DIFFUSIVE LOGISTIC EQUATIONS WITH TIME DELAY AND IMPULSES: EXISTENCE OF SOLUTION AND ATTRACTORS | Widjaja | Journal of the Indonesian Mathematical Society 75 292 1 PB
J. Indones. Math. Soc. (MIHMI)
Vol. 13, No. 1 (2007), pp. 39–50.
DIFFERENTIAL SANDWICH THEOREMS
FOR SOME SUBCLASS OF
ANALYTIC FUNCTIONS
ASSOCIATED WITH LINEAR OPERATORS
T.N. Shanmugam, M.P. Jeyaraman and A. Singaravelu
Abstract. Let q1 and q2 be univalent in ∆ := {z : |z| < 1} with q1 (0) = q2 (0) = 1.
We give some applications of first order differential subordination and superordination
to obtain sufficient conditions for a normalized analytic functions f with f (0) = 0 ,
f ′ (0) = 1 to satisfy
zf ′ (z) λ
q1 (z) ≺
≺ q2 (z).
f (z)
1. INTRODUCTION
Let H be the class of functions analytic in ∆ := {z : |z| < 1} and 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 A be the subclass of H consisting of functions of the form
f (z) = z + a2 z 2 + · · · . Let p, h ∈ H and let φ(r, s, t; z) : C3 × ∆ → C. If p and
φ(p(z), zp′ (z), z 2 p′′ (z); z) are univalent and if p satisfies the second order superordination
h(z) ≺ φ(p(z), zp′ (z), z 2 p′′ (z); z),
(1)
then p is a solution of the differential superordination (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). An univalent subordinant q¯ that satisfies q ≺ q¯ for
all subordinants q of (1) is said to be the best subordinant. Recently Miller and
Received 6 February 2006, revised 25 October 2006, accepted 15 November 2006.
2000 Mathematics Subject Classification: Primary 30C45, Secondary 30C80.
Key words and Phrases: Differential Subordination, Differential Superordination, Subordinant.
39
40
T.N. Shanmugam, M.P. Jeyaraman and A. Singaravelu
Mocanu [10] obtained conditions on h, q and φ for which the following implication
holds:
h(z) ≺ φ(p(z), zp′ (z), z 2 p′′ (z); z) ⇒ q(z) ≺ p(z).
Using the results of Miller and Mocanu [10], Bulboaca [3] considered certain classes
of first order differential superordinations as well as superordination-preserving operators [2]. Using the results of [3], Shanmugam et al. [12] obtained sufficient
conditions for a normalized analytic function f (z) to satisfy
q1 (z) ≺
and
q1 (z) ≺
f (z)
≺ q2 (z),
zf ′ (z)
z 2 f ′ (z)
2
{f (z)}
≺ q2 (z) ,
respectively where q1 and q2 are given univalent functions in ∆.
For αj ∈ C (j = 1, 2, . . . , l) and βj ∈ C \ Z−
0 := {0, −1, −2, . . .}, j = 1, 2, . . . m),
the generalized hypergeometric function l Fm (α1 , . . . , αl ; β1 , . . . , βm ; z) is defined by
the infinite series
l Fm (α1 , . . . , αl ; β1 , . . . , βm ; z) :=
∞
X
(α1 )n . . . (αl )n z n
.
(β1 )n . . . (βm )n n!
n=0
(l ≤ m + 1; l, m ∈ N0 := N ∪ {0}),
where (a)n is the Pochhammer symbol defined by
½
Γ(a + n)
1,
(n = 0);
(a)n :=
=
a(a
+
1)(a
+
2)
.
.
.
(a
+
n
−
1),
(n ∈ N).
Γ(a)
Corresponding to the function
h(α1 , . . . , αl ; β1 , . . . , βm ; z) := z l Fm (α1 , . . . , αl ; β1 , . . . , βm ; z),
l
the Dziok-Srivastava operator [5] (see also [13]) Hm
(α1 , . . . , αl ; β1 , . . . , βm ; z) is
defined by the Hadamard product
l
Hm
(α1 , . . . , αl ; β1 , . . . , βm ; z)f (z)
:= h(α1 , . . . , αl ; β1 , . . . , βm ; z) ∗ f (z)
∞
X
(α1 )n−1 . . . (αl )n−1 an z n
.(2)
= z+
(β1 )n−1 . . . (βm )n−1 (n − 1)!
n=2
It is well known [5] that
l
α1 Hm
(α1 + 1, . . . , αl ; β1 , . . . , βm ; z)f (z)
l
= z[Hm
(α1 , . . . , αl ; β1 , . . . , βm ; z)f (z)]′
l
+(α1 − 1)Hm
(α1 , . . . , αl ; β1 , . . . , βm ; z)f (z).
(3)
Differential Sandwich Theorems
41
To make the notation simple, we write
l
l
Hm
[α1 ]f (z) := Hm
(α1 , . . . , αl ; β1 , . . . , βm ; z)f (z).
Special cases of the Dziok-Srivastava linear operator includes the Hohlov linear operator [6] , the Carlson-Shaffer linear operator[4], the Ruscheweyh derivative
operator [11], the generalized Bernardi-Libera-Livingston linear integral operator
(cf. [1], [7], [8]).
2. PRELIMINARIES
In our present investigation, we shall need the following definition and results.
In this paper unless otherwise mentioned α and β are complex numbers.
Definition 2.1: [10, Definition 2, p. 817] Let Q be the set of all functions f that
¯ − E(f ), where
are analytic and injective on ∆
½
¾
E(f ) = ζ ∈ ∂∆ : lim f (z) = ∞ ,
z→ζ
and are such that f ′ (ζ) 6= 0 for ζ ∈ ∂∆ − E(f ).
Theorem 2.1 : [9, Theorem 3.4h , p. 132] Let q be univalent in the 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 ′ (z)φ(q(z)), h(z) = θ(q(z)) + ξ(z). Suppose that,
1. ξ(z) is starlike univalent in ∆ and
′
(z)
> 0 for z ∈ ∆.
2. ℜ zhξ(z)
If p is analytic in ∆ with p(∆) ⊆ D, and
θ(p(z)) + zp′ (z)φ(p(z)) ≺ θ(q(z)) + zq ′ (z)φ(q(z)),
(4)
then p ≺ q and q is the best dominant.
Lemma 2.1 : [12] Let q be univalent in ∆ with q(0) = 1. Further assuming that
·
¸
α
zq ′′ (z)
ℜ
+1+ ′
> 0.
β
q (z)
If p is analytic in ∆, with p(∆) ⊆ D and
αp(z) + βzp′ (z) ≺ αq(z) + βzq ′ (z),
42
T.N. Shanmugam, M.P. Jeyaraman and A. Singaravelu
then p ≺ q and q is the best dominant.
Theorem 2.2 : [3] Let q be univalent in the unit disk ∆ and ϑ and ϕ be analytic
in a domain D containing q(∆). Suppose that
i
h ′
(q(z))
> 0 for z ∈ ∆, and
1. ℜ ϑϕ(q(z))
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)) + zp′ (z)ϕ(p(z)) is univalent in
∆, and
ϑ(q(z)) + zq ′ (z)ϕ(q(z)) ≺ ϑ(p(z)) + zp′ (z)ϕ(p(z)),
(5)
then q ≺ p and q is the best subordinant.
Lemma
i2.2 : [12] Let q be univalent in ∆ , q(0) = 1. Further assuming that
h
α ′
ℜ β q (z) > 0.
If p ∈ H [q(0), 1] ∩ Q, and αp + βzp′ is univalent in ∆, and
αq(z) + βzq ′ (z) ≺ αp(z) + βzp′ (z) ,
then q ≺ p and q is the best subordinant.
3. SUBORDINATION RESULTS FOR ANALYTIC FUNCTIONS
By making use of Lemma 2.3, we prove the following results.
Theorem 3.1 : Let q be univalent in ∆ with q(0) = 1 and satisfying
·
¸
α
zq ′′ (z)
ℜ
> 0.
+1+ ′
β
q (z)
(6)
Let
Ψ(α, β, λ; z) := α
µ
zf ′ (z)
f (z)
¶λ
+ βλ
µ
zf ′ (z)
f (z)
¶λ ½
¾
zf ′′ (z) zf ′ (z)
1+ ′
.
−
f (z)
f (z)
(7)
If f ∈ A satisfies
Ψ(α, β, λ; z) ≺ αq(z) + βzq ′ (z),
then
µ
and q is the best dominant.
zf ′ (z)
f (z)
¶λ
≺ q(z) ,
(8)
Differential Sandwich Theorems
43
Proof. Define the function p(z) by
p(z) :=
µ
zf ′ (z)
f (z)
¶λ
.
Then by means of simple computation we can show that
Ψ(α, β, λ; z) = αp(z) + βzp′ (z).
Now (8) becomes
αp(z) + βzp′ (z) ≺ αq(z) + βzq ′ (z),
and Theorem 3.1 follows by an application of Lemma 2.1.
1 + Az
By taking q(z) =
(−1 ≤ B < A ≤ 1) we have the following Example.
1 + Bz
1 + Az
Example 3.1 : Let q(z) =
(−1 ≤ B < A ≤ 1) in Theorem 3.1. Further
1 + Bz
assuming that (6) holds. If f ∈ A, then
¶
µ
(A − B)z
1 + Az
+β
Ψ(α, β, λ; z) ≺ α
,
1 + Bz
(1 + Bz)2
⇒
and
zf ′ (z)
f (z)
¶λ
≺
1 + Az
,
1 + Bz
1 + Az
is the best dominant.
1 + Bz
1+z
, then for f ∈ A we have
Also if q(z) =
1−z
µ
¶
2βz
1+z
Ψ(α, β, λ; z) ≺ α
+
,
1−z
(1 − z)2
⇒
and
µ
1+z
is the best dominant.
1−z
µ
zf ′ (z)
f (z)
¶λ
≺
1+z
,
1−z
4. SUPERORDINATION RESULTS FOR ANALYTIC FUNCTIONS
Theorem 4.1 :
Let q be convex univalent in ∆ with q(0) = 1. Let f ∈ A,
µ ′ ¶λ
zf (z)
∈ H [1, 1] ∩ Q, with
f (z)
¸
·
α ′
q (z) > 0.
(9)
ℜ
β
44
T.N. Shanmugam, M.P. Jeyaraman and A. Singaravelu
If Ψ(α, β, λ; z) as defined by (7) is univalent in ∆, with
αq(z) + βzq ′ (z) ≺ Ψ(α, β, λ; z) ,
then
q(z) ≺
µ
zf ′ (z)
f (z)
¶λ
,
and q is the best subordinant.
Proof. Theorem 4.1 follows by an application of Lemma 2.2.
1 + Az
(−1 ≤ B < A ≤ 1) in Theorem 4.1, we have the
By taking q(z) =
1 + Bz
following Example.
Example 4.1 : Let q be convex univalent in ∆.
µ ′ ¶λ
zf (z)
Also let f ∈ A,
∈ H [1, 1] ∩ Q. Further assuming that (9) holds. If
f (z)
Ψ(α, β, λ; z) as defined by (7) is univalent in ∆, and
¶
µ
β(A − B)z
1 + Az
α
+
≺ Ψ(α, β, λ; z),
1 + Bz
(1 + Bz)2
then
1 + Az
≺
1 + Bz
and
µ
zf ′ (z)
f (z)
¶λ
,
1 + Az
is the best subordinant.
1 + Bz
Inparticular, we have
µ
¶
1+z
2βz
α
+
≺ Ψ(α, β, λ; z),
1−z
(1 − z)2
implies
1+z
≺
1−z
and
µ
zf ′ (z)
f (z)
¶λ
,
1+z
is the best subordinant.
1−z
5. SANDWICH THEOREMS
By combining the results of subordination and superordination, we get the
following “Sandwich theorems”.
Differential Sandwich Theorems
45
Theorem 5.1 : Let q1 and q2 be convex univalent in ∆ and satisfying (9) and
(6) respectively.
µ ′ ¶λ
zf (z)
∈ H [1, 1] ∩ Q and Ψ(α, β, λ; z) as defined by (7) is univalent
Let f ∈ A,
f (z)
in ∆. Further if
αq1 (z) + βzq1′ (z) ≺ Ψ(α, β, λ; z) ≺ αq2 (z) + βq2′ (z),
then
q1 (z) ≺
µ
zf ′ (z)
f (z)
¶λ
≺ q2 (z),
and q1 and q2 are respectively the best subordinant and best dominant.
1 + A2 z
1 + A1 z
, q2 (z) =
(−1 ≤ B2 ≤ B1 < A1 ≤ A2 ≤ 1), we
For q1 (z) =
1 + B1 z
1 + B2 z
have the following Example.
Example 5.1 :
If f ∈ A,
µ
zf ′ (z)
f (z)
by (7) is univalent in ∆ , and
¶λ
∈ H[1, 1] ∩ Q and Ψ(α, β, λ; z) as defined
Ψ1 (A1 , B1 , α, β, λ; z) ≺ Ψ(α, β, λ; z) ≺ Ψ2 (A2 , B2 , α, β, λ; z) ,
then
1 + A1 z
≺
1 + B1 z
where
µ
zf ′ (z)
f (z)
¶λ
≺
1 + A2 z
,
1 + B2 z
¶
β(A1 − B1 )z
1 + A1 z
,
+
1 + B1 z
(1 + B1 z)2
¶
µ
β(A2 − B2 )z
1 + A2 z
+
.
Ψ2 (A2 , B2 , α, β, λ; z) := α
1 + B2 z
(1 + B2 z)2
Ψ1 (A1 , B1 , α, β, λ; z) := α
The functions
dominant.
µ
1 + A2 z
1 + A1 z
and
are respectively the best subordinant and best
1 + B1 z
1 + B2 z
6. APPLICATION TO DZIOK-SRIVASTAVA OPERATOR
Theorem 6.1 : Let q be univalent in ∆ with q(0) = 1. Let
³ l
´λ
Hm [α1 +1]f (z)
η(α, β, λ, l, m; z) :=
×
l [α ]f (z)
Hm
1
"
(
)#
¡ l
¢
µ l
¶
(α1 + 1) Hm
[α1 + 2]f (z)
Hm [α1 + 1]f (z)
(α + βλ)
−1
.
− α1
l [α + 1]f (z)
l [α ]f (z)
Hm
Hm
1
1
(10)
46
T.N. Shanmugam, M.P. Jeyaraman and A. Singaravelu
If f ∈ A satisfies
η(α, β, λ, l, m; z) ≺ αp(z) + βzq ′ (z) ,
then
µ
l
[α1 + 1]f (z)
Hm
l [α ]f (z)
Hm
1
¶λ
≺ q(z) ,
and q is the best dominant.
Proof. Define the function p(z) by
p(z) :=
µ
l
Hm
[α1 + 1]f (z)
l [α ]f (z)
Hm
1
¶λ
.
By taking logarithmic derivative of (11) we get
" ¡
¢′
¡ l
¢′ #
l
z Hm
[α1 + 1]f (z)
z Hm
[α1 ]f (z)
zp′ (z)
=λ
−
.
l [α + 1]f (z)
l [α ]f (z)
p(z)
Hm
Hm
1
1
(11)
(12)
By using identity
¡ l
¢′
l
l
z Hm
[α1 ]f (z) = α1 Hm
[α1 + 1]f (z) − (α1 − 1)Hm
[α1 ]f (z),
and (11) in (12) we get
³ l
´λ
Hm [α1 +1]f (z)
αp(z) + βzp′ (z) =
×
l [α ]f (z)
Hm
1
µ l
¶
¾¸
·
½
l
(α1 + 1)Hm
[α1 + 2]f (z)
Hm [α1 + 1]f (z)
− α1
−1 .
(α + βλ)
l [α + 1]f (z)
l [α ]f (z)
Hm
Hm
1
1
Now Theorem 6.1 follows as an application of Lemma 2.1.
By taking l = 2, m = 1 and α2 = 1 in Theorem 6.1 we have the following
corollary.
Corollary 6.1 : Let q be univalent in ∆ with q(0) = 1 . Let
³
´λ
(z)
φ(a, c, α, β, λ : z) := L(a+1,c)f
×
L(a,c)f (z)
·
α + βλ
½
¾¸
(a + 1)L(a + 2, c)f (z) aL(a + 1, c)f (z)
−
−1 .
L(a + 1, c)f (z)
L(a, c)f (z)
If f ∈ A satisfies
φ(a, c, α, β, λ : z) ≺ αq(z) + βzq ′ (z) ,
then
µ
and q is the best dominant.
L(a + 1, c)f (z)
L(a, c)f (z)
¶λ
≺ q(z) ,
47
Differential Sandwich Theorems
Taking a = 1 and c = 1 in corollary 6.1 we get the following corollary.
Corollary 6.2 : Let q be univalent in ∆ with q(0) = 1. If f ∈ A and
µ
Dn+1 f (z)
Dn f (z)
¶λ ·
½
¾¸
(a + 1)Dn+2 f (z) aDn+1 f (z)
α + βλ
−
−
1
≺ αq(z)+βzq ′ (z),
Dn+1 f (z)
Dn f (z)
then
µ
Dn+1 f (z)
Dn f (z)
¶λ
≺ q(z) ,
and q is the best dominant.
Since the superordination results are a dual of the subordination here we
state only the results pertaining to the superordination.
Theorem 6.2 : Let q be convex univalent in ∆ with q(0) = 1. Let f ∈ A,
µ l
¶λ
i
h
Hm [α1 + 1]f (z)
α ′
q
(z)
> 0. Further if η(α, β, λ, l, m; z)
∈
H[1,
1]∩Q,
with
ℜ
β
l [α ]f (z)
Hm
1
as defined by (10) is univalent in ∆, with
αq(z) + βzq ′ (z) ≺ η(α, β, λ, l, m; z) ,
then
q(z) ≺
µ
l
Hm
[α1 + 1]f (z)
l [α ]f (z)
Hm
1
¶λ
,
and q is the best subordinant.
Theorem 6.3 : Let q be convex univalent in ∆.
¶λ
µ
L(a + 1, c)f (z)
∈ H[1, 1] ∩ Q and φ(a, c, α, β, λ : z) as defined by
Let f ∈ A,
L(a, c)f (z)
(13) is univalent in ∆. If
αq(z) + βzq ′ (z) ≺ φ(a, c, α, β, λ : z) ,
then
q(z) ≺
µ
L(a + 1, c)f (z)
L(a, c)f (z)
¶λ
,
and q is the best subordinant.
1 + Az 1 + z
Taking q(z) =
,
in Theorem 6.1 we can get more results and
1 + Bz 1 − z
we omit the details involved.
Combining the results of subordination and superordination, we state the
following Sandwich Theorems.
48
T.N. Shanmugam, M.P. Jeyaraman and A. Singaravelu
Theorem 6.4 : Let q1 and q2 be convex univalent in ∆ satisfying (9) and (6)
¶λ
µ l
Hm [α1 + 1]f (z)
∈ H[1, 1] ∩ Q and η(α, β, λ, l, m; z) as
respectively. If f ∈ A,
l [α ]f (z)
Hm
1
defined by (10) is univalent in ∆, and
αq1 (z) + βzq1′ (z) ≺ η(α, β, λ, l, m; z) ≺ αq2 (z) + βzq2′ (z) ,
then
q1 (z) ≺
µ
l
Hm
[α1 + 1]f (z)
l [α ]f (z)
Hm
1
¶λ
≺ q2 (z) ,
and q1 (z) and q2 (z) are respectively the best subordinant and best dominant.
1 + A2 z
1 + A1 z
, q2 (z) =
(−1 ≤ B2 < B1 < A1 < A2 ≤ 1), we
For q1 (z) =
1 + B1 z
1 + B2 z
have the following corollary.
l
Hm
[α1 + 1]f (z)
Corollary 6.3 : If f ∈ A,
l [α ]f (z)
Hm
1
as defined by (10) is univalent in ∆, and
µ
¶λ
∈ H[1, 1] ∩ Q and η(α, β, λ, l, m; z)
Φ1 (A1 , B1 , α, β; z) ≺ η(α, β, λ, l, m; z) ≺ Φ2 (A2 , B2 , α, β; z),
where
Φ1 (A1 , B1 , α, β; z) := α
µ
1 + A1 z
1 + B1 z
¶
+
β(A1 − B1 )z
,
(1 + B1 z)2
Φ2 (A2 , B2 , α, β; z) := α
µ
1 + A1 z
1 + B1 z
¶
+
β(A2 − B2 )z
,
(1 + B2 z)2
¶λ
≺
⇒
The functions
dominant.
1 + A1 z
≺
1 + B1 z
µ
l
[α1 + 1]f (z)
Hm
l [α ]f (z)
Hm
1
1 + A2 z
.
1 + B2 z
1 + A1 z
1 + A2 z
and
are respectively the best subordinant and best
1 + B1 z
1 + B2 z
Theorem 6.5 : Let q1 and q2 be convex univalent in ∆ and satisfing (9) and (6)
µ
¶λ
L(a + 1, c)f (z)
respectively. If f ∈ A,
∈ H[1, 1] ∩ Q and φ(a, c, α, β, λ : z) as
L(a, c)f (z)
defined by (13) is univalent in ∆, and
αq1 (z) + βzq1′ (z) ≺ φ(a, c, α, β, λ : z) ≺ αq2 (z) + βzq2′ (z) ,
then
q1 (z) ≺
µ
L(a + 1, c)f (z)
L(a, c)f (z)
¶λ
≺ q2 (z) ,
and q1 (z) and q2 (z) are respectively the best subordinant and best dominant.
Differential Sandwich Theorems
49
Let q1 (z) and q2 (z) be convex univalent in ∆ and satisfing (9)
¶λ
µ n+1
D
f (z)
∈ H[1, 1] ∩ Q,
and (6) respectively. Let f ∈ A,
Dn f (z)
Theorem 6.6 :
µ
Dn+1 f (z)
Dn f (z)
¾¸
¶λ ·
½
(a + 1)Dn+2 f (z) aDn+1 f (z)
−
−
1
,
α + βλ
Dn+1 f (z)
Dn f (z)
is univalent in ∆. Further if
´λ
³ n+1
(z)
×
αq1 (z) + βzq1′ (z) ≺ DDn f f(z)
·
α + βλ
½
¾¸
(a + 1)Dn+2 f (z) aDn+1 f (z)
−
−1
≺ αq2 (z) + βzq2′ (z) ,
Dn+1 f (z)
Dn f (z)
then
q1 (z) ≺
µ
Dn+1 f (z)
Dn f (z)
¶λ
≺ q2 (z),
and q1 and q2 are respectively the best subordinant and best dominant.
REFERENCES
1. S.D. Bernardi, “Convex and starlike univalent functions”, Trans. Amer. Math. Soc.
135 (1969), 429–446.
˘ , “A class of superordination-preserving integral operators”, Indag. Math.
2. T. Bulboaca
(N.S.) 13(3) (2002), 301–311.
˘ , “Classes of first-order differential superordinations”, Demonstratio Math.
3. T. Bulboaca
35(2) (2002), 287–292.
4. B.C. Carlson and D.B. Shaffer, “Starlike and prestarlike hypergeometric functions”, SIAM J. Math. Anal. 15(4) (1984), 737–745.
5. J. Dziok and H.M. Srivastava, “Certain subclasses of analytic functions associated
with the generalized hypergeometric function”, Integral Transforms Spec. Funct. 14(1)
(2003), 7–18.
6. Ju. E. Hohlov, “Operators and operations on the class of univalent functions”, Izv.
Vyssh. Uchebn. Zaved. Mat. 10(197) (1978), 83–89.
7. R.J. Libera, “Some classes of regular univalent functions”, Proc. Amer. Math. Soc.
16 (1965), 755–758.
8. A.E. Livingston, “On the radius of univalence of certain analytic functions”, Proc.
Amer. Math. Soc. 17 (1966), 352–357.
9. S.S. Miller and P.T. Mocanu, Differential Subordinations: Theory and Applications. Monographs and Textbooks in Pure and Applied Mathematics 225, Marcel
Dekker, New York, 2000.
10. S.S. Miller and P.T. Mocanu, “Subordinants of differential superordinations”,
Complex Var. Theory Appl. 48(10) (2003), 815–826.
50
T.N. Shanmugam, M.P. Jeyaraman and A. Singaravelu
11. St. Ruscheweyh, “New criteria for univalent functions”, Proc. Amer. Math. Soc.
49 (1975), 109–115.
12. T.N. Shanmugam, V. Ravichandran and S. Sivasubramanian, “Differential sandwich theorems for some subclasses of analytic functions”, Austra. J. Math. Anal. Appl.
3(1), (2006), 1-11.
13. H.M. Srivastava, “Some families of fractional derivative and other linear operators
associated with analytic, univalent, and multivalent functions”, in Analysis and its
applications (Chennai, 2000), 209–243, Allied Publ., New Delhi.
T.N. Shanmuguam: Department of Mathematics, Anna University, Chennai - 25, Tamilnadu, India
M.P. Jeyaraman: Department of Mathematics, Eswari Engineering College, Chennai 89, Tamilnadu, India
E-mail : jeyaraman-mp@yahoo.co.in
A. Singaravelu: Department of Mathematics, Valliammai Engineering College, Kattankolathur, Kancheepuram District, Tamilnadu, India.
E-mail : asing-59@yahoo.com
Vol. 13, No. 1 (2007), pp. 39–50.
DIFFERENTIAL SANDWICH THEOREMS
FOR SOME SUBCLASS OF
ANALYTIC FUNCTIONS
ASSOCIATED WITH LINEAR OPERATORS
T.N. Shanmugam, M.P. Jeyaraman and A. Singaravelu
Abstract. Let q1 and q2 be univalent in ∆ := {z : |z| < 1} with q1 (0) = q2 (0) = 1.
We give some applications of first order differential subordination and superordination
to obtain sufficient conditions for a normalized analytic functions f with f (0) = 0 ,
f ′ (0) = 1 to satisfy
zf ′ (z) λ
q1 (z) ≺
≺ q2 (z).
f (z)
1. INTRODUCTION
Let H be the class of functions analytic in ∆ := {z : |z| < 1} and 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 A be the subclass of H consisting of functions of the form
f (z) = z + a2 z 2 + · · · . Let p, h ∈ H and let φ(r, s, t; z) : C3 × ∆ → C. If p and
φ(p(z), zp′ (z), z 2 p′′ (z); z) are univalent and if p satisfies the second order superordination
h(z) ≺ φ(p(z), zp′ (z), z 2 p′′ (z); z),
(1)
then p is a solution of the differential superordination (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). An univalent subordinant q¯ that satisfies q ≺ q¯ for
all subordinants q of (1) is said to be the best subordinant. Recently Miller and
Received 6 February 2006, revised 25 October 2006, accepted 15 November 2006.
2000 Mathematics Subject Classification: Primary 30C45, Secondary 30C80.
Key words and Phrases: Differential Subordination, Differential Superordination, Subordinant.
39
40
T.N. Shanmugam, M.P. Jeyaraman and A. Singaravelu
Mocanu [10] obtained conditions on h, q and φ for which the following implication
holds:
h(z) ≺ φ(p(z), zp′ (z), z 2 p′′ (z); z) ⇒ q(z) ≺ p(z).
Using the results of Miller and Mocanu [10], Bulboaca [3] considered certain classes
of first order differential superordinations as well as superordination-preserving operators [2]. Using the results of [3], Shanmugam et al. [12] obtained sufficient
conditions for a normalized analytic function f (z) to satisfy
q1 (z) ≺
and
q1 (z) ≺
f (z)
≺ q2 (z),
zf ′ (z)
z 2 f ′ (z)
2
{f (z)}
≺ q2 (z) ,
respectively where q1 and q2 are given univalent functions in ∆.
For αj ∈ C (j = 1, 2, . . . , l) and βj ∈ C \ Z−
0 := {0, −1, −2, . . .}, j = 1, 2, . . . m),
the generalized hypergeometric function l Fm (α1 , . . . , αl ; β1 , . . . , βm ; z) is defined by
the infinite series
l Fm (α1 , . . . , αl ; β1 , . . . , βm ; z) :=
∞
X
(α1 )n . . . (αl )n z n
.
(β1 )n . . . (βm )n n!
n=0
(l ≤ m + 1; l, m ∈ N0 := N ∪ {0}),
where (a)n is the Pochhammer symbol defined by
½
Γ(a + n)
1,
(n = 0);
(a)n :=
=
a(a
+
1)(a
+
2)
.
.
.
(a
+
n
−
1),
(n ∈ N).
Γ(a)
Corresponding to the function
h(α1 , . . . , αl ; β1 , . . . , βm ; z) := z l Fm (α1 , . . . , αl ; β1 , . . . , βm ; z),
l
the Dziok-Srivastava operator [5] (see also [13]) Hm
(α1 , . . . , αl ; β1 , . . . , βm ; z) is
defined by the Hadamard product
l
Hm
(α1 , . . . , αl ; β1 , . . . , βm ; z)f (z)
:= h(α1 , . . . , αl ; β1 , . . . , βm ; z) ∗ f (z)
∞
X
(α1 )n−1 . . . (αl )n−1 an z n
.(2)
= z+
(β1 )n−1 . . . (βm )n−1 (n − 1)!
n=2
It is well known [5] that
l
α1 Hm
(α1 + 1, . . . , αl ; β1 , . . . , βm ; z)f (z)
l
= z[Hm
(α1 , . . . , αl ; β1 , . . . , βm ; z)f (z)]′
l
+(α1 − 1)Hm
(α1 , . . . , αl ; β1 , . . . , βm ; z)f (z).
(3)
Differential Sandwich Theorems
41
To make the notation simple, we write
l
l
Hm
[α1 ]f (z) := Hm
(α1 , . . . , αl ; β1 , . . . , βm ; z)f (z).
Special cases of the Dziok-Srivastava linear operator includes the Hohlov linear operator [6] , the Carlson-Shaffer linear operator[4], the Ruscheweyh derivative
operator [11], the generalized Bernardi-Libera-Livingston linear integral operator
(cf. [1], [7], [8]).
2. PRELIMINARIES
In our present investigation, we shall need the following definition and results.
In this paper unless otherwise mentioned α and β are complex numbers.
Definition 2.1: [10, Definition 2, p. 817] Let Q be the set of all functions f that
¯ − E(f ), where
are analytic and injective on ∆
½
¾
E(f ) = ζ ∈ ∂∆ : lim f (z) = ∞ ,
z→ζ
and are such that f ′ (ζ) 6= 0 for ζ ∈ ∂∆ − E(f ).
Theorem 2.1 : [9, Theorem 3.4h , p. 132] Let q be univalent in the 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 ′ (z)φ(q(z)), h(z) = θ(q(z)) + ξ(z). Suppose that,
1. ξ(z) is starlike univalent in ∆ and
′
(z)
> 0 for z ∈ ∆.
2. ℜ zhξ(z)
If p is analytic in ∆ with p(∆) ⊆ D, and
θ(p(z)) + zp′ (z)φ(p(z)) ≺ θ(q(z)) + zq ′ (z)φ(q(z)),
(4)
then p ≺ q and q is the best dominant.
Lemma 2.1 : [12] Let q be univalent in ∆ with q(0) = 1. Further assuming that
·
¸
α
zq ′′ (z)
ℜ
+1+ ′
> 0.
β
q (z)
If p is analytic in ∆, with p(∆) ⊆ D and
αp(z) + βzp′ (z) ≺ αq(z) + βzq ′ (z),
42
T.N. Shanmugam, M.P. Jeyaraman and A. Singaravelu
then p ≺ q and q is the best dominant.
Theorem 2.2 : [3] Let q be univalent in the unit disk ∆ and ϑ and ϕ be analytic
in a domain D containing q(∆). Suppose that
i
h ′
(q(z))
> 0 for z ∈ ∆, and
1. ℜ ϑϕ(q(z))
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)) + zp′ (z)ϕ(p(z)) is univalent in
∆, and
ϑ(q(z)) + zq ′ (z)ϕ(q(z)) ≺ ϑ(p(z)) + zp′ (z)ϕ(p(z)),
(5)
then q ≺ p and q is the best subordinant.
Lemma
i2.2 : [12] Let q be univalent in ∆ , q(0) = 1. Further assuming that
h
α ′
ℜ β q (z) > 0.
If p ∈ H [q(0), 1] ∩ Q, and αp + βzp′ is univalent in ∆, and
αq(z) + βzq ′ (z) ≺ αp(z) + βzp′ (z) ,
then q ≺ p and q is the best subordinant.
3. SUBORDINATION RESULTS FOR ANALYTIC FUNCTIONS
By making use of Lemma 2.3, we prove the following results.
Theorem 3.1 : Let q be univalent in ∆ with q(0) = 1 and satisfying
·
¸
α
zq ′′ (z)
ℜ
> 0.
+1+ ′
β
q (z)
(6)
Let
Ψ(α, β, λ; z) := α
µ
zf ′ (z)
f (z)
¶λ
+ βλ
µ
zf ′ (z)
f (z)
¶λ ½
¾
zf ′′ (z) zf ′ (z)
1+ ′
.
−
f (z)
f (z)
(7)
If f ∈ A satisfies
Ψ(α, β, λ; z) ≺ αq(z) + βzq ′ (z),
then
µ
and q is the best dominant.
zf ′ (z)
f (z)
¶λ
≺ q(z) ,
(8)
Differential Sandwich Theorems
43
Proof. Define the function p(z) by
p(z) :=
µ
zf ′ (z)
f (z)
¶λ
.
Then by means of simple computation we can show that
Ψ(α, β, λ; z) = αp(z) + βzp′ (z).
Now (8) becomes
αp(z) + βzp′ (z) ≺ αq(z) + βzq ′ (z),
and Theorem 3.1 follows by an application of Lemma 2.1.
1 + Az
By taking q(z) =
(−1 ≤ B < A ≤ 1) we have the following Example.
1 + Bz
1 + Az
Example 3.1 : Let q(z) =
(−1 ≤ B < A ≤ 1) in Theorem 3.1. Further
1 + Bz
assuming that (6) holds. If f ∈ A, then
¶
µ
(A − B)z
1 + Az
+β
Ψ(α, β, λ; z) ≺ α
,
1 + Bz
(1 + Bz)2
⇒
and
zf ′ (z)
f (z)
¶λ
≺
1 + Az
,
1 + Bz
1 + Az
is the best dominant.
1 + Bz
1+z
, then for f ∈ A we have
Also if q(z) =
1−z
µ
¶
2βz
1+z
Ψ(α, β, λ; z) ≺ α
+
,
1−z
(1 − z)2
⇒
and
µ
1+z
is the best dominant.
1−z
µ
zf ′ (z)
f (z)
¶λ
≺
1+z
,
1−z
4. SUPERORDINATION RESULTS FOR ANALYTIC FUNCTIONS
Theorem 4.1 :
Let q be convex univalent in ∆ with q(0) = 1. Let f ∈ A,
µ ′ ¶λ
zf (z)
∈ H [1, 1] ∩ Q, with
f (z)
¸
·
α ′
q (z) > 0.
(9)
ℜ
β
44
T.N. Shanmugam, M.P. Jeyaraman and A. Singaravelu
If Ψ(α, β, λ; z) as defined by (7) is univalent in ∆, with
αq(z) + βzq ′ (z) ≺ Ψ(α, β, λ; z) ,
then
q(z) ≺
µ
zf ′ (z)
f (z)
¶λ
,
and q is the best subordinant.
Proof. Theorem 4.1 follows by an application of Lemma 2.2.
1 + Az
(−1 ≤ B < A ≤ 1) in Theorem 4.1, we have the
By taking q(z) =
1 + Bz
following Example.
Example 4.1 : Let q be convex univalent in ∆.
µ ′ ¶λ
zf (z)
Also let f ∈ A,
∈ H [1, 1] ∩ Q. Further assuming that (9) holds. If
f (z)
Ψ(α, β, λ; z) as defined by (7) is univalent in ∆, and
¶
µ
β(A − B)z
1 + Az
α
+
≺ Ψ(α, β, λ; z),
1 + Bz
(1 + Bz)2
then
1 + Az
≺
1 + Bz
and
µ
zf ′ (z)
f (z)
¶λ
,
1 + Az
is the best subordinant.
1 + Bz
Inparticular, we have
µ
¶
1+z
2βz
α
+
≺ Ψ(α, β, λ; z),
1−z
(1 − z)2
implies
1+z
≺
1−z
and
µ
zf ′ (z)
f (z)
¶λ
,
1+z
is the best subordinant.
1−z
5. SANDWICH THEOREMS
By combining the results of subordination and superordination, we get the
following “Sandwich theorems”.
Differential Sandwich Theorems
45
Theorem 5.1 : Let q1 and q2 be convex univalent in ∆ and satisfying (9) and
(6) respectively.
µ ′ ¶λ
zf (z)
∈ H [1, 1] ∩ Q and Ψ(α, β, λ; z) as defined by (7) is univalent
Let f ∈ A,
f (z)
in ∆. Further if
αq1 (z) + βzq1′ (z) ≺ Ψ(α, β, λ; z) ≺ αq2 (z) + βq2′ (z),
then
q1 (z) ≺
µ
zf ′ (z)
f (z)
¶λ
≺ q2 (z),
and q1 and q2 are respectively the best subordinant and best dominant.
1 + A2 z
1 + A1 z
, q2 (z) =
(−1 ≤ B2 ≤ B1 < A1 ≤ A2 ≤ 1), we
For q1 (z) =
1 + B1 z
1 + B2 z
have the following Example.
Example 5.1 :
If f ∈ A,
µ
zf ′ (z)
f (z)
by (7) is univalent in ∆ , and
¶λ
∈ H[1, 1] ∩ Q and Ψ(α, β, λ; z) as defined
Ψ1 (A1 , B1 , α, β, λ; z) ≺ Ψ(α, β, λ; z) ≺ Ψ2 (A2 , B2 , α, β, λ; z) ,
then
1 + A1 z
≺
1 + B1 z
where
µ
zf ′ (z)
f (z)
¶λ
≺
1 + A2 z
,
1 + B2 z
¶
β(A1 − B1 )z
1 + A1 z
,
+
1 + B1 z
(1 + B1 z)2
¶
µ
β(A2 − B2 )z
1 + A2 z
+
.
Ψ2 (A2 , B2 , α, β, λ; z) := α
1 + B2 z
(1 + B2 z)2
Ψ1 (A1 , B1 , α, β, λ; z) := α
The functions
dominant.
µ
1 + A2 z
1 + A1 z
and
are respectively the best subordinant and best
1 + B1 z
1 + B2 z
6. APPLICATION TO DZIOK-SRIVASTAVA OPERATOR
Theorem 6.1 : Let q be univalent in ∆ with q(0) = 1. Let
³ l
´λ
Hm [α1 +1]f (z)
η(α, β, λ, l, m; z) :=
×
l [α ]f (z)
Hm
1
"
(
)#
¡ l
¢
µ l
¶
(α1 + 1) Hm
[α1 + 2]f (z)
Hm [α1 + 1]f (z)
(α + βλ)
−1
.
− α1
l [α + 1]f (z)
l [α ]f (z)
Hm
Hm
1
1
(10)
46
T.N. Shanmugam, M.P. Jeyaraman and A. Singaravelu
If f ∈ A satisfies
η(α, β, λ, l, m; z) ≺ αp(z) + βzq ′ (z) ,
then
µ
l
[α1 + 1]f (z)
Hm
l [α ]f (z)
Hm
1
¶λ
≺ q(z) ,
and q is the best dominant.
Proof. Define the function p(z) by
p(z) :=
µ
l
Hm
[α1 + 1]f (z)
l [α ]f (z)
Hm
1
¶λ
.
By taking logarithmic derivative of (11) we get
" ¡
¢′
¡ l
¢′ #
l
z Hm
[α1 + 1]f (z)
z Hm
[α1 ]f (z)
zp′ (z)
=λ
−
.
l [α + 1]f (z)
l [α ]f (z)
p(z)
Hm
Hm
1
1
(11)
(12)
By using identity
¡ l
¢′
l
l
z Hm
[α1 ]f (z) = α1 Hm
[α1 + 1]f (z) − (α1 − 1)Hm
[α1 ]f (z),
and (11) in (12) we get
³ l
´λ
Hm [α1 +1]f (z)
αp(z) + βzp′ (z) =
×
l [α ]f (z)
Hm
1
µ l
¶
¾¸
·
½
l
(α1 + 1)Hm
[α1 + 2]f (z)
Hm [α1 + 1]f (z)
− α1
−1 .
(α + βλ)
l [α + 1]f (z)
l [α ]f (z)
Hm
Hm
1
1
Now Theorem 6.1 follows as an application of Lemma 2.1.
By taking l = 2, m = 1 and α2 = 1 in Theorem 6.1 we have the following
corollary.
Corollary 6.1 : Let q be univalent in ∆ with q(0) = 1 . Let
³
´λ
(z)
φ(a, c, α, β, λ : z) := L(a+1,c)f
×
L(a,c)f (z)
·
α + βλ
½
¾¸
(a + 1)L(a + 2, c)f (z) aL(a + 1, c)f (z)
−
−1 .
L(a + 1, c)f (z)
L(a, c)f (z)
If f ∈ A satisfies
φ(a, c, α, β, λ : z) ≺ αq(z) + βzq ′ (z) ,
then
µ
and q is the best dominant.
L(a + 1, c)f (z)
L(a, c)f (z)
¶λ
≺ q(z) ,
47
Differential Sandwich Theorems
Taking a = 1 and c = 1 in corollary 6.1 we get the following corollary.
Corollary 6.2 : Let q be univalent in ∆ with q(0) = 1. If f ∈ A and
µ
Dn+1 f (z)
Dn f (z)
¶λ ·
½
¾¸
(a + 1)Dn+2 f (z) aDn+1 f (z)
α + βλ
−
−
1
≺ αq(z)+βzq ′ (z),
Dn+1 f (z)
Dn f (z)
then
µ
Dn+1 f (z)
Dn f (z)
¶λ
≺ q(z) ,
and q is the best dominant.
Since the superordination results are a dual of the subordination here we
state only the results pertaining to the superordination.
Theorem 6.2 : Let q be convex univalent in ∆ with q(0) = 1. Let f ∈ A,
µ l
¶λ
i
h
Hm [α1 + 1]f (z)
α ′
q
(z)
> 0. Further if η(α, β, λ, l, m; z)
∈
H[1,
1]∩Q,
with
ℜ
β
l [α ]f (z)
Hm
1
as defined by (10) is univalent in ∆, with
αq(z) + βzq ′ (z) ≺ η(α, β, λ, l, m; z) ,
then
q(z) ≺
µ
l
Hm
[α1 + 1]f (z)
l [α ]f (z)
Hm
1
¶λ
,
and q is the best subordinant.
Theorem 6.3 : Let q be convex univalent in ∆.
¶λ
µ
L(a + 1, c)f (z)
∈ H[1, 1] ∩ Q and φ(a, c, α, β, λ : z) as defined by
Let f ∈ A,
L(a, c)f (z)
(13) is univalent in ∆. If
αq(z) + βzq ′ (z) ≺ φ(a, c, α, β, λ : z) ,
then
q(z) ≺
µ
L(a + 1, c)f (z)
L(a, c)f (z)
¶λ
,
and q is the best subordinant.
1 + Az 1 + z
Taking q(z) =
,
in Theorem 6.1 we can get more results and
1 + Bz 1 − z
we omit the details involved.
Combining the results of subordination and superordination, we state the
following Sandwich Theorems.
48
T.N. Shanmugam, M.P. Jeyaraman and A. Singaravelu
Theorem 6.4 : Let q1 and q2 be convex univalent in ∆ satisfying (9) and (6)
¶λ
µ l
Hm [α1 + 1]f (z)
∈ H[1, 1] ∩ Q and η(α, β, λ, l, m; z) as
respectively. If f ∈ A,
l [α ]f (z)
Hm
1
defined by (10) is univalent in ∆, and
αq1 (z) + βzq1′ (z) ≺ η(α, β, λ, l, m; z) ≺ αq2 (z) + βzq2′ (z) ,
then
q1 (z) ≺
µ
l
Hm
[α1 + 1]f (z)
l [α ]f (z)
Hm
1
¶λ
≺ q2 (z) ,
and q1 (z) and q2 (z) are respectively the best subordinant and best dominant.
1 + A2 z
1 + A1 z
, q2 (z) =
(−1 ≤ B2 < B1 < A1 < A2 ≤ 1), we
For q1 (z) =
1 + B1 z
1 + B2 z
have the following corollary.
l
Hm
[α1 + 1]f (z)
Corollary 6.3 : If f ∈ A,
l [α ]f (z)
Hm
1
as defined by (10) is univalent in ∆, and
µ
¶λ
∈ H[1, 1] ∩ Q and η(α, β, λ, l, m; z)
Φ1 (A1 , B1 , α, β; z) ≺ η(α, β, λ, l, m; z) ≺ Φ2 (A2 , B2 , α, β; z),
where
Φ1 (A1 , B1 , α, β; z) := α
µ
1 + A1 z
1 + B1 z
¶
+
β(A1 − B1 )z
,
(1 + B1 z)2
Φ2 (A2 , B2 , α, β; z) := α
µ
1 + A1 z
1 + B1 z
¶
+
β(A2 − B2 )z
,
(1 + B2 z)2
¶λ
≺
⇒
The functions
dominant.
1 + A1 z
≺
1 + B1 z
µ
l
[α1 + 1]f (z)
Hm
l [α ]f (z)
Hm
1
1 + A2 z
.
1 + B2 z
1 + A1 z
1 + A2 z
and
are respectively the best subordinant and best
1 + B1 z
1 + B2 z
Theorem 6.5 : Let q1 and q2 be convex univalent in ∆ and satisfing (9) and (6)
µ
¶λ
L(a + 1, c)f (z)
respectively. If f ∈ A,
∈ H[1, 1] ∩ Q and φ(a, c, α, β, λ : z) as
L(a, c)f (z)
defined by (13) is univalent in ∆, and
αq1 (z) + βzq1′ (z) ≺ φ(a, c, α, β, λ : z) ≺ αq2 (z) + βzq2′ (z) ,
then
q1 (z) ≺
µ
L(a + 1, c)f (z)
L(a, c)f (z)
¶λ
≺ q2 (z) ,
and q1 (z) and q2 (z) are respectively the best subordinant and best dominant.
Differential Sandwich Theorems
49
Let q1 (z) and q2 (z) be convex univalent in ∆ and satisfing (9)
¶λ
µ n+1
D
f (z)
∈ H[1, 1] ∩ Q,
and (6) respectively. Let f ∈ A,
Dn f (z)
Theorem 6.6 :
µ
Dn+1 f (z)
Dn f (z)
¾¸
¶λ ·
½
(a + 1)Dn+2 f (z) aDn+1 f (z)
−
−
1
,
α + βλ
Dn+1 f (z)
Dn f (z)
is univalent in ∆. Further if
´λ
³ n+1
(z)
×
αq1 (z) + βzq1′ (z) ≺ DDn f f(z)
·
α + βλ
½
¾¸
(a + 1)Dn+2 f (z) aDn+1 f (z)
−
−1
≺ αq2 (z) + βzq2′ (z) ,
Dn+1 f (z)
Dn f (z)
then
q1 (z) ≺
µ
Dn+1 f (z)
Dn f (z)
¶λ
≺ q2 (z),
and q1 and q2 are respectively the best subordinant and best dominant.
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T.N. Shanmuguam: Department of Mathematics, Anna University, Chennai - 25, Tamilnadu, India
M.P. Jeyaraman: Department of Mathematics, Eswari Engineering College, Chennai 89, Tamilnadu, India
E-mail : jeyaraman-mp@yahoo.co.in
A. Singaravelu: Department of Mathematics, Valliammai Engineering College, Kattankolathur, Kancheepuram District, Tamilnadu, India.
E-mail : asing-59@yahoo.com