On Some Subclasses of $m$-fold Symmetric Bi-univalent Functions associated with the Sakaguchi Type Functions

Ismaila O Ibrahim,Amol B Patil,Timilehin G Shaba

doi:10.34198/ejms.8122.115

Ismaila O Ibrahim, Amol B Patil + Show 1 more

Open Access

https://doi.org/10.34198/ejms.8122.115

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Abstract

In the present investigation, we introduce the subclasses $\Lambda_{\Sigma_m}^{\rightthreetimes}(\sigma,\phi,\upsilon)$ and $\Lambda_{\Sigma_m}^{\rightthreetimes}(\sigma,\gamma,\upsilon)$ of $m$-fold symmetric bi-univalent function class $\Sigma_m$, which are associated with the Sakaguchi type of functions and defined in the open unit disk. Further, we obtain estimates on the initial coefficients $b_{m+1}$ and $b_{2m+1}$ for the functions of these subclasses and find out connections with some of the familiar classes.

Highlights

Let N and C represent the sets of natural numbers and complex numbers respectively and A be the family of analytic functions that are defined in U = {z ∈ C : |z| < 1} and have the series expansion
With reference to the Sakaguchi type function classes defined by Lokesh and Keerthi [11], we obtain estimates on initial coefficients |bm+1| and |b2m+1| for functions belong to the new subclasses ΛΣm(σ, φ, υ) and ΛΣm(σ, γ, υ) of the function class Σm
Initial coefficient bounds for a subclass of m-fold symmetric bi-univalent functions, Tbilisi Math

Summary

Introduction

Let Sm be the class of univalent and m-fold symmetric functions in U, which are of the form (1.3). On Some Subclasses of m-fold Symmetric Bi-univalent Functions. For each m ∈ N , every f ∈ Σ produces a m-fold symmetric bi-univalent function. For a function p of the form (1.3), an univalent continuation of p−1 to U is given by (see Srivastava et al [28]) the series expansion q(w) = w − bm+1wm+1 + (m + 1) b2m+1 − b2m+1 w2m+1−. With reference to the Sakaguchi type function classes defined by Lokesh and Keerthi [11], we obtain estimates on initial coefficients |bm+1| and |b2m+1| for functions belong to the new subclasses ΛΣm(σ, φ, υ) and ΛΣm(σ, γ, υ) of the function class Σm. We have pointed out connections with certain familiar subclasses of the class Σ

Coefficient Bounds for the Function Class

Corollaries and Consequences