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Abstract
The object of the present paper is to investigate various argument results of analytic and multiva-
lent functions which are defined by using a certain fractional derivative operator. Some interest-
ing applications are also considered.
Results & Lemmas (6)
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Lemma 1
Lemma 1 [6]. Let ( ) h z be analytic in , with ( ) 0 1 h = and ( ) 0 h z ≠ ( ) z ∈. Further suppose that (
Lemma 1 [6]. Let ( ) h z be analytic in , with ( ) 0 1 h = and ( ) 0 h z ≠ ( ) z ∈. Further suppose that (
Theorem 1.
Theorem 1. Let 0 λ ≥, min 1, p p µ ν < + +
Theorem 1. Let 0 λ ≥ , { } min 1, p p µ ν < + +
Corollary 1.
Corollary 1. Let 0 λ ≥, min 1, p p µ ν < + +
Corollary 1. Let 0 λ ≥ , { } min 1, p p µ ν < + +
Theorem 2.
Theorem 2. Let 0 λ ≥, min 1, p p µ ν < + +
Theorem 2. Let 0 λ ≥ , { } min 1, p p µ ν < + +
Corollary 2.
Corollary 2. Let α + ∈. Suppose that ( ) ( ) ( )
Corollary 2. Let α + ∈. Suppose that ( ) ( ) ( )
Theorem 3.
Theorem 3. Let 0 λ ≥, min 1, p p µ ν < + +
Theorem 3. Let 0 λ ≥ , { } min 1, p p µ ν < + +
Definitions (2)
Def 1.
Definition 1. Let 0 1 λ ≤ < and, µ ν ∈. Then the generalized fractional derivative operator,, 0,z λ µ ν
Definition 1. Let 0 1 λ ≤ < and , µ ν ∈. Then the generalized fractional derivative operator , , 0,z λ µ ν
Def 2.
Definition 2. Under the hypotheses of Definition 1, the fractional derivative operator,, 0, m m m z λ µ
Definition 2. Under the hypotheses of Definition 1, the fractional derivative operator , , 0, m m m z λ µ