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In this paper, we define a new class R t , δ , υ m , n , σ (A , B) of holomorphic functions in the open unit disk defined connected with the combination binomial series and Babalola operator using the differential subordination with Janowski-type functions. Using the well-known Carathéodory's inequality for function with real positive parts and the Keogh-Merkes and Ma-Minda's in equalities, we determined the upper bound for the first two initial coefficients of the Taylor-Maclaurin power series expansion. Then, we found an upper bound for the Fekete-Szegö functional for the functions in this family. Further, a similar result for the first two coefficients and for the Fekete-Szegő inequality have been done the function G − 1 when G ∈ R t , δ , υ m , n , σ (A , B) . Next, for the functions of these newly defined family we determine coefficient estimates, distortion bounds, radius problems, and the radius of starlikeness and close-to-convexity. The novelty of the results is that we were able to investigate basic properties of these new classes of functions using simple methods and these classes are connected with the new convolution operator and the Janowski functions. [ABSTRACT FROM AUTHOR]

The aim of this article is to establish two new and qualitative subfamilies F(ε, κ, ℵ) and G(ε, κ, ℵ) of biunivalent functions. For functions in these subfamilies, we determine the first two Maclaurin coefficient estimations |C2| and |C3|, and address the Fekete-Szegö problem. Additionally, we mention some corollaries related to the main results. [ABSTRACT FROM AUTHOR]

Our research introduces new subclasses of analytical functions that are defined by Euler polynomials. We then proceed to estimate the Fekete-Szegö functional problem and the Maclaurin coefficients for this specific subfamily, denoted as |a2| and |a3|. Furthermore, we demonstrate several new results that emerge when we specialize the parameters used in our main findings. [ABSTRACT FROM AUTHOR]

This study established upper bounds for the second and third coefficients of analytical and bi-univalent functions belonging to a family of particular classes of analytic functions utilizing q−Ultraspherical polynomials under q−Saigo’s fractional integral operator. We also discussed the Fekete-Szegö family function problem. As a result of the specialization of the parameters used in our main results, numerous novel outcomes were demonstrated. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Bi-univalent functions associated with the leaf-like domain within the open unit disk are investigated and a new subclass is introduced and studied in the research presented here. This is achieved by applying the subordination principle for analytic functions in conjunction with q -calculus. The class is proved to be not empty. By proving its existence, generalizations can be given to other sets of functions. In addition, coefficient bounds are examined with a particular focus on | α 2 | and | α 3 | coefficients, and Fekete–Szegö inequalities are estimated for the functions in this new class. To support the conclusions, previous works are cited for confirmation. [ABSTRACT FROM AUTHOR]

A new algorithm is proposed in this paper to explain how the modified homotopy perturbation approach can be successfully implemented based on the Pade approximants and the Laplace transform in order to acquire the accurate solutions of a nonlinear system of multi-photograph delay differential equations. The method that has been suggested has the benefit of giving exact solutions, and it is simple to implement analytically on the issues that have been presented. Examples have been provided to demonstrate that this strategy may be utilized and is successful in its application. The results show that the method that was described could be used to solve many different types of differential equations. [ABSTRACT FROM AUTHOR]

We introduce a new subclass J Σ η , 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 a 2 and a 3 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. [ABSTRACT FROM AUTHOR]

In the present paper, a subclass of analytic and bi-univalent functions is defined using a symmetric q-derivative operator by means of Gegenbauer polynomials. Coefficients bounds 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. [ABSTRACT FROM AUTHOR]

In this paper, a subclass of analytic and bi-univalent functions by means of (p, q) − Lucas polynomials is introduced. Certain coefficients bounds for functions belonging to this subclass are obtained. Furthermore, the Fekete-Szeg¨o problem for this subclass is solved. [ABSTRACT FROM AUTHOR]

By means of Gegenbauer polynomials, two subclasses of analytic and bi-univalent functions are introduced. Coefficient bounds and Fekete-Szegö inequalities for functions belong to these subclasses are obtained. [ABSTRACT FROM AUTHOR]

In this paper, a subclass TΣ ζ (m, γ,λ, p, q) of analytic and bi-univalent functions by means of (p, q)-Chebyshev polynomials is introduced. Certain coefficient bounds for functions belong to this subclass are obtained. In addition, the Fekete-Szego problem is solved in this subclass. [ABSTRACT FROM AUTHOR]