Coefficient Bounds for Certain Classes of Analytic Functions Associated with Faber Polynomial

Faber polynomials, which were introduced by Faber in 1903 [1], play an important role in the theory of functions of a complex variable and different areas of mathematics and there is a rich literature [2,3,4,5,6,7] describing their properties and their applications. Given a function $h\left(z\right)$ of the form consider the expansion valid for all $\zeta$ in some neighborhood of $\infty$. The function ${\Psi }_n\left(w\right)=w^n+{\displaystyle \sum _{k=1}^n}{a}_{nk}{w}^{n-k}$ is a polynomial of degree n, called the n-th Faber polynomial with respect to the function $h\left(z\right)$. In particular,

Let ${\Psi }_n\left(0\right)=F_n(b_0,b_1,\dots ,b_n),n\ge 0,$ see ([8], p. 118). Let A denote the class of all functions of the form: which are analytic in the open unit disc $U=\{z:z\in \mathbb{C}$ and $\left|z\right|<1\}$ and let S be the class of all functions in A which are univalent in U. By using the Faber polynomial expansion of functions of the form (1), Airault and Bouali [9], p. 184 showed that where $F_{j-1}(a_2,a_3,\cdots ,a_j)$ is the Faber polynomial given by: and

The first few terms of the Faber polynomials $F_{j-1},j\ge 2,$ are given by (e.g., see [10], p. 52)

The Koebe one-quarter theorem [8], p. 31 ensures the range of every function of the class S contains the disc $\{w:\left|w\right|<\frac{1}{4}\}$. Thus every univalent function $f\in S$ has an inverse $f^{-1},$ which is defined by and

The inverse map $g:=f^{-1}$ of the function f$\in A$ has Taylor expansion given by (see [9], p. 185) where the coefficients $K_n^p(a_2,a_3,\cdots ,a_n)$ are given by and $V_j$ is homogeneous polynomial of degree j in the variables $a_3,\cdots ,a_n,$ see ([11], p. 349 and [9], p. 183 and p. 205).

If f and g are analytic functions in U, we say that f is subordinate to g, written $f\left(z\right)\prec g\left(z\right)$ if there exists a Schwarz function $\omega \left(z\right)$ such that $f\left(z\right)=g\left(\omega \right(z\left)\right).$ Let $\varphi$ be an analytic function with positive real part in U, satisfying $\varphi \left(0\right)=1,{\varphi }^{{}^{\prime }}\left(0\right)>0,$ and $\varphi \left(U\right)$ is symmetric with respect to the real axis. Such a function has a Taylor series of the form

Using this $\varphi$, Ma and Minda [12] considered the classes and

Several well-known classes can be obtained by specializing of the function $\varphi ,$ for instance

An interesting families of the domains that are bounded by a conic sections were introduced and studied by Shams et al. [18], they introduced the class $SD(\alpha ,\beta )$ of $\beta$- uniformly starlike functions of order $\alpha (0\le \alpha <1)$ in U consisting of functions $f\in A$ which satisfy the following inequality and class $KD(\alpha ,\beta )$ of $\beta$-uniformly convex of order $\alpha (0\le \alpha <1),$ defined by

Since $Rew>\alpha |w-1|+\gamma$ if and only if $Re\{w(1+\alpha e^{i\theta })-\alpha e^{i\theta }\}>\gamma$(see [19]), then the condition (5) is equivalent to

Motivated by the classes $SD(\alpha ,\beta )$ and $KD(\alpha ,\beta )$ we now introduce and investigate the following subclasses of A, and obtain some interesting results.

We note that:

We note that:

A single-valued function f analytic in a domain $D\subset C$ is said to be univalent there if it never take the same value twice; that is, if $f\left(z_1\right)\ne$$f\left(z_2\right)$ for all points $z_1$ and $z_2$ in D with $z_1\ne z_2$ (see [8], p. 26). A function $f\in A$ is said to be bi-univalent in $U$ if f and its inverse map $f^{-1}$ are univalent in U. Let $\sigma$ denote the class of bi-univalent functions in U given by (1). The class of analytic bi-univalent functions was first introduced and studied by Lewin [21] and showed that $|a_2|<1.51$. Recently, many authors found non-sharp estimates on the first two Taylor–Maclaurin coefficients $\left|a_2\right|$ and $\left|a_3\right|$ for various subclasses of bi-univalent functions, see for example, ([22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43]). For other related topics see also, ([44,45,46,47]).

We note that:

We note that:

In this paper, we use the Faber polynomial expansions to obtain bounds for the general coefficients $|a_n|$ of bi-univalent functions in $M_{\sigma }(\lambda ,\beta ,\gamma ,\varphi )$ and $S_{\sigma }(\lambda ,\beta ,\gamma ,\varphi )$ as well as we provide estimates for the initial coefficients of these functions.

To prove our next theorem, we shall need the following lemma.

If we set in Definition 3 of the bi-univalent function class $M_{\sigma }(\lambda ,\beta ,\gamma ,\varphi ),$ we obtain a new class $M_{\sigma }(\lambda ,\beta ,\gamma ,A,B)$ given by Definition 5 below.