Filomat 30:13 (2016), 3615–3625
DOI 10.2298/FIL1613615X
Published by Faculty of Sciences and Mathematics,
University of Niš, Serbia
Available at: http://www.pmf.ni.ac.rs/filomat
Sufficient Conditions for a Certain General
Class of Carathéodory Functions
Qing-Hua Xua , Ting Yanga , H. M. Srivastavab
a College
of Mathematics and Information Science, JiangXi Normal University,
NanChang 330027, People’s Republic of China
b Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada
and
China Medical University, Taichung 40402, Taiwan, Republic of China
Abstract. In the present paper, the authors derive sufficient conditions for functions to be in a certain
general class of Carathéodory functions in the open unit disk by using the Miller-Mocanu lemma. As an
µ
application of our main result, we deduce sufficient conditions for functions belonging to the class ST Ŝβ
which is introduced here. The various results presented here would generalize and extend many known
results.
1. Introduction and Definitions
Let H[a0 , n] denote the class of functions p(z) of the form:
p(z) = a0 +
∞
X
ak zk
(n ∈ N = {1, 2, 3, ...}; a0 ∈ C),
k=n
which are analytic in the open unit disk
U = {z : z ∈ C and |z| < 1},
C being, as usual, the set of complex numbers.
Definition 1. If the function p(z) ∈ H[a0 , n] satisfies the following argument inequality:
| arg[λp(z) + 1 − λ]| <
π
µ
2
(z ∈ U; 0 < µ 5 1; λ ∈ C \ {0}),
then we say that p(z) is a strongly Carathéodory function of type λ and order µ in U and we write
p(z) ∈ P(λ, µ).
2010 Mathematics Subject Classification. Primary 30C45; Secondary 30C80
Keywords. Analytic functions; Univalent functions; Starlike functions; Strongly starlike functions; Spirallike functions; Principle
of differential subordination; Carathéodory functions; Argument inequalities; Analytical characterization.
Received: 24 October 2014; Accepted: 02 May 2015
Communicated by Dragan S. Djordjević
Email addresses: [email protected] (Qing-Hua Xu), [email protected] (Ting Yang), [email protected] (H. M.
Srivastava)
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3616
We note that, in the special case when λ = 1, the class P(λ, µ) reduces to the class ST P(µ) which was
studied by Shiraishi et al. [5] and Kim et al. [2] (see also [1], [4] and [9]).
Definition 2. Let An denote the class of functions of the form:
f (z) = z + an+1 zn+1 + an+2 zn+2 + · · ·
(n ∈ N),
which are analytic in U. Also let
A = A1 .
A function f ∈ An is called strongly starlike of order µ (0 < µ 5 1) if and only if
!
z f 0 (z) π
< µ (z ∈ U; 0 < µ 5 1).
arg
f (z) 2
We denote by ST S(µ) the class of all strongly starlike functions of order µ in U. We also write
S∗ := ST S(1)
for the familiar class of starlike functions in U.
The above-defined function class ST S(µ) was investigated by Shiraishi et al. [5] and Kim et al. [2]
Spaček [6] extended the class of starlike functions by introducing the class of spirallike functions of type
β in U and gave the following analytical characterization of spirallikeness functions of type β in U (see also
the recent work [8]).
Theorem 1 (see Spaček [6]). Let the function f ∈ A and suppose that the parameter β is constrained by − π2 < β < π2 .
Then f (z) is a spirallike function of type β in U if and only if
< e
iβ
!
z f 0 (z)
>0
f (z)
z ∈ U; −
π
π
<β<
.
2
2
We denote the class of spiralike functions of type β in U by Ŝβ .
From Definition 1 and Theorem 1, it is easy to see that strongly starlike functions of order µ and spirallike
functions of type β have some geometric relationships. Spirallike functions of type β map U into the rightz f 0 (z)
half complex plane by the mapping eiβ f (z) , while strongly starlike functions of order µ map U into the
angular region of the right-half complex plane by the mapping
lim eiβ
z→0
z f 0 (z)
f (z) .
Since
z f 0 (z)
= eiβ ,
f (z)
z f 0 (z)
we can deduce that, if we restrict the image of the mapping eiβ f (z) in the angular region of right-half
complex plane, then the vertex of the angular region lies on imaginary axis, and the angular region must be
symmetric with respect to the straight line parallel to the real axis and passing through the point eiβ . This
means that
π
| arg(ζ − i sin β)| < µ,
2
where ζ ∈ C and µ ∈ (0, 1]. In view of this observation, we extend the classes ST S(µ) and Ŝβ by introducing
µ
the analytic function class ST Ŝβ in U as follows.
Definition 3. Let the function f ∈ A. Suppose also that the parameters µ and β are constrained by
0<µ51
and
−
π
π
<β< .
2
2
Qing-Hua Xu et al. / Filomat 30:13 (2016), 3615–3625
3617
µ
We then say that f ∈ ST Ŝβ if and only if
!
0
π
iβ z f (z)
− i sin β < µ
arg e
2
f (z)
π
π
.
z ∈ U; 0 < µ 5 1; − < β <
2
2
Obviously, we have
ST Ŝ1β = Ŝβ
µ
ST Ŝ0 = ST S(µ).
and
Definition 4 (see, for example, [3]; see also [7]). For two functions f (z) and 1(z), analytic in U, we say that
the function f (z) is subordinate to 1(z) in U, and write
f (z) ≺ 1(z)
(z ∈ U),
if there exists a Schwarz function w(z), analytic in U with
w(0) = 0
|w(z)| < 1
and
(z ∈ U),
such that
f (z) = 1 w(z)
(z ∈ U).
In particular, if the function 1(z) is univalent in U, the above subordination is equivalent to
f (0) = 1(0)
and
f (U) ⊂ 1(U).
We denote by Q the class of functions q(z) which are analytic and injective on U \ E(q), where
(
)
E(q) = ζ : ζ ∈ ∂U and lim q(z) = ∞ ,
z→ζ
and are such that
q0 (ζ) , 0
ζ ∈ ∂U \ E(q) .
Here, as usual, we write
U := U ∪ ∂U
and
∂U = {z : z ∈ C
and
|z| = 1}.
Finally, let the subclass of Q for which q(0) = a0 be denoted by Q(a0 ).
The main object of this investigation is to derive sufficient conditions for functions to be in the general
class H[a0 , n] of Carathéodory functions in the open unit disk U. We also apply our main result (Theorem
2 below) in order to deduce the corresponding sufficient conditions for functions belonging to the class
µ
ST Ŝβ which we have introduced here. The various results presented here would generalize and extend
many known results.
2. Main Results
To prove our main results, we need the following lemma due to Miller and Mocanu [3].
Lemma. Let q(z) ∈ Q(a0 ) and let h(z) ∈ H[a0 , n] with h(z) , a0 . If h(z) ⊀ q(z), then there exist points z0 ∈ U and
ζ0 ∈ ∂U \ E(q) for which
h(z0 ) = q(ζ0 )
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and
z0 h0 (z0 ) = mζ0 q0 (ζ0 )
(m = n = 1).
By applying the above Lemma, we derive the following theorem.
Theorem 2. Let the parameters α and λ be constrained by
−
π
π
<α<
2
2
λ ∈ C \ {0},
and
respectively. Suppose also that the function 1(z) is analytic in U with
nh i
o
A := inf < 1(z) <(λ) + =(1(z))=(λ) cos α − <(1(z))=(λ) + =(1(z))<(λ) sin α > 0.
z∈U
(1)
If p(z) ∈ H[1, n] satisfies the following conditions:
p(z) , 1
(z ∈ U)
and
<{p(z) + 1(z)zp0 (z)} > max{A1 , A2 },
where
cos α nA tan α + <(λ) sin α + =(λ) cos α
A1 =
2nA
|λ|
!2
cos α nA tan α + <(λ) sin α − =(λ) cos α
A2 =
2nA
|λ|
!2
+
|λ|2 − <(λ)
nA
−
|λ|2
2|λ|2 cos α
(z ∈ U)
+
|λ|2 − <(λ)
nA
−
2
2
|λ|
2|λ| cos α
(z ∈ U),
and
then
| arg[λp(z) + 1 − λ]| <
π
− |α|
2
π
π
z ∈ U; − < α < ; λ ∈ C \ {0} .
2
2
Proof. Consider the functions h1 (z) and q1 (z) given by
π
π
h1 (z) = eiα [λp(z) + 1 − λ]
z ∈ U; − < α < ; λ ∈ C \ {0}
2
2
(2)
and
eiα + e−iα z
q1 (z) =
1−z
π
π
z ∈ U; − < α <
.
2
2
(3)
Then, clearly, the functions h1 (z) and q1 (z) are analytic in U with
h1 (0) = q1 (0) = eiα ∈ C
and
q1 (U) = {w : w ∈ C
and
<(w) > 0}.
We now suppose that h1 (z) ⊀ q1 (z). Then, by using the above Lemma, we can deduce that there exist points
z1 ∈ U
and
ζ1 ∈ ∂U \ {1}
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such that
h1 (z1 ) = q1 (ζ1 ) = iρ1
(ρ1 ∈ R)
(4)
z1 h0 (z1 ) = mζ1 q0 (ζ1 )
(m = n = 1).
(5)
and
We note that
h1 (z1 ) − eiα
ζ1 = q−1
1 h1 (z1 ) =
h1 (z1 ) + e−iα
(6)
and
ζ1 q01 (ζ1 ) = −
ρ21 − 2ρ1 sin α + 1
2 cos α
=: σ1 (ρ1 ) < 0.
(7)
For such points z1 ∈ U and ζ1 ∈ ∂U \ {1}, we obtain
<{p(z1 ) + 1(z1 )z1 p0 (z1 )}


e−iα z1 h01 (z1 ) 
 λ − 1 + e−iα h1 (z1 )


= < 
+ 1(z1 )

λ
λ


e−iα mζ1 q01 (ζ1 ) 
 λ − 1 + e−iα q1 (ζ1 )


= < 
+ 1(z1 )

λ
λ
!
e−iα mσ1 (ρ1 )
λ − 1 + e−iα iρ1
+ 1(z1 )
=<
λ
λ
|λ|2 − <(λ) + [<(λ) sin α + =(λ) cos α]ρ1
|λ|2
m{< 1(z1 ) [<(λ) cos α − =(λ) sin α] + = 1(z1 ) [<(λ) sin α + =(λ) cos α]}σ1 (ρ1 )
+
|λ|2
|λ|2 − <(λ) + [<(λ) sin α + =(λ) cos α])ρ1 nAσ1 (ρ1 )
5
+
|λ|2
|λ|2


2
|λ| − <(λ) + [<(λ) sin α + =(λ) cos α]ρ1 nA  ρ21 − 2ρ1 sin α + 1 


=
− 2 

2 cos α
|λ|2
|λ|
=
= Bρ21 + Cρ1 + D
=: κ(ρ1 ),
where
B=−
nA
,
2|λ|2 cos α
C=
nA tan α + <(λ) sin α + =(λ) cos α
|λ|2
D=
|λ|2 − <(λ)
nA
−
.
2
2
|λ|
2|λ| cos α
and
(8)
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We can thus see that the function κ(ρ1 ) in (8) takes on the maximum value at ρ∗1 given by
ρ∗1 =
nA sin α + [<(λ) sin α + =(λ) cos α] cos α
.
nA
Hence we have
<{p(z1 ) + 1(z1 )z1 p0 (z1 )} 5 κ(ρ∗1 )
=
cos α nA tan α + <(λ) sin α + =(λ) cos α
2nA
|λ|
+
!2
|λ|2 − <(λ)
nA
−
|λ|2
2 cos α|λ|2
=: A1 ,
where A is given by (1). This evidently contradicts the assumption of Theorem 1. Therefore, we have
π
π
< h(z) = <{eiα [λp(z) + 1 − λ]} > 0
z ∈ U; − < α <
.
2
2
(9)
We next put
h2 (z) = e−iα [λp(z) + 1 − λ]
z ∈ U; −
π
π
<α<
2
2
(10)
and
q2 (z) =
e−iα + eiα z
1−z
π
π
z ∈ U; − < α <
.
2
2
(11)
We then see that the functions h2 (z) and q2 (z) are analytic in U with
h2 (0) = q2 (0) = e−iα ∈ C
and
q2 (U) = {w : w ∈ C
and
<(w) > 0}.
We now suppose that
h2 (z) ⊀ q2 (z)
(z ∈ U).
Therefore, by using the above Lemma, we deduce that there exist points
z2 ∈ U
and
ζ2 ∈ ∂U \ {1}
such that
h2 (z2 ) = q2 (ζ2 ) = iρ2
(ρ1 ∈ R)
(12)
z2 h02 (z2 ) = mζ2 q02 (ζ2 )
(m = n = 1).
(13)
and
Furthermore, we note that
h2 (z2 ) − e−iα
ζ2 = q−1
h
(z
)
=
2
2
2
h2 (z2 ) + eiα
(14)
Qing-Hua Xu et al. / Filomat 30:13 (2016), 3615–3625
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and
ζ2 q02 (ζ2 ) = −
ρ22 + 2ρ2 sin α + 1
2 cos α
=: σ2 (ρ2 ) < 0.
(15)
For such points z2 ∈ U and ζ2 ∈ ∂U \ {1}, we obtain
<{p(z2 ) + 1(z2 )z2 p0 (z2 )}


eiα z2 h02 (z2 ) 
 λ − 1 + eiα h2 (z2 )

+ 1(z2 )
= < 

λ
λ


eiα mζ2 q02 (ζ2 ) 
 λ − 1 + eiα q2 (ζ2 )

+ 1(z2 )
= < 

λ
λ
!
eiα mσ2 (ρ2 )
λ − 1 + eiα iρ2
+ 1(z2 )
=<
λ
λ
|λ|2 − <(λ) + [=(λ) cos α − <(λ) sin α]ρ2
|λ|2
m{< 1(z2 ) [<(λ) cos α + =(λ) sin α] + =(1(z2 ))[=(λ) cos α − <(λ) sin α]}σ2 (ρ2 )
+
|λ|2
2
|λ| − <(λ) + [=(λ) cos α − <(λ) sin α]ρ2 nAσ2 (ρ2 )
5
+
|λ|2
|λ|2
!
|λ|2 − <(λ) + [=(λ) cos α − <(λ) sin α]ρ2 nA ρ22 + 2ρ2 sin α + 1
=
− 2
2 cos α
|λ|2
|λ|
=
= Bρ22 + Cρ2 + D
=: κ(ρ2 ),
(16)
where
B=−
nA
,
2 cos α|λ|2
C=−
nA tan α + <(λ) sin α − =(λ) cos α
|λ|2
and
D=
|λ|2 − <(λ)
nA
−
.
|λ|2
2|λ|2 cos α
We can see that the function κ(ρ2 ) in (16) takes on the maximum value at ρ∗2 given by
ρ∗2 = −
nA sin α − [=(λ) cos α − <(λ) sin α] cos α
.
nA
Hence we have
<{p(z2 ) + 1(z2 )z2 p0 (z2 )} 5 κ(ρ∗2 )
cos α nA tan α + <(λ) sin α − =(λ) cos α
=
2nA
|λ|
+
=: A2 ,
|λ|2 − <(λ)
nA
−
2
2
|λ|
2|λ| cos α
!2
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3622
where A is given by (1). This evidently contradicts the assumption of Theorem 1. Therefore, we have
<{h(z)} = <{e−iα [λp(z) + 1 − λ]} > 0
z ∈ U; −
π
π
<α<
.
2
2
(17)
Consequently, by suitably combining the inequalities (9) and (17), we complete the proof of Theorem 1.
3. Corollaries and Consequences
By setting λ = 1 in Theorem 1, we obtain the following corollary, which was obtained by Shiraishi et al.
[5].
Corollary 1. Let the function 1(z) be analytic in U with
A := inf {< 1(z) cos α − |= 1(z) sin α|} > 0.
(18)
z∈U
If p(z) ∈ H[1, n] satisfies the following conditions:
p(z) , 1
(z ∈ U)
and
<{p(z) + 1(z)zp0 (z)} >
1
[(cos α + 2nA) sin2 α − n2 A2 cos α]
2nA
(z ∈ U),
then
| arg{p(z)}| <
π
− |α|
2
z ∈ U; −
π
π
<α<
.
2
2
If we let
α=
π
(1 − µ)
2
(0 < µ 5 1)
in Theorem 1, we obtain the following corollary.
Corollary 2. Let the parameters µ and λ be constrained by
0<µ51
and
λ ∈ C \ {0},
respectively. Suppose also that the function 1(z) is analytic in U with
i
π h A := inf sin µ < 1(z) <(λ) + = 1(z) =(λ)
z∈U
2 i
π h − cos µ < 1(z) =(λ) + = 1(z) <(λ) > 0.
2
If p(z) ∈ H[1, n] satisfies the following conditions:
p(z) , 1
and
<{p(z) + 1(z)zp0 (z)} > max{A1 , A2 },
(z ∈ U)
(19)
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3623
where
A1 =

2
 nA cot π2 µ + <(λ) cos π2 µ + =(λ) sin π2 µ 





2nA 
|λ|
sin
π
2µ
+
|λ|2 − <(λ)
nA
−
2
|λ|2
2|λ| sin π2 µ
(z ∈ U),
(20)
and
A2 =

2
 nA cot π2 µ + <(λ) cos π2 µ − =(λ) sin π2 µ 





2nA 
|λ|
sin
π
2µ
+
|λ|2 − <(λ)
nA
−
2
|λ|
2|λ|2 sin π2 µ
(z ∈ U),
(21)
then
p(z) ∈ P(λ, µ).
If, in Corollary 2, we put
λ = 1 + i tan β
−
π
π
<β<
2
2
and
p(z) =
z f 0 (z)
= 1 + nan+1 zn + · · ·
f (z)
(z ∈ U)
for f (z) ∈ An , we are led to the following result.
Corollary 3. Let the parameters µ, β and λ be constrained by
0 < µ 5 1,
−
π
π
<β<
2
2
and
λ ∈ C \ {0},
respectively. Suppose also that the function 1(z) is analytic in U with
i
π h A := inf sin µ < 1(z) + = 1(z) tan β
z∈U
2 i
π h − cos µ < 1(z) tan β + = 1(z) > 0.
2
If f (z) ∈ An satisfies the following conditions:
z f 0 (z)
,1
f (z)
(z ∈ U)
and
"
#!
z f 0 (z)
z f 0 (z)
z f 0 (z) z f 00 (z)
<
+ 1(z)
1−
+ 0
> max{A1 , A2 },
f (z)
f (z)
f (z)
f (z)
(22)
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3624
where
A1 =

2
 nA cot π2 µ + cos π2 µ + sin π2 µ tan β 




2nA 
sec β
sin
π
2µ
+ sin2 β −
nA cos2 β
2 sin π2 µ
(z ∈ U)
(23)
and
A2 =
sin π2 µ nA cot π2 µ + cos π2 µ − sin π2 µ tan β
sec β
2nA
!2
+ sin2 β −
nA cos2 β
2 sin π2 µ
(z ∈ U),
(24)
then
µ
f (z) ∈ ST Ŝβ .
Proof. By applying Corollary 2, we deduce that
!
z f 0 (z)
π
− i tan β < µ
arg (1 + i tan β)
2
f (z)
or, equivalently, that
!
0
π
iβ z f (z)
− i sin β < µ,
arg e
2
f (z)
which implies that
µ
f (z) ∈ ST Ŝβ ,
as claimed. This completes the proof of Corollary 3.
If we set β = 0 in Corollary 3, we are led easily to Corollary 4 which was proven by Shiraishi et al. [5].
Corollary 4. Let the function 1(z) be analytic in U with
π
π A := inf < 1(z) sin µ − = 1(z) cos µ > 0
z∈U
2
2
(z ∈ U; 0 < µ 5 1).
If f (z) ∈ An satisfies the following conditions:
z f 0 (z)
,1
f (z)
(z ∈ U)
and
"
#!
z f 0 (z)
z f 0 (z)
z f 0 (z) z f 00 (z)
<
+ 1(z)
1−
+ 0
f (z)
f (z)
f (z)
f (z)
1
π
π
π
>
sin µ + 2nA cos2 µ − n2 A2 sin µ
2nA
2
2
2
then
f (z) ∈ ST S(µ).
(z ∈ U),
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3625
Corollary 5. Let the function 1(z) be analytic in U with
n o
π
π
.
A := inf < 1(z) + = 1(z) tan β > 0
z ∈ U; − < β 5
z∈U
2
2
If f (z) ∈ An satisfies the following conditions:
z f 0 (z)
,1
f (z)
(z ∈ U)
and
"
#!
z f 0 (z)
z f 0 (z)
z f 0 (z) z f 00 (z)
<
+ 1(z)
1−
+ 0
f (z)
f (z)
f (z)
f (z)
1
>
sec2 β − nA cos2 β + sin2 β
(z ∈ U),
2nA
then
f (z) ∈ Ŝβ .
Proof. If we set µ = 1 in Corollary 3, we are led easily to Corollary 5.
4. Concluding Remarks and Observations
In the present investigation, we have derived sufficient conditions for functions to be in a certain general
class of Carathéodory functions in the open unit disk U by using the Miller-Mocanu lemma (see the Lemma
in Section 1). As an application of our main result (Theorem 2 above), we have deduced sufficient conditions
µ
for functions belonging to the class ST Ŝβ which we have introduced in this paper. By suitably specializing
the parameters involved in the general result (Theorem 2), several interesting corollaries and consequences
have also been derived (see Corollaries 1 to 5 in Section 3). The various results presented here are shown
to generalize and extend many known results.
Acknowledgements
This work was supported by NNSF of the People’s Republic of China (Grant No. 11261022), the Jiangxi
Provincial Natural Science Foundation of the People’s republic of China (Grant No. 20132BAB201004 ), and
the Natural Science Foundation of Department of Education of Jiangxi Province of the People’s Republic of
China (Grant No. GJJ12177).
References
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