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On (j,k)-symmetrical functions
... Regular symmetric functions treat all variables alike, but (u, v)-symmetrical functions introduce a twist. Here, v denotes a fixed positive integer, and u can range from 0 to v − 1, (see [1]). A domain D is said to be v-fold symmetric if a rotation of D about the origin through an angle 2π v carries D onto itself. ...
... In our work we need the following decomposition theorem Lemma 1.1. [1] For every mapping ℏ : Ω → C and a v-fold symmetric set Ω, there exists a unique sequence of (u, v)-symmetrical functions ℏ u,v , such that ...
... Denote be F (u,v) for the family of all (u, v)-symmetric functions. Let us observe that the classes F (1,2) , F (0,2) and F (1,v) are well-known families of odd, even and of v−symmetrical functions, respectively. ...
This research paper addressed a significant knowledge gap in the field of complex analysis by introducing a pioneering category of q -starlike and q -convex functions intricately interconnected with -symmetrical functions. Recognizing the limited exploration of these relationships in existing literature, the authors delved into the new classes and . The main contribution of this work was the establishment of a framework that amalgamates q -starlikeness and q -convexity with the symmetry conditions imposed by -symmetrical functions. This comprehensive study include coefficient estimates, convolution conditions, and the properties underpinning the -neighborhood, thereby enriching the understanding of these novel function classes.
... Notably, when γ = 2, we obtain the class of odd univalent functions. Liczberski and Polubinski, in [3], extended this notion by developing the theory of (λ, γ)-symmetrical functions, where λ = 0, 1, 2, . . . , γ − 1 and γ = 2, 3, . . . . ...
... We observe that Ψ is a γ-fold symmetric domain with the γ of any integer. We use the unique decomposition [3] of every mapping h : Ψ → C, as follows: ...
... where α m λ is given by (3). ...
In this paper, we introduce new classes of functions defined within the open unit disk by integrating the concepts of (λ,γ)-symmetrical functions, generalized Janowski functions, and quantum calculus. We derive a structural formula and a representation theorem for the class Sqλ,γ(x,y,z). Utilizing convolution techniques and quantum calculus, we investigate convolution conditions supported by examples and corollary, establishing sufficient conditions. Additionally, we derive properties related to coefficient estimates, which further elucidate the characteristics of the defined function classes.
... Moreover, if the function g 2 is univalent in D, then the following equivalence hold true: g 1 ≺g 2 ⇔g 1 (0) � g 2 (0), g 1 (D) ⊂ g 2 (D). (6) Let P be the class of Carathedory function, an analytic function χ ∈ P if χ(z) � 1 + ∞ n�1 χ n z n , (7) such that ...
... ; ), the (j, i)-symmetric functions are an extension of the concept of even, odd, and i-symmetric functions. Several applications of the theory of (j, i) -symmetric functions may be found in [7]. For e � e (2πI/i) , the functions f: D ⟶ C is known to be (j, i) -symmetric if f e j z � e j f(z). ...
... Theorem 1 (see [7]). For mapping f: D ⟶ C, there is just one series of (j, i) -symmetrical functions f j,i (z) exists that given as follows: ...
In this paper, we make use of a certain Ruscheweyh-type q -differential operator to introduce and study a new subclass of q -starlike symmetric functions, which are associated with conic domains and the well-known celebrated Janowski functions in D . We then investigate many properties for the newly defined functions class, including for example coefficients inequalities, the Fekete–Szegö Problems, and a sufficient condition. There are also relevant connections between the results provided in this study and those in a number of other published articles on this subject.
... . In 1995, Liczberski and Polubinski [3] constructed the theory of ðα, βÞ-symmetrical functions for ðα = 0, 1, 2, ⋯, β − 1Þ and ðβ = 2, 3, ⋯Þ. If G is β-fold symmetric domain and α any integer, then a function h : G ⟶ ℂ is called ðα, βÞ -symmetrical if for each ω ∈ G, hðεωÞ = ε α hðωÞ: We note that the ðα, βÞ-symmetrical functions are a generalization of the notions of even, odd, and β-symmetrical functions. ...
... In [3], we observe that the theory of ðα, βÞ-symmetrical functions has many interesting applications; we now investigate some results in the classes of analytic functions. ...
... Theorem 1 (see [3], page 16). For every mapping h : Δ ↦ ℂ and a β-fold symmetric set Δ, there exists exactly one sequence of ðα, βÞ-symmetrical functions h α,β such that ...
In this note, we use the notions of α , β -symmetrical, generalized Janowski-type and spirallike functions to define a new class S α , β λ N , M , μ defined in the open unit disk. In particular, we obtain a structural formula, a representation theorem, Marx-Strohacker inequality. Our results continue to hold the covering and distortion properties.
... The theory of functions exhibiting (x, y)-symmetry has a wide range of intriguing applications. For instance, these functions are useful in exploring the set of fixed points of mappings, estimating the absolute value of certain integrals, and deriving results akin to Cartan's uniqueness theorem for holomorphic mappings, as demonstrated in [1]. The intrinsic properties of (x, y)-symmetrical functions are of great interest in the field of Geometric Function Theory. ...
... In 1995, Liczberski and Polubinski [1] introduced the notion of (x, y)-symmetrical functions for (y = 2, 3, . . . ) and (x = 0, 1, 2, . . . , y − 1). ...
In this article, the present study employs the utilization of the concepts pertaining to (x,y)-symmetrical functions, Janowski type functions, and q-calculus in order to establish a novel subclass within the open unit disk. Specifically, we delve into the examination of convolution properties, which serve as a tool for investigating and inferring adequate and equivalent conditions. Moreover, we also explore specific characteristics of the class S˜qx,y(α,β,λ), thereby further scrutinizing the convolution properties of these newly defined classes.
... The following result from [12] was used in this and the aforementioned article: ...
... Now, we will show that N k G M k G . For this purpose, let us remind that for the function f ∈ N k G there hold the sharp estimates µ G (Q f, m ) given by (12), while for function f ∈ M k G these estimates are presented in the formula (14). Therefore, the function f ∈ M k G that meets equality (14) does not belong to N k G . ...
In the [1], [4], [3] and [2] there were examined the Bavrin?s families (of holomorphic functions on bounded complete n? circular domains G ?Cn) in which the Temljakov operator Lf was presented as a product of a holomorphic function h with a positive real part and the (0, k)?symmetrical part of the function f,(k ? 2 is a positive integer). In [17] there was investigated the family of the above mentioned type, where the operator LLf was presented as a product of the same function h ? CG and (0, 2)-symmetrical part of the operator Lf. These considerations can be completed by the case of the factorization LLf by the same function h and the (0, k)-symmetrical part of operator Lf. In this article we will discuss the above case. In particular, we will present some estimates of a generalization of the norm of m-homogeneous polynomials Qf,m in the expansion of function f and we will also give a few relations between the different Bavrin?s families of the above kind.
... This definition applies concepts of odd, even, and planar Sakaguchi's functions to the -dimensional case. Liczberski and Polubinski [13] constructed the concept of ( , )-symmetrical functions for the positive integer and ( = 0, 1, 2, . . . , − 1). ...
... Theorem 1.1. [13] For the -fold symmetric set Σ, then for every function : Σ → C, can be written in the form, ...
In this paper, the concepts of -symmetrical functions and the concept of q -calculus are combined to define a new subclasses defined in the open unit disk. In particular. We look into a convolution property, and we'll use the results to look into our task even more, we deduce the sufficient condition, coefficient estimates investigate related neighborhood results for the class and some interesting convolution results are also pointed out.
... Liczberski and Polubinski [4] constructed the notion of (η, µ)-symmetrical functions for any integer (µ ≥ 2 and η = 0, 1, 2, . . . , µ − 1). ...
... We observe that K is µ-fold symmetric. We use the below unique decomposition [4] of every mappings h : K → C by ...
In our present work, the concepts of symmetrical functions and the concept of spirallike Janowski functions are combined to define a new class of analytic functions. We give a structural formula for functions in Sη,μ(F,H,λ), a representation theorem, the radius of starlikeness estimates, covering and distortion theorems and integral mean inequalities are obtained.
... The study of the ( j, k)-symmetrical function class S ( j,k) was initiated by Liczberski and Polubinski [17]. In their paper [17], they proved that, for every function defined over k-fold symmetric domain, there exists a unique sequence f j,k (z) of ( j, k)-symmetric functions such that ...
... The study of the ( j, k)-symmetrical function class S ( j,k) was initiated by Liczberski and Polubinski [17]. In their paper [17], they proved that, for every function defined over k-fold symmetric domain, there exists a unique sequence f j,k (z) of ( j, k)-symmetric functions such that ...
Let and denote, respectively, the integral means and the area of the image of the subdisk of a function f, which is analytic in . For with we introduce the family of the Janowski type (j, k)-symmetric starlike functions, which is denoted by Here, in this article, we first derive the bounds on for every The necessary coefficient condition for functions in the class is then presented. Our investigation leads us to get the sharp bounds on Yamashita’s functional of the form Finally, we provide the sharp estimate of the nth logarithmic coefficient.
... Liczberski and Po lubiński in [7] introduced the notion (j, k) symmetrical function (k = 2, 3, . . . ; j = 0, 1, . . . ...
... We observe that F 1 2 , F 0 2 and F 1 k are well-known families of odd functions, even functions and k-symmetrical functions respectively. It was further proved in [7] that each function defined on a symmetrical set can be uniquely represented as the sum of an even function and an odd function. ...
In this paper, the author introduces a new subclasses of analytic functions with respect to (j, k)-symmetric points and investigate various inclusion properties for these classes.Integral representation for functions in these classes and some interesting applications.
... The theory of (j, k) symmetrical functions has many interesting applications, for instance in the investigation of the set of fixed points of mappings, for the estimation of the absolute value of some integrals, and for obtaining some results of the type of Cartan's uniqueness theorem for holomorphic mappings [10]. ...
... Also, S (0,2) , S(1,2) and S (1,k) the classes of even, odd and k-symmetric functions respectively. We have the following decomposition theorem.Theorem 1[10] For every mapping f : D → C, where D is a k-fold symmetric set, there exists exactly the sequence of (j, k)symmetrical functions f j,k ,f(z) = k−1 j=0 f j,k (z) where f j,k (z) = 1 k k−1 v=0 ε −vj f(ε v z),(4)(f ∈ A; k = 1, 2, . . . ; j = 0, 1, 2, . . . ...
The aim of the present article is to introduce and study new subclass of Janowski type functions defined using notions of Janowski functions and (j, k)-symmetrical functions. Certain interesting coefficient inequalities, sufficiency criteria, distortion theorem, neighborhood property are investigated for this class.
... In the papers [3,6,11] there are considered the consequences of a modification of the above starlikeness factorization (1), using a unique decomposition of mappings f : B n → C n with respect to the cyclic group of k-th. roots of unity, k ≥ 2. Below we present such partition for mappings f : X ⊃ → Y, where X, Y are normed complex vector spaces and is a k-symmetric nonempty subset of X (ε = for the generator ε = exp 2πi k of the above group) [14]. By F j,k ( , Y) , j = 0, ..., k −1, let us denote the collection of ( j, k)-symmetrical maps f : → Y, i.e., maps f satisfying the condition ...
It is known that the starlikeness plays a central role in complex analysis, similarly as the convexity in functional analysis. However, if we consider the biholomorphisms between domains in C n , apart from starlikeness of domains, various symmetries are also important. This follows from the Poincaré theorem showing that the Euclidean unit ball is not biholomorphically equivalent to a polydisc in C n , n > 1 . From this reason the second author in 2003 considered some families of locally biholomorphic mappings defined in the Euclidean open unit ball using starlikeness factorization and a notion of k -fold symmetry. The 2017 paper of both authors contains some results on the absorption by a family S ( k ) , k ≥ 2 , of the above kind, the families of mappings biholomorphic starlike (convex) and vice versa. In the present paper there is given a new sufficient criterion for a locally biholomorphic mapping f , from the Euclidean ball B n into C n , to belong to the family S ( k ) , k ≥ 2 . The result is obtained using an n -dimensional version of Jack’s Lemma.
... In 1995, Liczberski and Polubinski [2] constructed the concept of (x, y)-symmetrical functions for (x = 0, 1, 2, . . . , y − 1), and ( y = 2, 3, . . . ...
The purpose of this paper is to define new classes of analytic functions by amalgamating the concepts of q-calculus, Janowski type functions and (x,y)-symmetrical functions. We use the technique of convolution and quantum calculus to investigate the convolution conditions which will be used as a supporting result for further investigation in our work, we deduce the sufficient conditions, Po´lya-Schoenberg theorem and the application. Finally motivated by definition of the neighborhood, we give analogous definition of neighborhood for the classes S˜qx,y(α,β) and K˜qx,y(α,β), and then investigate the related neighborhood results, which are also pointed out.
... F k represents the family including all k-fold symmetric functions. e concept of k-symmetrical function was protracted to so-called (j, k)-symmetrical function by Liczberski and Połubiński in [2]. To be specific, a function f(ξ) is reported for being (j, k)-symmetrical if ...
In this study, we familiarise a novel class of Janowski-type star-like functions of complex order with regard to (j, k)-symmetric points based on quantum calculus by subordinating with pedal-shaped regions. We found integral representation theorem and conditions for starlikeness. Furthermore, with regard to (j, k)-symmetric points, we successfully obtained the coefficient bounds for functions in the newly specified class. We also quantified few applications as special cases which are new (or known).
... Now we present a functions decomposition theorem [19]. ...
In the paper there is considered a generalization of the well-known Fekete–Szegö type problem onto some Bavrin’s families of complex valued holomorphic functions of several variables. The definitions of Bavrin’s families correspond to geometric properties of univalent functions of a complex variable, like as starlikeness and convexity. First of all, there are investigated such Bavrin’s families which elements satisfy also a ( j , k )-symmetry condition. As application of these results there is given the solution of a Fekete–Szegö type problem for a family of normalized biholomorphic starlike mappings in C n .
... The mentioned functional decomposition appears in the following result from [14]. ...
The paper concerns investigations of holomorphic functions of several complex variables with a factorization of their Temljakov transform. Firstly, there were considered some inclusions between the families of such holomorphic functions on complete n-circular domain of in some papers of Bavrin, Fukui, Higuchi, Michiwaki. A motivation of our investigations is a condensation of the mentioned inclusions by some new families of Bavrin’s type. Hence we consider some families of holomorphic functions f : defined also by a factorization of onto factors from and We present some interesting properties and extremal problems on .
In this paper, we introduce and study a new subclass of starlike functions with respect to [Formula: see text]-symmetric conjugate points using the principle of subordination. Several relationship with the well-known classes have been established. We have focussed on conic regions when it pertained to applications of our main results. Inclusion results, subordination property and coefficient inequality of the defined class are the main results of this paper. The applications of the results are presented here as corollaries, most of which are extensions of well-known results.
In the present article, the class Ss∗(j,m)(μ,η) is introduced using (j, m)-symmetrical functions. Further, we discuss certain interesting properties for the functions of this class.
We have introduced a comprehensive subclass of analytic functions with respect to (j,k) - symmetric points. We have obtained the interesting coefficient bounds for the newly defined classes of functions. Further, we have extended the study using quantum calculus. Our main results have several applications, here we have presented only a few of them.
The paper concerns holomorphic functions in complete bounded n-circular domains of the space . The object of the present investigation is to solve majorization problem of Temljakov operator. This type of problem has been studied earlier in Liczberski and Żywień (Folia Sci Univ Tech Res 33:37–42, 1986), Liczberski (Bull Technol Sci Univ Łódź 20:29–37, 1988) and Leś-Bomba and Liczberski (Demonstratio Math 42(3):491–503, 2009). In this paper we considered the family , i.e. the functions of the Bavrin family , which are (0, k)-symmetrical.
The aim in the present work is to introduce and study new subclasses of analytic functions that are defined by using the generalized classes of Janowski functions combined with the (n, m)-symmetrical functions, that generalize many others defined by different authors. We gave a representation theorem for these classes, certain inherently properties, while covering and distortion properties are also pointed out. ARTICLE HISTORY
The paper is devoted to the investigations of holomorphic functions on complete n-circular domains G of ℂn which are solutions of some partial differential equations in G. Our considerations concern a collection M
Gk, k ≥ 2, of holomorphic solutions of equations corresponding to planar Sakaguchi’s conditions for starlikeness with respect to k-symmetric points. In an earlier paper of the first author some embedding theorems for M
Gk were given. In this paper we solve the problem of finding some sharp estimates of m-homogeneous polynomials in a power series expansion of f from M
Gk. We obtain a formula of the extremal function which includes some special functions. Moreover, its construction is based on properties of hypergeometric functions and (j, k)-symmetric functions. The (j, k)-symmetric functions were considered in several papers of the second author and his co-author, J. Połubiński.
The theory of (j, k)-symmetric functions has many applications in the investigation of fixed points, estimation of the absolute values of some integrals and in obtaining results of the type of Cartan’s uniqueness theorem. The concept of -symmetric functions extends the idea of even, odd, k, 2k, -symmetric and conjugate functions. In this paper, we introduce a new class of analytic functions using the notion of -symmetric conjugate points. It unifies the classes and of starlike functions with respect to symmetric conjugate points and convex functions with respect to symmetric conjugate points, respectively. We also derive some inclusion results, integral representations and convolution conditions for functions belonging to the general function class . The various results presented in this paper may apply to yield the corresponding (new or known) results for a number of simpler known classes.
The subject of the paper concerns the existence of different inclusions between families St, Sc of C-n - biholomorphic starlike or convex mappings and a family S(k), k >= 2, of C-n - locally biholomorphic mappings. In the definition of S(k) we used a combination of a Kikuchi-Matsuno-Snifridge's starlikeness condition and a property of (j, k)-symmetrical mappings in C-n. This definition generalizes a notion of planar Sakaguchi's functions onto n-dimensional case. A motivation for the family S(k) comes from the paper [11] of the second author and the paper [6] of Hamada and Kohr. The family S(k) is a superset of a family which is connected with a problem posed in the first paper and solved in the second one.
Let B be the unit ball in ℂ n with respect to an arbitrary norm on ℂ n . In this paper, we give a necessary and sufficient condition that a Loewner chain f(z,t), such that {e -t f(z,t)} t≥0 is a normal family on B, is k-fold symmetrical. As a corollary, we give a necessary and sufficient condition that a normalized locally biholomorphic mapping on B is spirallike of type α and k-fold symmetrical. When α=0, this result solves a natural problem that is similar to an open problem posed by Liczberski. We also give two examples of k-fold symmetrical Loewner chains.
The paper concerns complex-valued functions which are holomorphic in bounded complete n-circular domains [Inline formula] and fulfil some geometric conditions. The families [Inline formula] of such kind of functions were considered for instance by Bavrin [1,2], Dobrowolska and Liczberski [4], Dziubiński and Sitarski [5], Fukui [6], Higuchi [8], Jakubowski and Kamiński [9], Liczberski and Wrzesień [14], Marchlewska [15,16], Michiwaki [17], and Stankiewicz [22]. The above functions were applied later to research some families of locally biholomorphic mappings in [Inline formula] (see for instance Pfaltzgraff and Suffridge [19], Liczberski [12], Hamada, Honda and Kohr [7]). In this paper, we consider an interesting family [Inline formula] of the type [Inline formula] which separates two Bavrin’s families [Inline formula]. These families correspond to the well-known families of convex univalent and close-to-convex univalent functions of one variable, respectively. We define [Inline formula] using the property of evenness of functions. We obtain for [Inline formula] some embedding theorems relevant to the mentioned separation question. Applying the Minkowski function of [Inline formula], we solve also some extremal problems for functions from [Inline formula]. As an application, we give a topologic property of the family [Inline formula].
In the paper the decomposition with respect to the group of roots of unity is applied to characterize some classes of biholomorphic mappings in There is considered the problem of subordination and majorization relations between convex mappings and their components in the above partition. There is also given a distortion theorem for convex mappings and a criterion for starlikeness in
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Authors' addresses: Piotr Liczberski
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Authors' addresses: Piotr Liczberski, Jerzy Polubiński, Institute of Mathematics, Technical University of Lódź, 90-924 Lódź ul. Źwirki 36, Poland.