On the Uniqueness of Meromorphic Functions on Annuli in terms of Deficiencies
Abstract: The purpose of this article is to study the uniqueness of meromorphic functions on annuli. Under a certain condition about deficiencies, we prove some new uniqueness theorems of meromorphic functions on the annulus A=z:1/R0<z<R0, where 1<R0≤+∞.
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In this work, using the differential operator and the concept of meromorephic analytic functions we introduce and investigate the class Σ β , λ α , m ( A , B , Y , β ) , of functions of the form f ( z ) = z − 1 + ∑ n = 1 ∞ a n z n , a n ≥ 0 , which are analytic and meromorphic univalent in the punctured unit disk U * = { z ∈ C : 0 < | z | < 1 } , satisfying | A [ γ z 2 ( D β , λ α , m f ( z ) ) ′ ′ ′ + z ( 3 Υ + β ) ( D β , λ α , m f ( z ) ) ′ ′ + ( Υ + β ) ( D β , λ α , m f ( z ) ) ′ ] B [ Υ z ( D β , λ α , m f ( z ) ) ′ ′ + ( Υ + β ) ( D β , λ α , m f ( z ) ) ′ ] | < 1 . Further, coefficient bounds, Hadamard product, radius of close to-convexity, inclusive properties and neighbourhoods of functions in our class are obtain.
In this work, using the differential operator and the concept of meromorephic analytic functions we introduce and investigate the class Σ β , λ α , m ( A , B , Y , β ) , of functions of the form f ( z ) = z − 1 + ∑ n = 1 ∞ a n z n , a n ≥ 0 , which are analytic and meromorphic univalent in the punctured unit disk U * = { z ∈ C : 0 < | z | < 1 } , satisfying | A [ γ z 2 ( D β , λ α , m f ( z ) ) ′ ′ ′ + z ( 3 Υ + β ) ( D β , λ α , m f ( z ) ) ′ ′ + ( Υ + β ) ( D β , λ α , m f ( z ) ) ′ ] B [ Υ z ( D β , λ α , m f ( z ) ) ′ ′ + ( Υ + β ) ( D β , λ α , m f ( z ) ) ′ ] | < 1 . Further, coefficient bounds, Hadamard product, radius of close to-convexity, inclusive properties and neighbourhoods of functions in our class are obtain.
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