Bounds for the Coefficient of Faber Polynomial of Meromorphic Starlike and Convex Functions

Let Σ be the class of meromorphic functions f of the form f ( ζ ) = ζ + ∑ n = 0 ∞ a n ζ − n which are analytic in Δ : = { ζ ∈ C : | ζ | > 1 } . For n ∈ N 0 : = N ∪ { 0 } , the nth Faber polynomial Φ n ( w ) of f ∈ Σ is a monic polynomial of degree n that is generated by a function ζ f ′ ( ζ ) / ( f ( ζ ) − w ) . For given f ∈ Σ , by F n , i ( f ) , we denote the ith coefficient of Φ n ( w ) . For given 0 ≤ α < 1 and 0 < β ≤ 1 , let us consider domains H α and S β ⊂ C defined by H α = { w ∈ C : Re ( w ) > α } and S β = { w ∈ C : | arg ( w ) | < β } , which are symmetric with respect to the real axis. A function f ∈ Σ is called meromorphic starlike of order α if ζ f ′ ( ζ ) / f ( ζ ) ∈ H α for all ζ ∈ Δ . Another function f ∈ Σ is called meromorphic strongly starlike of order β if ζ f ′ ( ζ ) / f ( ζ ) ∈ S β for all ζ ∈ Δ . In this paper we investigate the sharp bounds of F n , n − i ( f ) , n ∈ N 0 , i ∈ { 2 , 3 , 4 } , for meromorphic starlike functions of order α and meromorphic strongly starlike of order β . Similar estimates for meromorphic convex functions of order α ( 0 ≤ α < 1 ) and meromorphic strongly convex of order β ( 0 < β ≤ 1 ) are also discussed.

Certain q-difference operators and their applications to the subclass of meromorphic q-starlike functions

The main aim of this work is to find some coefficient inequalities and sufficient condition for some subclasses of meromorphic starlike functions by using q-difference operator. Here we also define the extended Ruscheweyh differential operator for meromorphic functions by using q-difference operator. Several properties such as coefficient inequalities and Fekete-Szego functional of a family of functions are investigated.

On sufficient conditions of meromorphic starlike functions

In this paper, we investigate interesting properties and sufficient conditions for meromorphic starlike functions in the punctured unit disc.

On a New Criterion for Meromorphic Starlike Functions

The main purpose of this paper is to derive a new criterion for meromorphic starlike functions of orderα.

ON A CLASS OF MEROMORPHIC STARLIKE FUNCTIONS WITH NEGATIVE COEFFICIENTS

Let $T^*_M(A, B, z_0)$ denote the class of functions \[f(z)=\frac{a}{z}-\sum_{n=1}^\infty a_nz^n, a\ge 1, a_n\ge 0\] regular and univalent in unit disc $U'=\{z:0<|z|<1\}$, satisfying the condition \[-z\frac{f'(z)}{f(z)}=\frac{1+Aw(z)}{1+Bw(z)}, \quad \text{ for } z\in U' \text{ and } w\in E\] (where $E$ is the class of analytic functions $w$ with $w(0) = 0$ and $|w(z)| \le 1$), where $-1\le A < B \le 1$, $0\le B \le 1$ and $f(z_0) =1/z_0$ ($0<z_0<1$). In this paper sharp coefficient estimates, distortion properties and radius of meromorphic convexity for functions in $T^*_M(A, B, z_0)$ have been obtained. We also study integral transforms of functions in $T^*_M(A, B, z_0)$. In the last, it is proved that the class $T^*_M(A, B, z_0)$ is closed under convex linear combinations.