Continue from this paper
Indexing is not verification. Extracted statements should be checked against the source.
Abstract
In this paper, by making use of binomial series, we define a new differential operator of
holomorphic functions in the open unit disk. Also, we introduce and investigate two new
classes containing this new operator associated with differential subordinations and
superordinations. Furthermore, we determine important properties for functions belonging
to these classes.
Results & Lemmas (9)
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 1.1
Lemma 1.1 [5]. Suppose that h is holomorphic and convex univalent function in, D ( ) 0, 0 ≠ λ = a h and ( ).0 Re
Lemma 1.1 [5]. Suppose that h is holomorphic and convex univalent function in , D ( ) 0 , 0 ≠ λ = a h and ( ) .0 Re
Lemma 1.2
Lemma 1.2 [6]. Suppose that the function h is convex in, D ( ), 0 a h =
Lemma 1.2 [6]. Suppose that the function h is convex in , D ( ) , 0 a h =
Lemma 2.1.
Lemma 2.1. Assume that 0,0,0,, 0 ∪ N N R = ∈ δ
Lemma 2.1. Assume that { } 0 ,0 ,0 , , 0 ∪ N N R = ∈ δ
Theorem 2.1.
Theorem 2.1. Suppose that ψ is a convex function in D with ( ) 1 0 = ψ and.0 > η
Theorem 2.1. Suppose that ψ is a convex function in D with ( ) 1 0 = ψ and .0 > η
Theorem 2.2.
Theorem 2.2. Suppose that ψ is a convex function in, D with ( ) 1 0 = ψ and.0 > η
Theorem 2.2. Suppose that ψ is a convex function in , D with ( ) 1 0 = ψ and .0 > η
Theorem 2.3.
Theorem 2.3. Assume that 1 ψ and 2 ψ are two convex functions in D with ( ) 0 1 ψ
Theorem 2.3. Assume that 1 ψ and 2 ψ are two convex functions in D with ( ) 0 1 ψ
Theorem 2.4.
Theorem 2.4. Suppose that ψ is a convex function in D with ( ) 1 0 = ψ and
Theorem 2.4. Suppose that ψ is a convex function in D with ( ) 1 0 = ψ and
Theorem 2.5.
Theorem 2.5. Suppose that ψ is a convex function in D with ( ) 1 0 = ψ and ( ) z T is defined by (2.6). If ( ),;,,
Theorem 2.5. Suppose that ψ is a convex function in D with ( ) 1 0 = ψ and ( ) z T is defined by (2.6). If ( ), ; , ,
Theorem 2.6.
Theorem 2.6. Assume that 1 ψ and 2 ψ are two convex functions in D with ( ) ( ) 1 0 0 2 1 = ψ =
Theorem 2.6. Assume that 1 ψ and 2 ψ are two convex functions in D with ( ) ( ) 1 0 0 2 1 = ψ =
Definitions (3)
Def 1.1
Definition 1.1 [6]. Let ( ) D H ∈ h p, and ( )
Definition 1.1 [6]. Let ( ) D H ∈ h p, and ( )
Def 1.2
Definition 1.2 [5]. Let Q be the family of all functions that are holomorphic and injective on ( ), q E D where ( )
Definition 1.2 [5]. Let Q be the family of all functions that are holomorphic and injective on ( ), \ q E D where ( ) {
Def 2.1.
Definition 2.1. Assume that ψ is an holomorphic and convex univalent function in D with ( ) 1 0 = ψ and, R ∈
Definition 2.1. Assume that ψ is an holomorphic and convex univalent function in D with ( ) 1 0 = ψ and , R ∈
Function classes studied:
Related Papers