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Studies of Sălăgean differential operator Dκ in connection with Stirling numbers are relatively new. In this paper, the differential operator Dκ involving Stirling numbers is considered. New necessary and sufficient conditions involving Stirling numbers for the series ϒθs(ς) written by the Pascal distribution are discussed for the subclass Tκ(ϵ,α). Also, we provide sufficient condition for the inclusion relation Iθs(Rϖ(E,D))⊂Tκ(ϵ,α). Further, we consider the properties of integral operator related to Pascal distribution series. New special cases as consequences of the main results are also obtained.

Sălăgean differential operator Dκ plays an important role in the geometric function theory, where many studies are using this operator to introduce new subclasses of analytic functions defined in the open unit disk. Studies of Sălăgean differential operator Dκ in connection with Stirling numbers are relatively new. In this paper, the differential operator Dκ involving Stirling numbers is considered. A new sufficient condition involving Stirling numbers for the series Υθs(ϰ) written with the Pascal distribution are discussed for the subclass Tκ(ϵ,♭). Also, we provide a sufficient condition for the inclusion relation IθsRϖ(E,D)⊂Tκ(ϵ,♭). Further, we consider the properties of an integral operator related to Pascal distribution series. New special cases as a consequences of the main results are also obtained.

This research paper introduces the novel subclass ϒΣϑ,β,μq˜ of bi-univalent functions that are connected to Fibonacci numbers. Our main contributions in this study involve establishing constraints on the absolute values of the second coefficient a2 and the third coefficient a3 for functions within this specific subclass. In addition, we provide solutions to Fekete–Szegö functional problems. Furthermore, our investigation reveals intriguing outcomes resulting from the specific parameter values used in our main findings.

This study establishes a new method to investigate bounds of ak; k≥n, for certain general classes of bi-univalent functions. The results include a number of improvements and generalizations for well-known estimations. We also discuss bounds of nan2−a2n−1 and consider several corollaries, remarks, and consequences of the results presented in this paper.

The aim of this paper is to find the necessary and sufficient conditions and inclusion relations for Pascal distribution series to be in the classes $\mathcal{SP}_p(\alpha,\beta)$ and $\mathcal{UCV}_p(\alpha,\beta)$ of uniformly spirallike functions. Further, we consider an integral operator related to Pascal distribution series. Several corollaries and consequences of the main results are also considered.

We introduce and study two certain classes of holomorphic and bi-univalent functions associating $\lambda$-pseudo-starlike functions with Sakaguchi-type functions. We determine upper bounds for the Taylor-Maclaurin coefficients $\vert a_2\vert$ and $\vert a_3\vert$ for functions belonging to these classes. Further we point out certain special cases for our results.

The objective of the current paper is to find the necessary and sufficient conditions for Miller–Ross-type Poisson distribution series to be in the classes ST*(γ,β) and KT(γ,β) of analytic functions with negative coefficients. Furthermore, we investigate several inclusion properties of the class Yσ(V,W) associated of the operator Iα,cε defined by this distribution. We also take into consideration an integral operator connected to series of Miller–Ross-type Poisson distributions. Special cases of the main results are also considered.

Our aim in this study is to consider a generalization of the concept of m-complex symmetric transformations to n-quasi-m-complex symmetric transformations. A map S∈B(Y) is said to be an n-quasi-m-complex symmetric transformation if there exists a conjugation C on Y such that S satisfies the condition S*n∑0≤k≤m(−1)m−kmkS*kCSm−kCSn=0, for some positive integers n and m. This class of transformation contains the class of m-complex symmetric transformations as a proper subset. Some basic structural properties of n-quasi-m-complex symmetric linear transformations are established with the help of transformation matrix representation. In particular, we obtain that a power of an n-quasi-m-complex symmetric is again an n-quasi-m-complex symmetric operator. Moreover, if T and S are such that T is an n1-quasi-m1-complex symmetric and S is an n2-quasi-m2-complex symmetric, their product TS is an max{n1,n2}-quasi-(m1+m2−1)-complex symmetric under suitable conditions. We examine the stability of n-quasi-m-complex symmetric operators under perturbation by nilpotent operators.

The purpose of this paper is to find new inclusion relations of the harmonic class HF(ϱ,γ) with the subclasses SHF*,KHF and TNHF(τ) of harmonic functions by applying the convolution operator Θ(ℑ) associated with the Mittag-Leffler function. Further for ϱ=0, several special cases of the main results are also obtained.

We introduce a new subclass JΣη,δ,μ(p˜) of bi-univalent functions, defined by shell-like curves connected with Fibonacci numbers. Our main results in this paper include estimates of the Taylor–Maclaurin coefficients a2 and a3 for functions in this subclass, as well as solutions to Fekete–Szegö functional problems. We also show novel outcomes resulting from the specialization of the parameters used in our main results.

Recently, some researchers determined lower bounds for the normalized version of some special functions to its sequence of partial sums, e.g., Struve and Dini functions, Wright functions and Miller–Ross functions. In this paper, we determine lower bounds for the normalized Le Roy-type Mittag-Leffler function Fα,βγ(z)=z+∑n=1∞Anzn+1, where An=ΓβΓα(n−1)+βγ and its sequence of partial sums (Fα,βγ(z))m(z)=z+∑n=1mAnzn+1. Several examples of the main results are also considered.

Utilizing the concepts from q-calculus in the field of geometric function theory, we introduce a subclass of p-valent meromorphic functions relating to the domain of lemniscate of Bernoulli. The well known problem of Fekete-Szegö for this class is evaluated. Also some geometric results related to subordinations are evaluated for this class in connection with Janowski functions.

In this paper, we derive several subordination results and integral means result for certain class of analytic functions defined by means of q-differential operator. Some interesting corollaries and consequences of our results are also considered.

Making use of Horadam polynomials, we propose a special family of regular functions of the type gz=z+∑j=2∞djzj which are bi-univalent (or bi-schlicht) in the disc z∈ℂ:z

The prime purpose of this article is to derive a necessary and sufficient condition for a linear operator associated with the Pascal distribution series to be in the class TSμ,σ,δ of analytic functions. Moreover, inclusion relation and an integral operator linked to the Pascal distribution series is considered. We have also provided some results as corollaries of our theorems.

In this paper, we introduce three new classes SkaM,N;μ,Rkaμ, and Tkaθ of analytic functions defined by Fournier–Ruscheweyh integral operator. For these classes, we investigate the majorization problem. Furthermore, a number of new results are shown to follow upon specializing the parameters involved in our main results.

In the present paper, we will observe that the Sălăgean differential operator can be written in terms of Stirling numbers. Furthermore, we find a necessary and sufficient condition and inclusion relation for Pascal distribution series to be in the class ℙkλ,α of analytic functions with negative coefficients defined by the Sălăgean differential operator. Also, we consider an integral operator related to Pascal distribution series. Several corollaries and consequences of the main results are also considered.

In this paper, the function classes SPp(σ,ν) and UCSP(σ,ν) are investigated for the normalized Wright functions. More precisely, several sufficient and necessary conditions are provided so that the aforementioned functions are in these classes. Furthermore, several corollaries will follow from our results.

In this study, a new class RΣμ(x,γ,α,δ,β) of bi-univalent functions studied by means of Gegenbauer polynomials (GP) with Rabotnov functions is introduced. The coefficient of the Taylor coefficients a2 and a3 and Fekete-Szegö problems for functions belonging to RΣμ(x,γ,α,δ,β) have been derived as well. Furthermore, a variety of new results will appear by considering the parameters in the main results.

The Miller–Ross-type Poisson distribution is an important model for plenty of real-world applications. In the present analysis, we study and introduce a new class of bi-univalent functions defined by means of Gegenbauer polynomials with a Miller–Ross-type Poisson distribution series. For functions in each of these bi-univalent function classes, we have derived and explored estimates of the Taylor coefficients a2 and a3 and Fekete-Szegö functional problems for functions belonging to these new subclasses.

The zero-truncated Poisson distribution is an important and appropriate model for many real-world applications. Here, we exploit the zero-truncated Poisson distribution probabilities to construct a new subclass of analytic bi-univalent functions involving Gegenbauer polynomials. For functions in the constructed class, we explore estimates of Taylor–Maclaurin coefficients a2 and a3, and next, we solve the Fekete–Szegő functional problem. A number of new interesting results are presented to follow upon specializing the parameters involved in our main results.

In the present paper, a subclass of analytic and biunivalent functions by means of Gegenbauer polynomials is introduced. Certain coefficients bound for functions belonging to this subclass are obtained. Furthermore, the Fekete-Szegö problem for this subclass is solved. A number of known or new results are shown to follow upon specializing the parameters involved in our main results.

By using k-Fibonacci numbers, we present a comprehensive family of regular and biunivalent functions of the type gz=z+∑j=2∞ djzj in the open unit disc D. We estimate the upper bounds on initial coefficients and also the functional of Fekete-Szegö for functions in this family. We also discuss few interesting observations and provide relevant connections of the result investigated.

In this paper, we introduce a special family Mσm(τ,ν,η,φ) of the function family σm of m-fold symmetric bi-univalent functions defined in the open unit disc D and obtain estimates of the first two Taylor–Maclaurin coefficients for functions in the special family. Further, the Fekete–Szegö functional for functions in this special family is also estimated. The results presented in this paper not only generalize and improve some recent works, but also give new results as special cases.

In the current work, we introduce a special family of the function family of analytic and m-fold symmetric bi-univalent functions and obtain estimates of the Taylor–Maclaurin coefficients |dm+1| and |d2m+1| for functions in the special family. For δ a real number, Fekete–Szegö functional |d2m+1−δdm+12| for functions in the special family is also estimated. We indicate several cases of the defined family and connections to existing results are also discussed.

The main object of the present paper is to apply the concepts of p,q-derivative by establishing a new subclass of analytic functions connected with symmetric circular domain. Further, we investigate necessary and sufficient conditions for functions belonging to this class. Convex combination, weighted mean, arithmetic mean, growth theorem, and convolution property are also determined.

In the present analysis, we aim to construct a new subclass of analytic bi-univalent functions defined on symmetric domain by means of the Pascal distribution series and Gegenbauer polynomials. Thereafter, we provide estimates of Taylor–Maclaurin coefficients a2 and a3 for functions in the aforementioned class, and next, we solve the Fekete–Szegö functional problem. Moreover, some interesting findings for new subclasses of analytic bi-univalent functions will emerge by reducing the parameters in our main results.