Dominant of functions satisfying a differential subordination and applications
R. Chandrashekar, Rosihan M. Ali, and K. G. Subramanian
Citation: AIP Conference Proceedings 1605, 580 (2014); doi: 10.1063/1.4887653
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Dominant of Functions Satisfying a Differential
Subordination and Applications
R. Chandrashekara, Rosihan M. Alib and K. G. Subramanianc
a
Department of Technology Management, Faculty of Technology Management and Business,
Universiti Tun Hussein Onn Malaysia, 86400 Parit Raja, Batu Pahat, Johor, Malaysia
b
School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM, Penang, Malaysia
c
School of Computer Sciences, Universiti Sains Malaysia, 11800 USM, Penang, Malaysia
Abstract. Best dominant is obtained for normalized analytic functions f satisfying (1 D ) f ( z) z D f c( z) E zf cc( z) E h( z)
in the unit disk , where h is a normalized convex function, and D , E are appropriate real parameters. This fundamental
result is next applied to investigate the convexity and starlikeness of the image domains f ( ) for particular choices of h.
Keywords: Starlike and convex functions, differential subordination, dominant.
PACS: 02.30.-f
INTRODUCTION
Let
be the class of analytic functions f defined in the open unit disk : { z  :| z | 1}. For a  , n
a positive integer, and z  , let
n
and
n
with 1
^
(a)
^
f
f
: f ( z)
a ¦ ak z
k n
f
f 
k
: f ( z)
z
¦az
k
k n 1
k
`
`
,
. The subclass of consisting of starlike functions in satisfying
§ zf c( z ) ·
¸ ! 0, z  ,
© f ( z) ¹
Re ¨
is denoted by +, , and . is the subclass of consisting of convex functions in satisfying
§ zf cc( z ) ·
Re ¨ 1 ¸ ! 0, z  .
f c( z ) ¹
©
For two analytic functions f and g , the function f is subordinate to g , written f ( z ) E g ( z ) if there is an
analytic self-map w of with w(0) 0 satisfying f ( z ) g ( w( z )). If g is univalent, then f subordinate to g
is equivalent to f (0) g (0) and f () Ž g ().
This paper considers a class of functions satisfying a second-order differential subordination to a given convex
function. Best dominant amongst the solutions to this differential subordination is determined. Further, sufficient
conditions are obtained that ensure these solutions are either starlike or convex functions in . Such conditions in
terms of differential inequalities have been investigated in several works, notably by [1, 2, 3, 4, 5, 6, 7]. In
particular, Kanas and Owa [8] studied connections between certain second-order differential subordination involving
expressions of the form f ( z ) z , f c( z ) and 1 zf cc( z ) f c( z ). The class studied in this paper presents a more
general framework.
Proceedings of the 21st National Symposium on Mathematical Sciences (SKSM21)
AIP Conf. Proc. 1605, 580-585 (2014); doi: 10.1063/1.4887653
© 2014 AIP Publishing LLC 978-0-7354-1241-5/$30.00
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The following lemma will be needed.
Lemma 1 [9, Theorem 1, p. 192] Let h be convex in with h(0)
p( z ) zp c( z )
J
a, J z 0 and Re J t 0. If p 
n (a)
and
E h ( z ),
then
p ( z ) E q ( z ) E h ( z ),
where
J
q( z )
J /n
nz
The function q is convex and is the best ( a, n) dominant.
³
z
0
h (t ) t
( J / n ) 1
dt.
SECOND – ORDER DIFFERENTIAL SUBORDINATION
In the following sequel, we shall assume that h is an analytic convex function in with h (0)
1. For E t 0
and D 2 E t 0 consider the class of functions f  n satisfying the second-order differential subordination
(1 D )
f ( z)
z
D f c( z ) E zf cc( z ) E h( z ).
(1)
Let P and Q satisfy
Q P
D E and PQ
E.
(2)
Note that Re P t 0 and ReQ t 0.
The following result gives the best dominant to solutions of the differential subordination (1).
Theorem 1 Let P and Q be given by (2), and D , E be real numbers such that E t 0 and D 2 E t 0 . If f  n
satisfies
(1 D )
f ( z)
z
D f c( z ) E zf cc( z ) E h( z ),
then
f ( z)
z
and q is the best ( a , n ) dominant.
1
E q( z ) :
En
2
1
1
0
0
³³
h ( rsz ) r
(1/ P n ) 1 (1/Q n ) 1
n
n 1
s
drds ,
Proof. Let
p( z )
f ( z)
z
1 an 1 z an 2 z
".
Evidently
(1 D )
f ( z)
z
D f c( z ) E zf cc( z )
E z pcc( z ) (D 2 E ) zpc( z ) p ( z ),
2
and (1) can be expressed as
2
E z pcc( z ) (D 2E ) zpc( z ) p( z ) E h( z ).
(3)
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Writing
F ( z ) Q zp c( z ) p ( z ),
it follows that
2
F ( z ) P zF c( z ) E z p cc( z ) (D 2 E ) zp c( z ) p ( z ) E h ( z ),
where P and Q are given by (2). Lemma 1 now yields
1
F ( z) E
P nz
1/ P n
³
z
0
h ( t )t
(1/ P n ) 1
dt ,
and thus
p ( z ) Q zpc( z )
1
³
Pn
1
0
h( rz ) r
(1/ P n ) 1
dr .
A second application of Lemma 1 shows that
p( z) E
1
Q nz
1/Q n
§ 1 1
· (1/Q n ) 1
(1/ P n ) 1
h ( rt ) r
dr ¸ t
dt ,
³
0 ¨
0
n
P
©
¹
³
z
which in view of (2) implies that
f ( z)
z
1
E q( z ) :
En
2
1
1
0
0
³³
h( rsz ) r
Since q ( z ) (D 2 E ) nzq c( z ) E ª¬n ( n 1) zq c( z ) n z q cc( z ) º¼
2 2
(1/ P n ) 1 (1/Q n ) 1
s
drds.
h ( z ), the function Q ( z )
zq ( z ) is a solution
of the differential subordination
f ( z)
D f c( z ) E zf cc( z ) E h( z ).
z
This shows that q E q for all ( a , n ) dominants q , and hence q is the best ( a , n ) dominant.
The following result is an immediate consequence of Theorem 1.
(1 D )
Ƒ
Corollary 1 Under the assumptions of Theorem 1, if
f ( z)
(1 D )
D f c( z ) E zf cc( z ) E 1+ Mz,
z
then
f ( z)
Mz
E 1
,
z
1 D n E n( n 1)
and the superordinate function is the best dominant.
An application of Corollary 1 gives the following sufficient condition for starlikeness.
Theorem 2 Let D
and
E
E t 2D . Further let
be real numbers with D t 1 and
(4)
(5)
f  n
and
0 M M (D , E , n), where
M (D , E , n )
2( E 2D )[1 D n E n ( n 1)]
D ( n 1) E [ n ( n 1) 2]
.
(6)
If f satisfies the differential subordination
(1 D )
f ( z)
z
D f c( z ) E zf cc( z ) E 1+ Mz,
then f  +, .
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Proof. Let
zwc( z ).
f ( z)
(7)
A brief computation shows that
§ zf c( z ) ·
§ zwcc( z ) ·
¸ Re ¨ 1 c ¸ .
w ( z) ¹
© f (z) ¹
©
Re ¨
(8)
In view of the analytical condition for starlikeness, that is, Re zf c( z )/ f ( z ) ! 0 in , it remains to show that
zwcc( z )
wc( z )
1.
(9)
Using (7), (4) can be rewritten as
(10)
2
wc( z ) (D 2 E ) zwcc( z ) E z wccc( z ) E 1 Mz.
Integrating (10), evidently
2
(1 D ) w( z ) D zwc( z ) E z wcc( z )
where I is an analytic self-map of with I (0)
zwcc( z )
wc( z )
d
1
1
1
(11)
0
0. It follows from (7) and (11) that
1
E wc( z )
³
z Mz I ( sz ) ds ,
1 M ³ I ( sz ) ds (D 1) w( z )
zwc( z )
E
0
D
E
.
For (9) to hold true, it is sufficient to prove
1
1
E wc( z )
1
1 M ³ I ( sz ) ds 0
(D 1) w( z )
E
zwc( z )
D
E
1.
(12)
Now the subordination (5), implies
1
wc( z )
1 D n E n ( n 1)
1 D n E n ( n 1) M
,
(13)
.
(14)
where M 1 D n E n( n 1), while
w( z )
z
1
M
2[1 D n E n ( n 1)]
Since I ( z ) z , a brief computation shows that
1
M
0
2
1 M ³ I ( sz ) ds 1 .
(15)
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Taking into account the inequalities (13), (14) and (15), the condition (12) is fulfilled whenever M M (D , E , n )
Ƒ
with M (D , E , n) given by (6). This completes the proof.
The following theorem which gives sufficient conditions for convexity is also a consequence of Corollary 1.
Theorem 3 Let D and E be real numbers with D t 1 and E t 2 2 D . Further let
f  n and
0 M M (D , E , n ), where
2
M (D , E , n )
2[1 D n E n ( n 1)][( E 2D ) 2D
2
]
2 E [2 D ( n 1) E n ( n 1)] ( E D )[D ( n 1) E n ( n 1)]
.
(16)
If f satisfies the differential subordination
(1 D )
f ( z)
D f c( z ) E zf cc( z ) E 1+ Mz,
z
then f  . .
Proof. In view of the fact
§
©
Re ¨ 1 zf cc( z ) ·
f c( z )
zf cc( z )
! 0,
¸ t 1 c
f ( z)
¹
it is sufficient to prove the inequality
zf cc( z )
f c( z )
1.
(17)
Let
f c( z )
ª§ zf cc( z ) · º
cc
«¬©¨ 1 f c( z ) ¸¹ 1»¼ zf ( z ).
(18)
Proceeding similarly as in the proof of Theorem 2, with I as an analytic self-map of the unit disk, it follows from
(18) and (4) that
(1 D )
f (z)
z
D f c( z ) E f c( z )
ª§ zf cc( z ) · º
«¬©¨ 1 f c( z ) ¸¹ 1»¼ 1 M I ( z ).
(19)
Subsequently,
zf cc( z )
f c( z )
d
1
1
E f c( z )
| 1 M I ( z ) | (D 1)
E
f
zf c( z )
D
E
,
which leads to the condition
1
1
E f c( z )
| 1 M I ( z ) | §¨ D 1 ·¸
© E ¹
f
zf c( z )
D
E
1
(20)
for (17) to hold true.
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Applying (7) and (11), as well as the inequalities (13), (14) and (15), yield
1
f c( z )
2 E [1 D n E n( n 1)]
(21)
2[1 D n E n( n 1)]( E 2D ) M [D ( n 1) E ( n( n 1) 2)]
and
f ( z)
zf c( z )
2 E [1 D n E n( n 1) M ]
2[1 D n E n ( n 1)]( E 2D ) M [D ( n 1) E ( n ( n 1) 2)]
,
(22)
where
M 2( E 2D )[1 D n E n ( n 1)]
D ( n 1) E [ n ( n 1) 2]
.
In view of (21), (22) and the fact that | 1 M I ( z ) | 1 M , (20) is fulfilled for M M (D , E , n ), where
Ƒ
M (D , E , n) is given by (16). This completes the proof.
ACKNOWLEDGMENTS
The work presented here was supported in part by a research university grant from Universiti Sains Malaysia.
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