Volume 8 (2007), Issue 1, Article 20, 6 pp.
SUBORDINATION AND SUPERORDINATION RESULTS FOR Φ-LIKE
FUNCTIONS
T.N. SHANMUGAM, S. SIVASUBRAMANIAN, AND MASLINA DARUS
D EPARTMENT OF I NFORMATION T ECHNOLOGY,
S ALALAH C OLLEGE OF E NGINEERING
S ALALAH , S ULTANATE OF O MAN
drtns2001@yahoo.com
D EPARTMENT OF M ATHEMATICS ,
E ASWARI E NGINEERING C OLLEGE
R AMAPURAM , C HENNAI -600 089
I NDIA
sivasaisastha@rediffmail.com
S CHOOL O F M ATHEMATICAL S CIENCES ,
FACULTY O F S CIENCES AND T ECHNOLOGY,
UKM, M ALAYSIA
maslina@pkrisc.cc.ukm.my
Received 24 June, 2006; accepted 29 December, 2006
Communicated by N.E. Cho
A BSTRACT. Let q1 be convex univalent and q2 be univalent in ∆ := {z : |z| < 1} with q1 (0) =
q2 (0) = 1. Let f be a normalized analytic function in the open unit disk ∆. Let Φ be an analytic
function in a domain containing f (∆), Φ(0) = 0 and Φ0 (0) = 1. We give some applications of
first order differential subordination and superordination to obtain sufficient conditions for the
function f to satisfy
z(f ∗ g)0 (z)
q1 (z) ≺
≺ q2 (z)
Φ(f ∗ g)(z)
where g is a fixed function.
Key words and phrases: Differential subordination, Differential superordination, Convolution, Subordinant.
2000 Mathematics Subject Classification. 30C45.
1. I NTRODUCTION AND M OTIVATIONS
Let A be the class of all normalized analytic functions f (z) in the open unit disk ∆ :=
{z : |z| < 1} satisfying f (0) = 0 and f 0 (0) = 1. Let H be the class of functions analytic in ∆
and for any a ∈ C and n ∈ N, H[a, n] be the subclass of H consisting of functions of the form
We would like to thank the referee for his insightful suggestions.
175-06
2
T.N. S HANMUGAM , S. S IVASUBRAMANIAN ,
AND
M ASLINA DARUS
f (z) = a + an z n + an+1 z n+1 + · · · . Let p, h ∈ H and let φ(r, s, t; z) : C3 × ∆ → C. If p and
φ(p(z), zp02 p00 (z); z) are univalent and if p satisfies the second order superordination
(1.1)
h(z) ≺ φ(p(z), zp02 p00 (z); z),
then p is a solution of the differential superordination (1.1). If f is subordinate to F , then F
is called a superordinate of f. An analytic function q is called a subordinant if q ≺ p for all p
satisfying (1.1). A univalent subordinant q̄ that satisfies q ≺ q̄ for all subordinants q of (1.1)
is said to be the best subordinant. Recently Miller and Mocanu [5] obtained conditions on h, q
and φ for which the following implication holds:
(1.2)
h(z) ≺ φ(p(z), zp02 p00 (z); z) ⇒ q(z) ≺ p(z).
Using the results of Miller and Mocanu [4], Bulboacă [2] considered certain classes of first
order differential superordinations as well as superordination-preserving integral operators [1].
In an earlier investigation, Shanmugam et al. [8] obtained sufficient conditions for a normalized
z 2 f 0 (z)
analytic function f (z) to satisfy q1 (z) ≺ zff (z)
≺ q2 (z) where q1
0 (z) ≺ q2 (z) and q1 (z) ≺
{f (z)}2
and q2 are given univalent functions in ∆ with q1 (0) = 1 and q2 (0) = 1. A systematic study of
the subordination and superordination has been studied very recently by Shanmugam et al. in
[9] and [10] (see also the references cited by them).
Let Φ be an analytic function in a P
domain containing f (∆)P
with Φ(0) = 0 and Φ0 (0) = 1. For
∞
n
n
any two analytic functions f (z) = ∞
n=0 an z and g(z) =
n=0 bn z , the Hadamard product
or convolution of f (z) and g(z), written as (f ∗ g)(z) is defined by
(f ∗ g)(z) =
∞
X
an b n z n .
n=0
The function f ∈ A is called Φ-like if
zf 0 (z)
(1.3)
<
>0
Φ(f (z))
(z ∈ ∆).
The concept of Φ− like functions was introduced by Brickman [3] and he established that a
function f ∈ A is univalent if and only if f is Φ-like for some Φ. For Φ(w) = w, the function f
is starlike. In a later investigation, Ruscheweyh [7] introduced and studied the following more
general class of Φ-like functions.
Definition 1.1. Let Φ be analytic in a domain containing f (∆), Φ(0) = 0, Φ0 (0) = 1 and
Φ(w) 6= 0 for w ∈ f (∆) \ {0} . Let q(z) be a fixed analytic function in ∆ , q(0) = 1. The
function f ∈ A is called Φ-like with respect to q if
(1.4)
zf 0 (z)
≺ q(z)
Φ(f (z))
(z ∈ ∆).
When Φ(w) = w, we denote the class of all Φ-like functions with respect to q by S ∗ (q).
Using the definition of Φ− like functions, we introduce the following class of functions.
Definition 1.2. Let g be a fixed function in A. Let Φ be analytic in a domain containing f (∆) ,
Φ(0) = 0, Φ0 (0) = 1 and Φ(w) 6= 0 for w ∈ f (∆) \ {0} . Let q(z) be a fixed analytic function
in ∆, q(0) = 1. The function f ∈ A is called Φ-like with respect to Sg∗ (q) if
(1.5)
z(f ∗ g)0 (z)
≺ q(z)
Φ(f ∗ g)(z)
J. Inequal. Pure and Appl. Math., 8(1) (2007), Art. 20, 6 pp.
(z ∈ ∆).
http://jipam.vu.edu.au/
S UBORDINATION AND
SUPERORDINATION RESULTS FOR
Φ- LIKE FUNCTIONS
3
We note that S ∗ z (q) := S ∗ (q).
1−z
In the present investigation, we obtain sufficient conditions for a normalized analytic function
f to satisfy
z(f ∗ g)0 (z)
q1 (z) ≺
≺ q2 (z).
Φ(f ∗ g)(z)
We shall need the following definition and results to prove our main results. In this sequel,
unless otherwise stated, α and γ are complex numbers.
Definition 1.3 ([4, Definition 2, p. 817]). Let Q be the set of all functions f that are analytic
¯ − E(f ), where
and injective on ∆
E(f ) = ζ ∈ ∂∆ : lim f (z) = ∞ ,
z→ζ
0
and are such that f (ζ) 6= 0 for ζ ∈ ∂∆ − E(f ).
Lemma 1.1 ([4, Theorem 3.4h, p. 132]). Let q be univalent in the open unit disk ∆ and
θ and φ be analytic in a domain D containing q(∆) with φ(ω) 6= 0 when ω ∈ q(∆). Set
ξ(z) = zq 0 (z)φ(q(z)), h(z) = θ(q(z)) + ξ(z). Suppose that
(1) ξ(z)
univalent in ∆, and
n is0 starlike
o
zh (z)
(2) < ξ(z) > 0 (z ∈ ∆).
If p is analytic in ∆ with p(∆) ⊆ D and
(1.6)
θ(p(z)) + zp0 (z)φ(p(z)) ≺ θ(q(z)) + zq 0 (z)φ(q(z)),
then p(z) ≺ q(z) and q is the best dominant.
Lemma 1.2. [2, Corollary 3.1, p. 288] Let q be univalent in ∆, ϑ and ϕ be analytic in a domain
D containing q(∆). Suppose that
i
h 0
(q(z))
(1) < ϑϕ(q(z))
> 0 for z ∈ ∆, and
0
(2) ξ(z) = zq (z)ϕ(q(z)) is starlike univalent function in ∆.
If p ∈ H [q(0), 1] ∩ Q, with p(∆) ⊂ D, and ϑ(p(z)) + zp0 (z)ϕ(p(z)) is univalent in ∆, and
(1.7)
ϑ (q(z)) + zq 0 (z)ϕ(q(z)) ≺ ϑ (p(z)) + zp0 (z)ϕ(p(z)),
then q(z) ≺ p(z) and q is the best subordinant.
2. M AIN R ESULTS
By making use of Lemma 1.1, we prove the following result.
0
(z)
Theorem 2.1. Let q(z) 6= 0 be analytic and univalent in ∆ with q(0) = 1 such that zqq(z)
is
starlike univalent in ∆. Let q(z) satisfy
αq(z) zq 0 (z) zq 00 (z)
(2.1)
< 1+
−
+ 0
> 0.
γ
q(z)
q (z)
Let
z(f ∗ g)0 (z)
z(f ∗ g)00 (z) z (Φ(f ∗ g)(z))0
(2.2)
Ψ(α, γ, g; z) := α
+γ 1+
−
.
Φ(f ∗ g)(z)
(f ∗ g)0 (z)
Φ(f ∗ g)(z)
If q satisfies
γzq 0 (z)
(2.3)
Ψ(α, γ, g; z) ≺ αq(z) +
,
q(z)
J. Inequal. Pure and Appl. Math., 8(1) (2007), Art. 20, 6 pp.
http://jipam.vu.edu.au/
4
T.N. S HANMUGAM , S. S IVASUBRAMANIAN ,
then
AND
M ASLINA DARUS
z(f ∗ g)0 (z)
≺ q(z)
Φ(f ∗ g)(z)
and q is the best dominant.
Proof. Let the function p(z) be defined by
p(z) :=
(2.4)
z(f ∗ g)0 (z)
.
Φ(f ∗ g)(z)
Then the function p(z) is analytic in ∆ with p(0) = 1. By a straightforward computation
zp0 (z)
z(f ∗ g)00 (z) z[Φ(f ∗ g)(z)]0
= 1+
−
p(z)
(f ∗ g)0 (z)
Φ(f ∗ g)(z)
which, in light of hypothesis (2.3) of Theorem 2.1, yields the following subordination
αp(z) +
(2.5)
γzp0 (z)
γzq 0 (z)
≺ αq(z) +
.
p(z)
q(z)
By setting
γ
,
ω
it can be easily observed that θ(ω) and φ(ω) are analytic in C \ {0} and that
θ(ω) := αω
φ(ω) 6= 0
and φ(ω) :=
(ω ∈ C \ {0}) .
Also, by letting
ξ(z) = zq 0 (z)φ(q(z)) =
(2.6)
γ
zq 0 (z).
q(z)
and
h(z) = θ{q(z)} + ξ(z) = αq(z) +
(2.7)
γ
zq 0 (z),
q(z)
we find that ξ(z) is starlike univalent in ∆ and that
αq(z) zq 0 (z) zq 00 (z)
< 1+
−
+ 0
>0
γ
q(z)
q (z)
by the hypothesis (2.1). The assertion of Theorem 2.1 now follows by an application of Lemma
1.1.
When Φ(ω) = ω in Theorem 2.1 we get:
Corollary 2.2. Let q(z) 6= 0 be univalent in ∆ with q(0) = 1. If q satisfies
z(f ∗ g)0 (z)
z(f ∗ g)00 (z)
γzq 0 (z)
(α − γ)
+γ 1+
≺
αq(z)
+
,
(f ∗ g)(z)
(f ∗ g)0 (z)
q(z)
then
z(f ∗ g)0 (z)
≺ q(z)
(f ∗ g)(z)
and q is the best dominant.
For g(z) =
z
1−z
and Φ(ω) = ω, we get the following corollary.
J. Inequal. Pure and Appl. Math., 8(1) (2007), Art. 20, 6 pp.
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S UBORDINATION AND
SUPERORDINATION RESULTS FOR
Φ- LIKE FUNCTIONS
5
Corollary 2.3. Let q(z) 6= 0 be univalent in ∆ with q(0) = 1. If q satisfies
zf 0 (z)
zf 00 (z)
γzq 0 (z)
(α − γ)
+γ 1+ 0
≺ αq(z) +
,
f (z)
f (z)
q(z)
then
zf 0 (z)
≺ q(z)
f (z)
and q is the best dominant.
1+Az
For the choice α = γ = 1 and q(z) = 1+Bz
(−1 ≤ B < A ≤ 1) in Corollary 2.3, we have
the following result of Ravichandran and Jayamala [6].
Corollary 2.4. If f ∈ A and
1+
zf 00 (z)
1 + Az
(A − B)z
≺
+
,
0
f (z)
1 + Bz (1 + Az)(1 + Bz)
then
and
zf 0 (z)
1 + Az
≺
f (z)
1 + Bz
1+Az
1+Bz
is the best dominant.
Theorem 2.5. Let γ 6= 0. Let q(z) 6= 0 be convex univalent in ∆ with q(0) = 1 such that
is starlike univalent in ∆. Suppose that q(z) satisfies
αq(z)
(2.8)
<
> 0.
γ
If f ∈ A,
z(f ∗g)0 (z)
Φ(f ∗g)(z)
zq 0 (z)
q(z)
∈ H[1, 1] ∩ Q, Ψ(α, γ, g; z) as defined by (2.2) is univalent in ∆ and
(2.9)
αq(z) +
γzq 0 (z)
≺ Ψ(α, γ, g; z),
q(z)
then
q(z) ≺
z(f ∗ g)0 (z)
Φ(f ∗ g)(z)
and q is the best subordinant.
Proof. By setting
γ
,
ω
it is easily observed that ϑ(w) is analytic in C, ϕ(w) is analytic in C \ {0} and that
ϑ(w) := αω
ϕ(w) 6= 0,
and ϕ(w) :=
(w ∈ C \ {0}).
The assertion of Theorem 2.5 follows by an application of Lemma 1.2.
For Φ(ω) = ω in Theorem 2.5, we get
Corollary 2.6. Let q(z) 6= 0 be convex univalent in ∆ with q(0) = 1. If f ∈ A and
γzq 0 (z)
z(f ∗ g)0 (z)
z(f ∗ g)00 (z)
αq(z) +
≺ (α − γ)
+γ 1+
,
q(z)
(f ∗ g)(z)
(f ∗ g)0 (z)
then
z(f ∗ g)0 (z)
q(z) ≺
(f ∗ g)(z)
and q is the best subordinant.
J. Inequal. Pure and Appl. Math., 8(1) (2007), Art. 20, 6 pp.
http://jipam.vu.edu.au/
6
T.N. S HANMUGAM , S. S IVASUBRAMANIAN ,
AND
M ASLINA DARUS
Combining Theorem 2.1 and Theorem 2.5 we get the following sandwich theorem.
Theorem 2.7. Let q1 be convex univalent and q2 be univalent in ∆ satisfying (2.8) and (2.1)
zq 0 (z)
zq 0 (z)
respectively such that q1 (0) = 1, q2 (0) = 1, q11(z) and q22(z) are starlike univalent in ∆ with
q1 (z) 6= 0 and
Let f ∈ A,
Further, if
z(f ∗g)0 (z)
Φ(f ∗g)(z)
q2 (z) 6= 0.
∈ H[1, 1] ∩ Q, and Ψ(α, γ, g; z) as defined by (2.2) be univalent in ∆.
αq1 (z) +
γzq10 (z)
γzq20 (z)
≺ Ψ(α, γ, g; z) ≺ αq2 (z) +
,
q1 (z)
q2 (z)
then
z(f ∗ g)0 (z)
≺ q2 (z)
Φ(f ∗ g)(z)
and q1 and q2 are respectively the best subordinant and best dominant.
q1 (z) ≺
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J. Inequal. Pure and Appl. Math., 8(1) (2007), Art. 20, 6 pp.
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