In this paper we first introduce a new class of derivative operators, termed g-derivative operators, which unifies the q-derivative, (p,q)-derivative and (α,β,γ)-derivative operators, even classical derivative in the literature. Based on this generalized framework, we define two novel function classes, specifically g-starlike functions and g-convex functions. Further, we employ the subordination principle of analytic functions to conduct the coefficient estimations for such function classes, and subsequently derive the bounds for the corresponding Fekete-Szegö inequality, Toeplitz determinants and Hankel determinants.

We introduce and study two new subclasses of the function class Σ of bi-univalent functions defined in the open unit disk and associated with the Sălăgean and q-derivative operators. Furthermore, we find estimates for the Taylor–Maclaurin coefficients |a_2| and |a_3| of the functions from these new subclasses. The accumulated results are quite interesting and new.

In this investigation, we derive Fekete–Szegè inequalities and upper bounds for the second Hankel determinant for a new subclass of analytic functions in the open unit disk formed by the close-to-convex functions and the Salagean derivative operator. Additionally, there is discussion of a number of intriguing applications of the findings given here.

In this paper, we establish the initial Taylor-Maclaurin coefficients for normalized analytic functions f in the open unit disk. We also assume that f and its inverse g=f ^-1 satisfy the following conditions e^iϕ[ f ^' (z)] ^τ[ z/f (z)] ^ν ≺ψ (z)cosϕ +isinϕ and e^iϕ[ g'(z)] ^τ[ z/g(z)] ^ν ≺ψ (z)cosϕ +isinϕ , for -π /2<ϕ <π /2 , where ψ is a univalent function whose range is symmetric with respect to the real axis, and τ and ν are non-zero real numbers. We also examine other classes of related functions and establish connections with previously known results.

In this study, a new subclass of analytic functions in the open unit disk was defined with the help of the function obtained by applying the Salagean derivative to the q-Ruscheweyh derivative operator. For the functions belonging to this subclass, the coefficient bounds for the Fekete-Szeg & ouml; functional were determined. The results of this paper are a generalization of the results of Kanas and Darwish [Fekete-Szeg & ouml; problem for starlike and convex functions of complex order, Appl. Math. Lett. 23(7) (2010) 777-782].

The purpose of this paper is to find initial coefficient bounds |a_2| and |a_3| and Fekete-Szegö estimates for the functions that belong to a newly defined subclass 𝒢ℬ^ν_κ(l) consisting of analytic functions normalized by the conditions f(0)=0 and f^'(0)=1 defined by the subordination to Limaçon-shaped domain. Similar results have been derived for the inverse function f^-1 , logf(z)z and zf(z) . Furthermore, applications of our results to certain distributions are defined and discussed using Hadamard product. Our findings generalize existing results and also introduce novel subclasses of univalent functions.

The main objective of this paper is to present and investigate a subclass 𝒞(b, q) of q-convex functions in the unit disk that is defined by the q-Bernoulli numbers. For this subclass, we find the upper bounds on the Fekete-Szeg functional, the coefficient bounds, and the second Hankel determinant.

In this research, we present and study a new subclass of bi-univalent functions related to the Krawtchouk polynomials that meet subordination requirements seen in the open unit disk, a symmetric domain. We derive estimates for the Fekete-Szegoinequality |a(3)-gamma a(2)(2)| and the Taylor-Maclaurin coeffcients |a(2)| , |a(3)| for this new subclass.

Communication networks are exposed to internal or external risks that can affect all or part of the system. The most important components that form the infrastructure of these systems are routers, which act as nodes. In the field of graph theory, there are sophisticated techniques that can be used to optimize the path of a packet as it travels through various routers from its origin to its destination. A notable example of such an algorithm is Dijkstra's algorithm, which is designed to efficiently determine the shortest path. The algorithm works under the assumption that the system operates under ideal conditions. Real-time systems can perform better if risk factors and optimal conditions are taken into account. The relationship between the nodes can be expressed by various metrics such as distance, delay, and bandwidth. The aforementioned metrics facilitate the calculation of the optimal path, with the ultimate objective of achieving low-latency networks characterized by rapid response times. Round-trip time (RTT) can be employed as a metric for measuring enhancements in a range of latency types, including those associated with processing, transmission, queuing, and propagation. The use of Z-numbers was employed in this study to incorporate risk into the optimal path metric. RTT was the preferred metric and reliability was represented by fuzzy linguistic qualifiers. A comparison of several scenarios was shown using a numerical example of a communication network. It is expected that this study will have a significant impact on the evolution from models that consider only ideal conditions to real-time systems that include risks using Z-numbers.

In this paper, we obtain upper bounds of the initial Taylor-Maclaurin coefficients |a2|, |a3| and |a4| and of the Fekete-Szego functional vertical bar a3 - eta a(2) (2)vertical bar for certain subclasses of analytic and 2 bi-starlike functions S sigma*(beta,theta, n, m) in the open unit disk. We have also obtained an upper bound of the functional vertical bar a(2)a(4) - a(2) vertical bar for the 3 functions in the class S sigma*(beta,theta, n, m). Moreover, several interesting applications of the results presented here are also discussed.

In the present paper, we obtain the upper bounds for the second Hankel determinant and Fekete-Szegö inequalities for a new subclass of analytic functions in the open unit disk defined by the hypergeometric functions. Moreover, several interesting applications of the results presented here are also discussed.

In this present investigation, with the help of the Sălăgean differential operator and the Hadamard product, we define new subclasses of analytic functions. We investigate sharp upper bounds for these subclasses. The results presented in this paper have been shown to generalize and improve some recent work of Frasin and Darus (Internat J Math Math Sci 24(9): 577–581, 2000).

The main goal of this investigation is to obtain sharp upper bounds for Fekete-Szegö functional and the third Hankel determinant for a certain subclass SL∗u,v,α of holomorphic functions defined by the Carlson-Shaffer operator in the unit disk. Finally, for some special values of parameters, several corollaries were presented.

We introduce a comprehensive subfamily of analytic and bi-univalent functions in this study using Horadam polynomials and the q-analog of the Noor integral operator. We establish upper bounds for the absolute values of the second and the third coefficients and the Fekete–Szegö functional for the functions belonging to this family. Various observations of the results presented here are also discussed.

Using $ (p, q) $-Lucas polynomials and bi-Bazilevic type functions of order $\rho +i\xi,$ we defined a new subclass of biunivalent functions. We obtained coefficient inequalities for functions belonging to the new subclass. In addition to these results, the upper bound for the Fekete-Szegö functional was obtained. Finally, for some special values of parameters, several corollaries were presented.

In this paper, we introduce some new subclasses of analytic functions in the open unit disk U with negative coefficients defined by generalized Alexander integral operator. The aim of the present paper is to determine coefficient inequalities, inclusion relations, neighborhoods, partial sums and integral means properties for functions f belonging to these subclasses.

This paper's main goal is to introduce and study a subclass S*(b, q) of q-starlike functions in the unit disk defined by the q-Bernoulli numbers. We determine the coefficient bounds, the upper bounds for the Fekete-Szego functional, and the second Hankel determinant for this subclass.

Our objective is to usher and investigate the subclass𝒮^*_∑^η_q(μ,λ;ϕ) of the function class ∑ of analytic and bi-univalent functions related with the symmetric q-derivative operator and the generalized Bernardi integral operator. On the one hand, without the generalized Bernardi integral operator we estimate the second Hankel determinants for the reduced subclasses 𝒮^*_∑_q(λ;ϕ) with respect to symmetric points. On the other hand, we also give the corresponding results of Fekete-Szegö functional inequalities and the upper bounds of the coefficients a_2 and a_3 for these subclasses.

In the discipline of geometric function theory, Lucas polynomials and other special polynomials have recently acquired traction. We establish a new class of bi-univalent functions and get coefficient estimates and Fekete–Szegö inequalities for this new class in this paper by connecting these polynomials, subordination and combination of Babalola and Opoola operator.

In the current work, we discuss certain stirring results of coe cient estimates of a uni ed class which is bridge between bi-starlike and bi-convex functions related to shell-like curves by means of subordination. Further, appropriate connections are discussed.