safa salehian

In this paper, we introduce and investigate a subclass  $\mathcal{G}_{\Sigma}^{h,p}(\lambda,m,n, \alpha,\gamma)$ of bi-univalent functions in the open unit disk $\mathbb{U}$. Upper bounds for this class's second and third coefficients of functions are found. The results, which we have presented in this paper, would generalize and improve some recent works of several earlier authors.

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.

‎In the present paper‎, ‎we define two subclasses $Sigma(lambda‎, ‎alpha‎, ‎beta)$‎, ‎$Sigma_{mathcal C}(alpha‎, ‎beta)$ of meromorphic univalent functions and subclass $Sigma_{{mathcal B},{mathcal C}}(alpha‎, ‎beta‎, ‎lambda)$ of meromorphic bi-univalent functions‎. ‎Furthermore‎, ‎we obtain estimates on the general coefficients $|b_n|~(n ge1)$ for functions in the subclasses $Sigma(lambda‎, ‎alpha‎, ‎beta)$‎, ‎$Sigma_{mathcal C}(alpha‎, ‎beta)$ and estimates for the early coefficients of functions in subclass $Sigma_{{mathcal B},{mathcal C}}(alpha‎, ‎beta‎, ‎lambda)$ by associated subordination‎. ‎The results obtained in this paper would generalize and improve those in related works of several earlier authors‎.

Let 𝝨 be the class of meromorphic bi-univalent functions f of the formf(z)=z+b_0+∑_(n=1)^∞▒b_n/z^n ,which are univalent (analytic and one to one) on the domain Δ={z∈C∶1