpha <1).\) Further, an upper bound for the inverse function of \(f\) to the non-linear functional \(|t_{2}t_{4}-t_{3}^{2}|\) was determined when it belongs to the same class of functions, using Toeplitz determinants.

New extensions for classes of analytic functions associated with close-to-convex and starlike of order [formula omitted

Let A be the class of functions f ( z ) with f ( 0 ) = 0 and f ′ ( 0 ) = 1 in the open unit disk U . For f ( z ) ∈ A , f ( z ) is said to be in the class A ( β 1 , β 2 , β 3 ; λ ) if it satisfies | β 1 z ( f ( z ) z ) ′ + β 2 z 2 ( f ( z ) z ) ″ + β 3 z 3 ( f ( z ) z ) ‴ | ≦ λ ( z ∈ U ) for some complex numbers β 1 , β 2 and β 3 , and for some real λ > 0 . Some interesting properties for f ( z ) ∈ A ( β 1 , β 2 , β 3 ; λ ) associated with close-to-convex functions and starlike functions of order α are discussed.

Löwner chains and a subclass of biholomorphic mappings

In this paper, we give characterization of almost starlike functions of order α (respectively almost starlike mappings of order α) on the unit disc in C (respectively the unit ball in a finite-dimensional complex Banach space) in terms of Löwner chains. Furthermore, using the properties of Löwner chains, we can easily prove that two classes of generalized Roper–Suffridge extension operators preserve almost starlikeness of order α on two important classes of Reinhardt domains in C n , respectively.

Convexity of integral transforms and function spaces

For β<1, let 𝒫γ(β) denote the class of all normalized analytic functions f in the unit disc Δ such that for some ϕ∈ℝ. Let 𝒮*(μ), 0≤μ<1, denote the usual class of starlike functions of order μ. Define 𝒦(μ)={f: zf′(z)∈𝒮*(μ)}, the class of all convex functions of order μ. In this paper, we consider integral transforms of the form The aim of this paper is to find conditions on λ (t) so that each of the transformations carries 𝒫γ(β) into 𝒮*(μ) or 𝒦(μ). A number of applications for certain special choices of λ (t) are also established. These results extend the previously known results by a number of authors.

Some properties of an integral transform

Let A be the class of normalized analytic functions in the unit disk U , and S , S ∗ ( α ) and K ( α ) denote the subclasses of A consisting of univalent functions, starlike functions of order α and convex functions of order α in U , respectively. Y.C. Kim gave the following conjecture: Let 0 ≤ α < 1 , β > 1 . If f ∈ S , or S ∗ ( α ) , or K ( α ) , then ϕ ( 3 , 3 + β ; z ) ∗ f ( z ) belongs to the same class, where ϕ ( a , c ; z ) is the incomplete beta function. In this work, we prove that Kim’s conjecture is true for f ∈ S ∗ ( α ) or f ∈ K ( α ) and improve some other results.

On a class of meromorphic p‐valent starlike functions involving certain linear operators

Let ∑p be the class of functions f(z) which are analytic in the punctured disk 𝔼* = {z ∈ ℂ : 0 &lt; |z| &lt; 1}. Applying the linear operator Dn+p defined by using the convolutions, the subclass 𝒯n+p(α) of ∑p is considered. The object of the present paper is to prove that 𝒯n+p(α) ⊂ 𝒯n+p−1(α). Since 𝒯0(α) is the class of meromorphic p‐valent starlike functions of order α, all functions in 𝒯n+p−1(α) are meromorphic p‐valent starlike in the open unit disk 𝔼. Further properties preserving integrals and convolution conditions are also considered.

Coefficient bounds and convolution properties for certain classes of close-to-convex functions

A number of authors (cf. Koepf [4], Ma and Minda [6]) have been studying the sharp upper bound on the coefficient functional $|a_3 - \mu a_2^2|$ for certain classes of univalent functions. In this paper, we consider the class $\mathcal{C}(\varphi, \psi)$ of normalized close-to-convex functions which is defined by using subordination for analytic functions $\varphi$ and $\psi$ on the unit disc. Our main object is to provide bounds of the quantity $a_3 - \mu a_2^2$ for functions $f(z) = z + a_2 z^2 + a_3 z^3 + \dotsb$ in $\mathcal{C}(\varphi, \psi)$ in terms of $\varphi$ and $\psi$, where $\mu$ is a real constant. We also show that the class $\mathcal{C}(\varphi, \psi)$ is closed under the convolution operation by convex functions, or starlike functions of order $1/2$ when $\varphi$ and $\psi$ satisfy some mild conditions.

Applications of duality principle to integral transforms of analytic functions

LetAbe the class of functions analytic in the unit disk and normalized by f(0)=f′(0)+1=0 Let S, S*(γ), and K(γ)be respectively the classes of normalized univalent functions, starlike functions of order γ and convex functions of order γ. In this paper we investigate the properties of the integral transform where λ is a non-negative real valued function normalized by ∫1 0 λ(t)dt=1. From our main results we get conditions on λ and the classFso that V λ (f) mapsF into various subclasses of the class of univalent functions. As a corollary to our results, we give an affirmative answer in support of a conjecture of Kim: f is a member ofSor S*(γ) or K(γ), then the function φ(3,3+ αz) * f(z) belongs to the same class for α >1, where φ(b,c;z) * f(z) stands for the convolution of incomplete beta function with f∈ A

Criteria for strongly starlike functions

Let A n be the family of normalized regular functions f, where f(z)=z+an+1zn+1+…, in the unit disc Δ and let S(α) and T λ be the families of functions f such that respectively. S(α) is the class of strongly starlike functions. Further, let S *(α) denote the well-known family of starlike functions of order α. The purpose of this paper is to find conditions so that 1. satisfying satisfying . 2. satisfying is in S(α) or in 3. satisfying is in S(α) or in S *(α); 4. satisfying Re f′(z)≥1−λ1,|g′(z)−1|≤λ2, imply z(f*g)′ is in S(α). The results of this paper extend the previously known results and improve some criteria for starlike functions. These are then used to obtain new information for uniformly convex functions and Bemardi integral transform.

On convex functions of order α and type β

Owa [1] gave three subordination theorems for convex functions of order α and starlike functions of order α. Unfortunately, none of the theorems is correct. In this paper, similar problems are discussed for a generalized class and sharp results are given.

A Conformal Mapping Inequality for Starlike Functions of order ½

A Conformal Mapping Inequality for Starlike Functions of order ½

On the Radius of β-Convexity of Starlike Functions of Order α

On the Radius of β-Convexity of Starlike Functions of Order α