SUBORDINATION AND SUPERORDINATION
RESULTS FOR Φ-LIKE FUNCTIONS
T.N. SHANMUGAM
S. SIVASUBRAMANIAN
Department of Information Technology,
Salalah College of Engineering
Salalah, Sultanate of Oman
EMail: drtns2001@yahoo.com
10/11 Department of Mathematics,
Easwari Engineering College
Ramapuram, Chennai-600 089, India
EMail: sivasaisastha@rediffmail.com
Subordination and
Superordination
T.N. Shanmugam,
S. Sivasubramanian and
Maslina Darus
vol. 8, iss. 1, art. 20, 2007
MASLINA DARUS
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School Of Mathematical Sciences,
Faculty Of Sciences and Technology,
UKM, Malaysia
EMail: maslina@pkrisc.cc.ukm.my
Contents
Received:
24 June, 2006
Accepted:
29 December, 2006
Communicated by:
N.E. Cho
2000 AMS Sub. Class.:
30C45.
Key words:
Differential subordination, Differential superordination, Convolution,
Subordinant.
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Abstract:
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
q1 (z) ≺
z(f ∗ g)0 (z)
≺ q2 (z)
Φ(f ∗ g)(z)
where g is a fixed function.
Acknowledgements: We would like to thank the referee for his insightful suggestions.
Subordination and
Superordination
T.N. Shanmugam,
S. Sivasubramanian and
Maslina Darus
vol. 8, iss. 1, art. 20, 2007
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Contents
1
Introduction and Motivations
4
2
Main Results
8
Subordination and
Superordination
T.N. Shanmugam,
S. Sivasubramanian and
Maslina Darus
vol. 8, iss. 1, art. 20, 2007
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1.
Introduction and Motivations
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 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
h(z) ≺ φ(p(z), zp02 p00 (z); z),
(1.1)
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)
02 00
h(z) ≺ φ(p(z), zp p (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 analytic function f (z) to satisfy q1 (z) ≺ zff (z)
0 (z) ≺
z 2 f 0 (z)
{f (z)}2
q2 (z) and q1 (z) ≺
≺ q2 (z) where q1 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 domain containingPf (∆) with Φ(0) = 0
∞
n
and Φ0 (0) = 1. For any two analytic functions f (z) =
n=0 an z and g(z) =
Subordination and
Superordination
T.N. Shanmugam,
S. Sivasubramanian and
Maslina Darus
vol. 8, iss. 1, art. 20, 2007
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P∞
n
, the Hadamard product or convolution of f (z) and g(z), written as (f ∗
g)(z) is defined by
∞
X
(f ∗ g)(z) =
an b n z n .
n=0 bn z
n=0
The function f ∈ A is called Φ-like if
zf 0 (z)
>0
(1.3)
<
Φ(f (z))
(z ∈ ∆).
Subordination and
Superordination
T.N. Shanmugam,
S. Sivasubramanian and
Maslina Darus
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 ∈ ∆).
vol. 8, iss. 1, art. 20, 2007
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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
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respect to Sg∗ (q) if
(1.5)
z(f ∗ g)0 (z)
≺ q(z)
Φ(f ∗ g)(z)
(z ∈ ∆).
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
q1 (z) ≺
z(f ∗ g)0 (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
¯ − E(f ), where
analytic and injective on ∆
E(f ) = ζ ∈ ∂∆ : lim f (z) = ∞ ,
Subordination and
Superordination
T.N. Shanmugam,
S. Sivasubramanian and
Maslina Darus
vol. 8, iss. 1, art. 20, 2007
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z→ζ
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and are such that f 0 (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) is starlike univalent in ∆, and
n 0 o
(z)
2. < zhξ(z)
> 0 (z ∈ ∆).
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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
h 0
i
(q(z))
1. < ϑϕ(q(z))
> 0 for z ∈ ∆, and
2. ξ(z) = zq 0 (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.
Subordination and
Superordination
T.N. Shanmugam,
S. Sivasubramanian and
Maslina Darus
vol. 8, iss. 1, art. 20, 2007
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2.
Main Results
By making use of Lemma 1.1, we prove the following result.
Theorem 2.1. Let q(z) 6= 0 be analytic and univalent in ∆ with q(0) = 1 such that
zq 0 (z)
is starlike univalent in ∆. Let q(z) satisfy
q(z)
αq(z) zq 0 (z) zq 00 (z)
(2.1)
< 1+
−
+ 0
> 0.
γ
q(z)
q (z)
Maslina Darus
Let
vol. 8, iss. 1, art. 20, 2007
(2.2)
Ψ(α, γ, g; z) := α
z(f ∗ g)0 (z)
Φ(f ∗ g)(z)
z(f ∗ g)00 (z) z (Φ(f ∗ g)(z))0
+γ 1+
−
.
(f ∗ g)0 (z)
Φ(f ∗ g)(z)
If q satisfies
(2.3)
γzq 0 (z)
,
Ψ(α, γ, g; z) ≺ αq(z) +
q(z)
then
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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
(2.4)
Subordination and
Superordination
T.N. Shanmugam,
S. Sivasubramanian and
p(z) :=
z(f ∗ g)0 (z)
.
Φ(f ∗ g)(z)
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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
γzp0 (z)
γzq 0 (z)
(2.5)
αp(z) +
≺ αq(z) +
.
p(z)
q(z)
Maslina Darus
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
vol. 8, iss. 1, art. 20, 2007
φ(ω) :=
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(ω ∈ C \ {0}) .
Also, by letting
(2.6)
Subordination and
Superordination
T.N. Shanmugam,
S. Sivasubramanian and
γ
ξ(z) = zq 0 (z)φ(q(z)) =
zq 0 (z).
q(z)
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and
(2.7)
h(z) = θ{q(z)} + ξ(z) = αq(z) +
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.
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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)
Subordination and
Superordination
T.N. Shanmugam,
S. Sivasubramanian and
Maslina Darus
and q is the best dominant.
For g(z) =
z
1−z
vol. 8, iss. 1, art. 20, 2007
and Φ(ω) = ω, we get the following corollary.
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Corollary 2.3. Let q(z) 6= 0 be univalent in ∆ with q(0) = 1. If q satisfies
zf 00 (z)
γzq 0 (z)
zf 0 (z)
(α − γ)
+γ 1+ 0
≺ αq(z) +
,
f (z)
f (z)
q(z)
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0
zf (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)
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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
0 (z)
that zqq(z)
is starlike univalent in ∆. Suppose that q(z) satisfies
αq(z)
(2.8)
<
> 0.
γ
If f ∈ A,
and
z(f ∗g)0 (z)
Φ(f ∗g)(z)
Subordination and
Superordination
T.N. Shanmugam,
S. Sivasubramanian and
Maslina Darus
vol. 8, iss. 1, art. 20, 2007
∈ H[1, 1] ∩ Q, Ψ(α, γ, g; z) as defined by (2.2) is univalent in ∆
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0
(2.9)
αq(z) +
γzq (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.
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Proof. By setting
γ
ϑ(w) := αω and ϕ(w) := ,
ω
it is easily observed that ϑ(w) is analytic in C, ϕ(w) is analytic in C \ {0} and that
ϕ(w) 6= 0,
(w ∈ C \ {0}).
The assertion of Theorem 2.5 follows by an application of Lemma 1.2.
Close
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
Subordination and
Superordination
T.N. Shanmugam,
S. Sivasubramanian and
z(f ∗ g)0 (z)
q(z) ≺
(f ∗ g)(z)
Maslina Darus
and q is the best subordinant.
vol. 8, iss. 1, art. 20, 2007
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)
zq 0 (z)
zq 0 (z)
and (2.1) respectively such that q1 (0) = 1, q2 (0) = 1, q11(z) and q22(z) are starlike
univalent in ∆ with
q1 (z) 6= 0 and q2 (z) 6= 0.
z(f ∗g)0 (z)
Let f ∈ A, Φ(f ∗g)(z) ∈ H[1, 1] ∩ Q, and Ψ(α, γ, g; z) as defined by (2.2) be univalent
in ∆. Further, if
αq1 (z) +
γzq10 (z)
q1 (z)
≺ Ψ(α, γ, g; z) ≺ αq2 (z) +
then
γzq20 (z)
q2 (z)
,
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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References
[1] T. BULBOACĂ, A class of superordination-preserving integral operators,
Indag. Math. (N.S.), 13(3) (2002), 301–311.
[2] T. BULBOACĂ, Classes of first-order differential superordinations, Demonstr.
Math., 35(2) (2002), 287–292.
[3] L. BRICKMAN, Φ-like analytic functions. I, Bull. Amer. Math. Soc., 79 (1973),
555–558.
Subordination and
Superordination
T.N. Shanmugam,
S. Sivasubramanian and
Maslina Darus
[4] S.S. MILLER AND P.T. MOCANU, Differential Subordinations, Dekker, New
York, 2000.
vol. 8, iss. 1, art. 20, 2007
[5] S.S. MILLER AND P.T. MOCANU, Subordinants of differential superordinations, Complex Var. Theory Appl., 48(10) (2003), 815–826.
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[6] V. RAVICHANDRAN AND M. JAYAMALA, On sufficient conditions for
Caratheodory functions, Far East J. Math. Sci., 12(2) (2004), 191–201.
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[7] St. RUSCHEWEYH, A subordination theorem for F -like functions, J. London
Math. Soc., 13(2) (1976), 275–280.
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[8] T.N. SHANMUGAM, V. RAVICHANDRAN AND S. SIVASUBRAMANIAN,
Differential sandwich theorems for some subclasses of analytic Functions,
Aust. J. Math. Anal. Appl., 3(1) (2006), Art. 8, 11 pp. (electronic).
[9] T.N. SHANMUGAM, S. SIVASUBRAMANIAN AND H.M. SRIVASTAVA,
Differential sandwich theorems for certain subclasses of analytic functions involving multiplier transformations, Int. Transforms Spec. Functions, 17(12)
(2006), 889–899.
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[10] T.N. SHANMUGAM, S. SIVASUBRAMANIAN AND M. DARUS, On certain subclasses of functions involving a linear Operator, Far East J. Math. Sci.,
23(3), (2006), 329–339.
[11] T.N. SHANMUGAM, S. SIVASUBRAMANIAN AND S. OWA, On sandwich
theorems for some subclasses of analytic functions involving a linear operator,
to appear in Integral Transforms and Special functions.
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Superordination
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S. Sivasubramanian and
Maslina Darus
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