Bulletin of Mathematical Analysis and Applications
ISSN: 1821-1291, URL: http://www.bmathaa.org
Volume 2 Issue 2(2010), Pages 42-49.
SUBORDINATION AND SUPERORDINATION FOR CERTAIN
ANALYTIC FUNCTIONS
SAIBAH SIREGAR, MASLINA DARUS AND BASEM A. FRASIN
Abstract. In this article, we investigate results on subordination and superordination given by some authors. Motivated by earlier work and by using
a method based upon the Briot-Bouquet differential subordination, we prove
several subordination results related to the class B(α). For this purpose, a
class denoted by Bb∗ is defined and some properties are obtained in the open
unit disk.
1. Introduction and Definition
Let F and G be analytic functions in the open unit disk D = {z : z ∈ C, |z| < 1}.
If f, F ∈ H(D) and F is univalent in D we say that the function f is subordinate to
F , or F is superordinate to f , written f (z) ≺ F (z), if f (0) = F (0) and f (D) ⊆ F (D).
In general, given two functions F and G, which are analytic in D, the function F
is said to be subordinate to G in D if there exists a function h, analytic in D with
h(0) = 0 and
|h(z)| < 1 f or all
z∈D
such that
F (z) = G(h(z)) f or all z ∈ D.
Let ϕ : C2 → C and let h be univalent in D. If p is analytic and satisfies the
differential subordination ϕ(p(z), zp0 (z)) ≺ h(z) then p is called a solution of the
differential subordination.
The univalent function q is called a dominant of the solutions of the differential
subordination, if p ≺ q. If p and ϕ(p(z), zp0 (z)) are univalent in D and satisfy the
differential superordination h(z) ≺ ϕ(p(z), zp0 (z)) then p is called a solution of the
differential superordination. An analytic function q is called subordinant of the
solution of the differential superordination if q ≺ p.
2000 Mathematics Subject Classification. 35A07, 35Q53.
Key words and phrases. Analytic function; Differential Subordination; Subordination and
Superordination.
c
2010
Universiteti i Prishtinës, Prishtinë, Kosovë.
Submitted March, 2010. Published April, 2010.
S. S. was supported by UKM-OUP-FST-2008 (postdoctoral research fellowship from Universiti
Kebangsaan Malaysia).
M. D. was supported by UKM-ST-06-FRGS0107-2009, MOHE Malaysia.
42
SUBORDINATION AND SUPERORDINATION
43
Let H be the class of functions analytic in D and H[a, n] be the subclass of H. For
a ∈ C and n ∈ N∗ we denote
H[a, n] = {f ∈ H(D) : f (z) = a + an z n + · · · }.
Let A denote the class of functions of the form
∞
X
f (z) = z +
an z n ,
(1.1)
n=2
and normalized by f (0) = f 0 (0) − 1 = 0, which are analytic in D.
Also, denote
(
S ∗ (α) =
zf 0 (z)
f : f ∈ A, Re
f (z)
)
!
> α,
z ∈ D;
0≤α<1 ,
(1.2)
and
!
zf 0 (z) π
f : f ∈ A, arg
< α
f (z) 2
(
∗
Sst
(α) =
)
z ∈ D;
0<α≤1
(1.3)
be the the familiar classes starlike function of order α in D and strongly starlike
functions of order α in D, respectively.
We note that
∗
Sst
(α) ⊂ S ∗ ,
(0 < α ≤ 1),
∗
and Sst
(1) = S ∗ .
A function f ∈ A is said to be a member of the class B(α) if only if
z 2 f 0 (z)
− 1 < 1 − α
(z ∈ D).
2
f (z)
(1.4)
Note that the condition (1.4) is equivalent to
)
(
z 2 f 0 (z)
>α
Re
f 2 (z)
(1.5)
for some α (0 ≤ α < 1) and for all z ∈ D.
Frasin and Darus [1] have defined the class B(α) and investigate some interesting
properties for this class. In this paper we shall give new additional results for
functions of the class B(α).
We denote by B ∗ the class of A define by
( (
)
2zf 0 (z)
2 0
00
z
f
(z)
1
zf
(z)
Bb∗ = Re
+
1+ 0
−
−1
> 0;
bf 2 (z) b
f (z)
f (z)
)
z∈D
and Bb∗ (α) the class of A define by
( (
)
2zf 0 (z)
2 0
00
f
(z)
(z)
z
1
zf
Bb∗ (α) = Re
+
1+ 0
−
−1
> α;
bf 2 (z)
b
f (z)
f (z)
(1.6)
)
z∈D
(1.7)
where b ∈ C = C \ {0}.
Srivastava and Lashin [3] investigated the starlike and convex functions of complex
order.
44
S. SIREGAR, M. DARUS, B.A. FRASIN
The main objective of the present paper to the aforementioned works is to apply a
method based upon the Briot-Bouquet differential subordination and superordination in order to derive several subordination and superordination results involving
analytic functions.
2. Preliminaries
In order to prove our main subordination results, we shall make use of the following
known results.
Lemma 2.1. (see [4]) Let the (nonconstant) function w be analytic in D and such
that w(0) = 0. If |w(z)| attains its maximum value on circle |z| = r < 1 at a point
zo ∈ D, we have
zo w0 (z) = kw(zo ),
where k ≥ 1 is a real number.
Lemma 2.2. (Miller and Mocanu [6].) Let the functions F and G be analytic in
the unit disk D and let
F (0) = G(0).
0
If the function H(z) := zG (z) is starlike in D and
zF 0 (z) ≺ zG0 (z),
then
Z
z
H(t)
dt,
t
0
The function G is convex and is the best dominant in (2.1).
F (z) ≺ G(z) = G(0) +
(2.1)
Lemma 2.3. (Eenigenburg et. al [5]). Let β and γ be complex constants. Also let
the function h be convex (univalent) in D with
h(0) = 1
and
Re(βh(z) + γ) > 0,
(z ∈ D).
Suppose that the function
p(z) = 1 + p1 z + p2 z 2 + . . . .
is analytic in D and satisfies the following differential subordination:
zp0 (z)
p(z) +
≺ h(z)
βp(z) + γ
If the differential equation:
zq 0 (z)
q(z) +
= h(z)
(q(0) := 1),
βq(z) + γ
has a univalent solution q, then
(2.2)
(2.3)
p(z) ≺ q(z) ≺ h(z)
and q is the best dominant in (2.2) (that is, p(z) ≺ q(z)) for all p(z) satisfying
(2.2) and if p(z) ≺ q̂(z) for all p(z) satisfying (2.2), then q(z) ≺ q̂(z)).
Lemma 2.4. (Miller and Mocanu [7]) Let q(z) be convex univalent in the unit
disk D and γ ∈ C. Further, assume that Re{γ} > 0. If p(z) ∈ H[q(0), 1] ∩ Q,
with p(z) + γzp0 (z) is univalent in D then q(z) + γzq 0 (z) ≺ p(z) + γzp0 (z) implies
q(z) ≺ p(z), and q is the best subordinate.
SUBORDINATION AND SUPERORDINATION
45
Remark 2.5. The conclusion of Lemma 2.3 can be written in the following form:
p(z) +
zp0 (z)
zq 0 (z)
≺ q(z) +
⇒ p(z) ≺ q(z)
βp(z) + γ
βq(z) + γ
Remark 2.6. The differential equation (2.3) has its formal solution given by
β
zF 0 (z)
γ
β + γ H(z)
q(z) =
− ,
=
F (z)
β
F (z)
β
where
F (z) =
β+γ
zγ
Z
β1
z
β γ−1
{H(t)} t
dt
,
0
and
Z
H(z) = z. exp
0
z
h(t) − 1
dt .
t
3. Main Result
We begin with the following theorem.
Theorem 3.1. Let the function h be univalent in D, let h and Re(bh(z)) > 0,
z ∈ D, b ∈ C = C \ {0}, also let f ∈ A.
a) If
!
zf 00 (z) 2zf 0 (z)
z 2 f 0 (z) 1 +
1+ 0
−1
≺ h(z),
(3.1)
−
bf 2
b
f (z)
f (z)
then
z 2 f 0 (z)
≺ h(z).
bf 2
b) If the following differential equation:
q(z) +
zq 0 (z)
= h(z)
βq(z) + γ
(3.2)
(q(0) := 1),
has a univalent solution q(z), then
z 2 f 0 (z) 1 zf 00 (z) 2zf 0 (z)
+
1
+
−
−
1
bf 2
b
f 0 (z)
f (z)
!
≺ h(z) ⇒
z 2 f 0 (z)
≺ q(z) ≺ h(z),
bf 2
(3.3)
and q is the best dominant in (3.3).
Proof. a) We begin by setting
z 2 f 0 (z)
=: p(z),
bf 2 (z)
(3.4)
so that p has the following expansion:
p(z) = 1 + p1 z + p2 z 2 + . . . .
By differentiating logarithmically (3.4), we obtain
zp0 (z)
z 2 f 0 (z) 1 zf 00 (z) 2zf 0 (z)
p(z) +
=
+
1
+
−
−
1
bp(z)
bf 2
b
f 0 (z)
f (z)
!
46
S. SIREGAR, M. DARUS, B.A. FRASIN
and the subordination (3.1) can be written as follows:
p(z) +
zp0 (z)
≺ h(z).
bp(z)
The conclusion of the theorem would follow from Lemma 2.3 by taking
β=b
γ = 0.
This evidently completes the proof of Theorem 3.1.
Theorem 3.2. Let f be analytic in D such that f (0) = 0, h be convex univalent
in D and h ∈ H[0, 1] ∩ Q. Assume that
!
zf 00 (z) 2zf 0 (z)
z 2 f 0 (z) 1 +
1+ 0
−
−1
bf 2
b
f (z)
f (z)
is univalent function in D, where Re{γ} > 0 and b ∈ C = C \ {0}. If h ∈ A and
the subordination
zq 0 (z)
z 2 f 0 (z) γ zf 00 (z) 2zf 0 (z)
h(z) = q(z) +
≺
+
1
+
−
1
,
−
βq(z) + γ
bf 2
b
f 0 (z)
f (z)
holds, then
h(z) ≺
where p(z) =
z 2 f 0 (z)
bf 2 (z)
implies
h(z) ≺ q(z) ≺ p(z),
z 2 f 0 (z)
and h is the best subordinant.
bf 2 (z)
Proof. Our aim is to apply Lemma 2.4. Setting p(z) :=
Now we must show that
z 2 f 0 (z)
bf 2 (z) .
q(z) + zq 0 (z) ≺ p(z) + zp0 (z).
By the assumption of the theorem we have
2 0
f (z)
h(z) = q(z) + γzq 0 (z) ≺ z bf
+ γb
1+
2
zf 00 (z)
f 0 (z)
−
2zf 0 (z)
f (z)
−1
= p(z) + γzp0 (z).
Thus in view of Lemma 2.3 and Lemma 2.4, h(z) ≺ q(z) ≺ p(z) and h is the best
subordinate.
If we combine Theorem 3.2 together with Theorem 3.1, then we obtain the differential sandwich-type theorem.
Next, applying Lemma 2.1, we prove the following:
Theorem 3.3. Let f ∈ A. If
z 2 f 0 (z)
1 − α
1 zf 00 (z) 2zf 0 (z)
−
−1+
1+ 0
−1 <
,
bf 2
b
f (z)
f (z)
2α
where
1
2
≤ α < 1 and b ∈ C = C \ {0}, then f ∈ Bb∗ (α).
(z ∈ D), (3.5)
SUBORDINATION AND SUPERORDINATION
47
Proof. We define w(z) by
z 2 f 0 (z)
1 + (1 − 2α)w(z)
=
bf 2 (z)
1 − w(z)
(w(z) 6= 1),
(3.6)
we see that w is regular in D and w(0) = 0. By the logarithmic differentiations, we
get from (3.6) that
!
zf 00 (z) 2zf 0 (z)
(1 − 2α)zw0 (z)
zw0 (z)
1 1+ 0
−
−1
=
+
.
(3.7)
b
f (z)
f (z)
1 + (1 − 2α)w(z) 1 − w(z)
It follows from (3.6) and (3.7) that
zf 00 (z) 2zf 0 (z)
z 2 f 0 (z) 1 +
1
+
−
−
1
bf 2
b
f 0 (z)
f (z)
=
!
(1 − 2α)zw0 (z)
zw0 (z)
1 + (1 − 2α)w(z)
+
+
1 − w(z)
1 + (1 − 2α)w(z) 1 − w(z)
or equivalently,
z 2 f 0 (z)
1 zf 00 (z) 2zf 0 (z)
−
1
+
1
+
−
−
1
bf 2
b
f 0 (z)
f (z)
!
!
2(1 − α)w(z)
zw0 (z)
=
1+
.
1 − w(z)
[1 + (1 − 2α)w(z)]w(z)
(3.8)
Suppose there exist zo ∈ D such that
max |w(z)| = |w(zo )| = 1,
|z|<|zo |
(w(zo ) 6= −1),
and then from Lemma 2.1, we have
zo w0 (z) = kw(zo ),
where k ≥ 1 is a real number. From (3.8), we have
z 2 f 0 (z )
1 zo f 00 (zo ) 2zo f 0 (zo )
o
o
−1+
1+
−
−1 2
bf (zo )
b
f 0 (zo )
f (zo )
!
2(1 − α)w(z )
zo w0 (zo )
o
1+
= 1 − w(zo )
[1 + (1 − 2α)w(zo )]w(zo ) 2(1 − α)w(z ) zo w0 (zo )
o ≥ 1 − w(zo ) [1 + (1 − 2α)w(zo )]w(zo ) (1 − α)k
2α
1−α
≥
2α
which contradicts our assumption (3.5). Therefore |w(z)| < 1 holds for all z ∈ D.
We finally have f ∈ Bb∗ (α).
≥
Putting α =
1
2
in Theorem 3.3, we have the following corollary:
48
S. SIREGAR, M. DARUS, B.A. FRASIN
Corollary 3.4. Let f ∈ A. If
z 2 f 0 (z)
1
1 zf 00 (z) 2zf 0 (z)
−1+
−
1+ 0
−1 < ,
bf 2
2
b
f (z)
f (z)
(z ∈ D).
(3.9)
Then f ∈ Bb∗ ( 21 ).
Remark 3.5. Setting b = 1 in Theorem 3.3, we arrive to Theorem 2.5 obtained by
Frasin el. al., [2].
Theorem 3.6. If f ∈ Bb∗ (α), (0 ≤ α < 1) and Re(bz + b) > 0; z ∈ D, then
z 2 f 0 (z)
≺ q(z)
bf 2 (z)
where q is the best dominant given by


bz
e
1
− 1 ,
q(z) = 
b (−bz)−b Γ(b) + Γ(b, −bz)
and
Γ(b, −bz) = Γ(b) + z b · 1 F1 (b, 1 + b, bz).
Proof. First of all, we observe that (1.4) is equivalent to the inequality:
!
z 2 f 0 (z) 1 zf 00 (z) 2zf 0 (z)
+
1
+
−
−
1
−
1
(z ∈ D),
< 1,
bf 2
b
f 0 (z)
f (z)
which implies that
z 2 f 0 (z) 1 zf 00 (z) 2zf 0 (z)
+
1
+
−
−
1
bf 2
b
f 0 (z)
f (z)
!
≺ 1 + z.
Thus, in Theorem 3.1, we choose
h(z) = 1 + z,
and note that
Re(bh(z)) > 0,
when z ∈ D , and h satisfies the hypotheses of Lemma 2.3. Consequently, in the
view of Lemma 2.3 and Remark 2.6, we have
Z z
h(t) − 1
H(z) = z · exp
dt ,
t
0
which, for h(z) = 1 + t, yields
H(z) = zez ,
and
1/b
{H(t)}b
F (z)
dt
t
0
Z z bt 1/b
e
=
b
dt
.
1−b
t
0
Z
=
b
z
(3.10)
SUBORDINATION AND SUPERORDINATION
By using the software MAPLE, F can be simplified to the following form:
1/b
F (z) = − z b−1 (−bz)−b − zbΓ(b) + zbΓ(b, −bz)
.
49
(3.11)
From (3.10) and (3.11), we obtain
"
#
1
ebz
q(z) =
−1 .
b (−bz)−b (Γ(b) + Γ(b, −bz))
The proof of Theorem 3.6 is complete.
Acknowledgement: The authors would like to thank the anonymous referee for
the informative and creative comments given to the article.
References
[1] B. A. Frasin and M. Darus, On certain analytic univalent function, Int.Jour. Math.and Math.
Sci. 2001, 25(5): 305-310.
[2] B.A. Frasin, M. Darus and S. Siregar. Some sufficient conditions for univalence and subordination results of certain analytic and univalent functions. Mathematica Cluj-Tome 50 (73),
(2008) no.1: 39-49.
[3] H. M. Srivastava and A. Y. Lashin, Some applications of the Briot-Bouquet differential subordination, Jour. Ineq. Pure and Appl. Math. 2005, 6(2): 1-7.
[4] I.S. Jack, Functions starlike and convex of order α, J. London Mth. Soc. (2)3, (1971), 469-474.
[5] P. Eenigenburg, S.S. Miller, P.T. Mocanu,and M. O. Read, On a Briot-Bouquet differential
subordination, General Inequalities 3, pp. 339-348, International Series of Numerical Mathematics, Vol.64, Birkhäuser Verlag, Basel, 1983; see also Rev. Roumaine Math. Pures Appl.
, 1984, 29: 567-573.
[6] S.S. Miller and P.T. Mocanu, Differential Subordinations: Theory and Applications Series
on Monographs and Textbooks in Pure and Applied Mathematics (No 225), Marcel Dekker,
New York and Basel, 2000.
[7] S.S. Miller and P.T. Mocanu, Subordinations of differential superordinations, Complex Variables, 48(10)(2003), 815-826.
Saibah Siregar and Maslina Darus
School of Mathematical Sciences
Faculty of Science and Technology
Universiti Kebangsaan Malaysia
Bangi 43600 Selangor D.E., Malaysia
E-mail address: saibahmath@yahoo.com
E-mail address: maslina@ukm.my (corresponding author)
Basem A. Frasin
Department of Mathematical Sciences
Al al-Bayt University
P.O. Box 130095, Mafraq, Jordan
E-mail address: bafrasin@yahoo.com