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Neighborhoods and partial sums of certain meromorphically multivalent functions

Jin-Lin Liu · 2007 📄 find paper →

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Abstract

By making use of the familiar concept of neighborhoods of analytic functions, the author proves an inclusion relations associated with the (n, δ)−neighborhoods of a subclass Qk[p, α; A, B] which was introduced by Srivastava, Hossen and Aouf. The partial sums of the functions in Qk[p, α; A, B] are also considered.

Results & Lemmas (2)

Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.

Theorem 1. Theorem 1. Let f ∈Qk[p, α; A, B] be given by (1.1). If f satisfies the inclusing condition (f(z) + εz−p)(1 + ε)−1 ∈Qk[p, α; A, B] (ε ∈C; |ε|…
Theorem 1. Let f ∈Qk[p, α; A, B] be given by (1.1). If f satisfies the inclusing condition (f(z) + εz−p)(1 + ε)−1 ∈Qk[p, α; A, B] (ε ∈C; |ε| < δ; δ > 0), (2.2) then Nδ(f) ⊂Qk[p, α; A, B]. (2.3)
Theorem 2. Theorem 2. Let f ∈P p,k be given by (1.1) and define the partial sums sm(z) by sm(z) =      z−p m = 1, 2,... k −1; z−p + m X n=k…
Theorem 2. Let f ∈P p,k be given by (1.1) and define the partial sums sm(z) by sm(z) =      z−p m = 1, 2, . . . k −1; z−p + m X n=k an+p−1zn+p−1

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