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Abstract
In this paper we investigate the new subclass of starlike functions in the unit disk
{
}
:
1
U
z
z
=
∈
<
via the generalized salagean differential operator. Basic proper-
ties of this new subclass are also discussed.
Subject Areas
Mathematical Analysis
Results & Lemmas (12)
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.
Lemma 1
Lemma 1 ([5] [7]) Let ψ ∈Ψ with corresponding domain Ω. If ( ) P Ψ is de- fined as the set of functions ( ) p z given as ( ) 2 1 2 1 p z c…
Lemma 1 ([5] [7]) Let ψ ∈Ψ with corresponding domain Ω. If ( ) P Ψ is de- fined as the set of functions ( ) p z given as ( ) 2 1 2 1 p z c z c z
Lemma 2
Lemma 2 ([8]). Let η and µ be complex constants and ( ) h z a convex univalent function in U satisfying ( ) 0 1 h =, and ( ) ( ) Re h z η
Lemma 2 ([8]). Let η and µ be complex constants and ( ) h z a convex univalent function in U satisfying ( ) 0 1 h = , and ( ) ( ) Re h z η
Lemma 3
Lemma 3 ([9]). Let 0 η ≠ and µ be complex constants and ( ) h z regular in U with ( ) 0 0 h′ ≠, then the solution ( ) q z of (7) given by…
Lemma 3 ([9]). Let 0 η ≠ and µ be complex constants and ( ) h z regular in U with ( ) 0 0 h′ ≠ , then the solution ( ) q z of (7) given by (8) is univalent in U if (i) Re
Theorem 1.
Theorem 1. Let ( ] 0,1 λ ∈ and ( ) h z a convex univalent function in U satisfying ( ) 0 1 h =, and ( )
Theorem 1. Let ( ] 0,1 λ ∈ and ( ) h z a convex univalent function in U satisfying ( ) 0 1 h = , and ( )
Theorem 2.
Theorem 2. Let ( ] 0,1 2 λ ∈ and ( ) h z a convex univalent function in U satisfying ( ) 0 1 h =, and ( ) 1
Theorem 2. Let ( ] 0,1 2 λ ∈ and ( ) h z a convex univalent function in U satisfying ( ) 0 1 h = , and ( ) 1
Theorem 3.
Theorem 3. 1 n n S S λ λ + ⊂.
Theorem 3. 1 n n S S λ λ + ⊂ .
Corollary 1.
Corollary 1. All functions in n Sλ are starlike univalent in U.
Corollary 1. All functions in n Sλ are starlike univalent in U.
Corollary 2.
Corollary 2. The class 1 n S “clone” the analytic representation of convex functions.
Corollary 2. The class 1 n S “clone” the analytic representation of convex functions.
Theorem 4.
Theorem 4. The class n Sλ is preserve under the Bernardi integral transformation: ( ) ( ) 1 0 1 d, 1. z c c c F z t
Theorem 4. The class n Sλ is preserve under the Bernardi integral transformation: ( ) ( ) 1 0 1 d , 1. z c c c F z t
Theorem 5.
Theorem 5. Let n f Sλ ∈. Then f has integral representation: ( ) ( ) 0 1 exp d z n p t
Theorem 5. Let n f Sλ ∈ . Then f has integral representation: ( ) ( ) 0 1 exp d z n p t
Theorem 6.
Theorem 6. Let n f Sλ ∈. Then ( ) ( ), 2. 1 1 k
Theorem 6. Let n f Sλ ∈ . Then ( ) ( ) , 2. 1 1 k
Theorem 7.
Theorem 7. Let n f Sλ ∈. Then ( ) ( ) ( ) 1 1 n n
Theorem 7. Let n f Sλ ∈ . Then ( ) ( ) ( ) 1 1 n n
Definitions (4)
Def 1
Definition 1 ([2]). Let ( ], 0,1 f λ ∈ ∈
Definition 1 ([2]). Let ( ] , 0,1 f λ ∈ ∈
Def 2
Definition 2 ([3]). Let f ∈, and 0 n∈. Then ( ) ( ) ( ) ( ) ( )
Definition 2 ([3]). Let f ∈, and 0 n∈. Then ( ) ( ) ( ) ( ) ( )
Def 3.
Definition 3. A function f ∈ belongs to the class n Sλ if and only if ( ) ( ) ( ) ( ] 1
Definition 3. A function f ∈ belongs to the class n Sλ if and only if ( ) ( ) ( ) ( ] 1
Def 4.
Definition 4. Let 1 2 u u u i = +, 1
Definition 4. Let 1 2 u u u i = + , 1