Exploiting Miller-Ross Poisson distribution to construct novel subclass of bi-univalent functions

This paper deals with two new subclasses of holomorphic and bi-univalent functions in the open unit disk defined by generalized bivariate Fibonacci polynomials. In this paper the coefficient bounds are estimated for [Formula: see text] and [Formula: see text] which [Formula: see text] and [Formula: see text] are the Taylor–Maclaurin coefficients of the functions belonging to these new subclasses. Then, the Fekete–Szegö problem is handled for the functions in these subclasses. Also, several remarks are presented. The results of this paper generalize certain earlier results in the literature.

<abstract><p>In the current article, we introduced new subclasses of bi-univalent functions associated with bounded boundary rotation. For these new classes, the authors first obtained two initial coefficient bounds. They also verified the special cases where the familiar Brannan and Clunie's conjecture were satisfied. Furthermore, the famous Fekete-Szegö inequality was obtained for the newly defined subclasses of bi-univalent functions, and some of the results improved the earlier results available in the literature.</p></abstract>

In mathematics, physics, and engineering, orthogonal polynomials and special functions play a vital role in the development of numerical and analytical approaches. This field of study has received a lot of attention in recent decades, and it is gaining traction in current fields, including computational fluid dynamics, computational probability, data assimilation, statistics, numerical analysis, and image and signal processing. In this paper, using q-Hermite polynomials, we define a new subclass of bi-univalent functions. We then obtain a number of important results such as bonds for the initial coefficients of |a2|, |a3|, and |a4|, results related to Fekete–Szegö functional, and the upper bounds of the second Hankel determinant for our defined functions class.

In 2010, Srivastava et al. [Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett.23(10) (2010) 1188–1192] reviewed the study of coefficient problems for bi-univalent functions. Inspired by the pioneering work of Srivastava et al. [Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett.23(10) (2010) 1188–1192], there has been triggering interest to study the coefficient problems for the different subclasses of bi-univalent functions. Motivated largely by Srivastava et al. [Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett.23(10) (2010) 1188–1192] and Halim et al. [Coefficient estimates for meromorphic bi-univalent functions, preprint (2011), arXiv:1108.4089], in this paper, we propose to investigate the coefficient estimates for two classes of meromorphic bi-univalent functions. Also, we find estimates on the coefficients |b0| and |b1| for functions in these new classes. Some interesting remarks of the results presented here are also discussed.

In this investigation, by using a relation of subordination, we define a new subclass of analytic bi-univalent functions associated with the Fibonacci numbers. Moreover, we survey the bounds of the coefficients for functions in this class.

In this article, the authors use the Faber polynomial expansions to find the general coefficient estimates for a few new subclasses of bi-univalent functions with bounded boundary rotation and bounded radius rotation. Some of the results improve the existing coefficient bounds in the literature.

In the present paper, introduction of new subclasses of bi-univalent functions in the open disk was defined. Moreover,by using Salagean operator,in these new subclasses for functions, upper bounds for the second and third coefficients were found. Presented results are a generalization of the results obtained by Srivastava et al.[12], Frasin and Aouf [7] and Caglar et al.[5].

In this paper, we introduce and investigate new subclasses of bi-univalent functions with respect to the symmetric points in U=z∈C:z<1 defined by Bernoulli polynomials. We obtain upper bounds for Taylor–Maclaurin coefficients a2,a3 and Fekete–Szegö inequalities a3−μa22 for these new subclasses.

In this study, we have delineated a novel subclass of bi-univalent functions in the unit disk and established on the coefficients also by using Bernoulli polynomials. Moreover, the Fekete-Szegö problem within this subclass has been resolved.

In this paper we introduce general subclasses of bi-univalent functions by using convolution. Bounds for the first two coefficients |a2| and |a3| for bi-univalent functions in these classes are obtained. The obtained results generalize the results which are given in [Murugusundaramoorthy, G., Magesh, M., Prameela, V., Coefficient bounds for certain subclasses of bi-univalent function, Abstr. Appl. Anal., (2013), Art. ID 573017, 3 pp.] and [Brannan, D. A. and Taha, T. S., On some classes of bi-univalent functions, Studia Univ. Babes¸ Bolyai Math., 31 (1986), No. 2, 70–77].

We introduce two subclasses of biunivalent functions and find estimates on the coefficientsa2anda3for functions in these new subclasses. Also, consequences of the results are pointed out.

In this article, two new subclasses of the bi-univalent function class σ related with Legendre polynomials are presented. Additionally, the first two Taylor–Maclaurin coefficients a2 and a3 for the functions belonging to these new subclasses are estimated.

The purpose of the present paper is to introduce a new subclass of the function class ∑ of bi-univalent functions defined in the open unit disc. Furthermore, we obtain estimates on the coefficients ∣a2∣ and ∣a3∣ for functions of this class. Relevant connections of the results presented here with various well-known results are briefly indicated.

New subclasses of bi-univalent functions

In this work, considering a new subclass of bi-univalent functions which are m-fold symmetric and analytic functions in the open unit disk, we determine estimates for the general Taylor-Maclaurin coefficient of the functions in this class. Furthermore, initial upper bounds of coefficients for m-fold symmetric, analytic and bi-univalent functions were found in this study. For this purpose, we used the Faber polynomial expansions. In certain cases, the coefficient bounds presented in this paper would generalize and improve some recent works in the literature. We hope that this paper will inspire future researchers in applying our approach to other related problems.