Initial coefficient bounds for a subclass of $m$-fold symmetric bi-univalent functions

Let $\Sigma$ denote the class of functions $$f(z)=z+\sum_{n=2}^{\infty}a_nz^n$$ belonging to the normalized analytic function class $\mathcal{A}$ in the open unit disk $\mathbb{U}$, which are bi-univalent in $\mathbb{U}$, that is, both the function $f$ and its inverse $f^{-1}$ are univalent in $\mathbb{U}$. The usual method for computation of the coefficients of the inverse function $f^{-1}(z)$ by means of the relation $f^{-1}\big(f(z)\big)=z$ is too difficult to apply in the case of $m$-fold symmetric analytic functions in $\mathbb{U}$. Here, in our present investigation, we aim at overcoming this difficulty by using a general formula to compute the coefficients of $f^{-1}(z)$ in conjunction with the residue calculus. As an application, we introduce two new subclasses of the bi-univalent function class $\Sigma$ in which both $f(z)$ and $f^{-1}(z)$ are $m$-fold symmetric analytic functions with their derivatives in the class $\mathcal{P}$ of analytic functions with positive real part in $\mathbb{U}$. For functions in each of the subclasses introduced in this paper, we obtain the coefficient bounds for $|a_{m+1}|$ and $|a_{2m+1}|$.

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