Abstract
In this paper , we obtain some subordination and superordination results involving
the integral operator
.Also,we get Differential sandwich results for classes of univalent
functions in the unit disk.
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 2.1
Lemma 2.1 [5]: Let q be univalent in the unit disk U and let θ and be analytic in a domain D containing q(U) with ( ) when ( ) Set ( ) ( )…
Lemma 2.1 [5] : Let q be univalent in the unit disk U and let θ and be analytic in a domain D containing q(U) with ( ) when ( ) Set ( ) ( ) ( ( )) ( ) ( ( )) ( ) Suppose that (i) ( ) is starlike univalent in ,
Lemma 2.2
Lemma 2.2 [6]: Let q be convex univalent in function in U and let * + with ( ( ) ( ) ) ( ( )) If is analytic in, and ( ) ( ) ( ) ( ) (2.3)…
Lemma 2.2 [6]: Let q be convex univalent in function in U and let * + with ( ( ) ( ) ) ( ( )) If is analytic in , and ( ) ( ) ( ) ( ) (2.3) then and is the best dominant of (2.3).
Lemma 2.3
Lemma 2.3 [6]: Let q be convex univalent in U and let, further assume that Re ( ). If, ( )- Q and ( ) ( ) is univalent in U, then ( ) ( ) (…
Lemma 2.3 [6]: Let q be convex univalent in U and let , further assume that Re ( ) . If , ( )- Q and ( ) ( ) is univalent in U, then ( ) ( ) ( ) ( ) (2.4) which implies that and q is the best subordinant of (2.4).
Lemma 2.4
Lemma 2.4 [3]: Let q be convex univalent in the unit disk U and let be analytic in domain D containing q ( ). Suppose that (i) Re ( ( )) (…
Lemma 2.4 [3]: Let q be convex univalent in the unit disk U and let be analytic in domain D containing q ( ) . Suppose that (i) Re { ( ( )) ( ( ))} (ii) Q( ) ( ) ( ( )) . If , ( ) - ( ) ( ( )) ( ) ( ) is univalent in U and ( ( )) ( ) ( ( )) ( ( ))
Theorem 3.1
Theorem 3.1: Let q be convex univalent function in U with ( ) also, assume that q Re( ( ) ( ) ) (.
Theorem 3.1 : Let q be convex univalent function in U with ( ) also, assume that q Re( ( ) ( ) ) ( .
Corollary 3.1
Corollary 3.1: Let also Re *
Corollary 3.1 : Let also Re *
Theorem 3.2
Theorem 3.2: Let q be convex univalent function in U with ( ) ( ) ( ) furthermore, accept that q fulfills Re (
Theorem 3.2 : Let q be convex univalent function in U with ( ) ( ) ( ) furthermore, accept that q fulfills Re (
Theorem 4.1
Theorem 4.1: Let q be convex univalent function in U with ( ) * +
Theorem 4.1: Let q be convex univalent function in U with ( ) * +
Corollary 4.1
Corollary 4.1: Let and Re * +
Corollary 4.1: Let and Re * +
Theorem 4.2
Theorem 4.2: Let q be convex univalent function in U with ( ) also, accept that q fulfills * ( )
Theorem 4.2: Let q be convex univalent function in U with ( ) also, accept that q fulfills * ( )
Theorem 5.1
Theorem 5.1: Let q1 be convex univalent function in U with q1(0)=1,Re and let q2 be univalent in U,q2(0)=1 and fulfills (3,1), let
Theorem 5.1 : Let q1 be convex univalent function in U with q1(0)=1,Re { } and let q2 be univalent in U ,q2(0)=1 and fulfills (3,1), let
Theorem 5.2
Theorem 5.2: Let q1 be convex univalent function in U with q1(0)=1, and fulfills (4.5), let q2 be
Theorem 5.2: Let q1 be convex univalent function in U with q1(0)=1, and fulfills (4.5), let q2 be
Definitions (1)
Def 2.1
Definition 2.1 [5]: Denote by Q the set of all functions that are analytic and injective on ̅ ( ) where ( ) * ( ) ∞ + (2.1) and are such…
Definition 2.1 [5] : Denote by Q the set of all functions that are analytic and injective on ̅ ( ) where ( ) * ( ) ∞ + (2.1) and are such that (ξ) ≠0 for ξ ∂U \ E( ).
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