General coefficient estimates for bi-univalent functions: a new approach

Authors: OQLAH ALREFAI, MOHAMMED ALI

Abstract: We prove for univalent functions $f(z)=z+\sum_{k=n}^{\infty}a_k z^k;(n\geq 2)$ in the unit disk $\mathbb{U}=\{z:\;|z|<1\})$ with $f^{-1}(w)=w+\sum_{k=n}^{\infty}b_k w^k;\;(|w|\lt r_0(f),\;r_0(f)\geq \frac{1}{4})$ that \[ b_{2n-1}=n a_n^2-a_{2n-1}\;\;\hbox{and}\;\; b_k=-a_k\;\;\hbox{for}\;\; (n\leq k\leq 2n-2) .\] As applications, we find estimates for $|a_n|$ whenever $f$ is bi-univalent, bi-close-to-convex, bi-starlike, bi-convex, or for bi-univalent functions having positive real part derivatives in $\mathbb{U}$. Moreover, we estimate $|na_n^2-a_{2n-1}|$ whenever $f$ is univalent in $\mathbb{U}$ or belongs to certain subclasses of univalent functions. The estimation method can be applied for various subclasses of bi-univalent functions in $\mathbb{U}$ and it helps to improve well-known estimates and to generalize some known results as shown in the last section. % You shouldn't use formulas and citations in the abstract.

Keywords: Univalent functions, bi-univalent functions, starlike functions, convex functions, close-to-convex functions, Faber polynomials, coefficient estimates} % Include keywords separeted by comma.

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