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Abstract
Let
(
)
*
, ,
,
c
S
A B
α δ
denote the class of analytic and univalent functions in an open unit disk
of the
form
( )
2
n
n
n
f
z
z
a z
∞
=
=
+∑
and satisfy
( )
( )
1
1
sin
1
i zf
z
Az
e
i
g z
t
Bz
α
αδ
δ
α
′
+
−
−
+
where
( )
( )
( )
,
2
f
z
f
z
g z
+
=
cos
0,
tαδ
α
δ
=
−
>
0
<1,
δ
≤
2
π
α <
and
1
Results & Lemmas (8)
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 (Duren 1983) (p. 41) For a function p P ∈ of the form (9), the sharp inequality 2 n p ≤ holds for each 1. n ≥ Equality holds for…
Lemma 1 (Duren 1983) (p. 41) For a function p P ∈ of the form (9), the sharp inequality 2 n p ≤ holds for each 1. n ≥ Equality holds for the function ( ) 1 .
Lemma 2
Lemma 2 (Efraimidis 2016) Let p P ∈ of the form (9) and Then
Lemma 2 (Efraimidis 2016) Let p P ∈ of the form (9) and Then
Theorem 1
Theorem 1 If ( ) *,,,, C f S A B α δ ∈ then where (
Theorem 1 If ( ) * , , , , C f S A B α δ ∈ then where (
Theorem 2
Theorem 2 If ( ) *,,,, C f S A B α δ ∈ then where
Theorem 2 If ( ) * , , , , C f S A B α δ ∈ then where
Theorem 3
Theorem 3 If ( ) *,,,, C f S A B α δ ∈ then ( )
Theorem 3 If ( ) * , , , , C f S A B α δ ∈ then ( )
Corollary 1
Corollary 1 a) For ( ) * 0,0,1, 1, C f S ∈ − we get 2 2 2 3
Corollary 1 a) For ( ) * 0,0,1, 1 , C f S ∈ − we get 2 2 2 3
Corollary 2
Corollary 2 a) For ( ) * 0,,1, 1, C f S δ ∈ − we get ( ) (
Corollary 2 a) For ( ) * 0, ,1, 1 , C f S δ ∈ − we get ( ) (
Corollary 3
Corollary 3 a) For ( ) * 0,0,,, C f S A B ∈ we get b) For (
Corollary 3 a) For ( ) * 0,0, , , C f S A B ∈ we get b) For (
Function classes studied:
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