m-fold symmetric bi-univalent functions

The purpose of the present paper is to introduce two new subclasses of the function class ∑m  of bi-univalent functions which both f  and f-1  are m-fold symmetric analytic functions. Furthermore, we obtain estimates on the initial coefficients for functions in each of these new subclasses. Also we explain the relation between our results with earlier known results.

In the present paper, we introduce a new subclass H∑m (λ,β)of the m-fold symmetric bi-univalent functions. Also, we find the estimates of the Taylor-Maclaurin initial coefficients |am+1| , |a2m+1| and general coefficients |amk+1| (k ≥ 2) for functions in this new subclass. The results presented in this paper would generalize and improve some recent works of several earlier authors.

The purpose of the present paper is to introduce and investigate two new subclasses 𝒦𝛴𝑚(𝜆,𝛾;𝛼) and 𝒦∗𝛴𝑚(𝜆,𝛾;𝛽) of 𝛴𝑚 consisting of analytic and 𝑚-fold symmetric bi-univalent functions defined in the open unit disk 𝑈. We obtain upper bounds for the coefficients |𝑎𝑚+1| and |𝑎𝑚| for functions belonging to these subclasses. Many of the well-known and new results are shown to follow as special cases of our results.