BOUNDS FOR CERTAIN LINEAR COMBINATIONS OF THE FABER COEFFICIENTS OF FUNCTIONS ANALYTIC IN AN ELLIPSE

Abstract

Let $\varOmega$ be a bounded, simply connected domain in $\mathbb{C}$ with $0\in\varOmega$ and $\partial\varOmega$ analytic. Let $S(\varOmega)$ denote the class of functions $F(z)$ which are analytic and univalent in $\varOmega$ with $F(0)=0$ and $F'(0)=1$. Let $\{\varPhi_{n}(z)\}_{n=0}^{\infty}$ be the Faber polynomials associated with $\varOmega$. If $F(z)\in S(\varOmega)$, then $F(z)$ can be expanded in a series of the form

$$ F(z)=\sum_{n=0}^{\infty}A_{n}\varPhi_{n}(z),\quad z\in\varOmega, $$

in terms of the Faber polynomials. Let

$$ E_{r}=\bigg\{(x,y)\in\mathbb{R}^{2}:\frac{x^{2}}{(1+(1/r^{2}))^{2}}+\frac{y^{2}}{(1-(1/r^{2}))^{2}}\lt1\bigg\}, $$

where $r\gt1$.

In this paper, we obtain sharp bounds for certain linear combinations of the Faber coefficients of functions $F(z)$ in $S(E_{r})$ and in certain related classes.