General Mathematics Vol. 17, No. 4 (2009), 83–95
On a Certain Differential Sandwich Theorem
Associated with a New Generalized
Derivative Operator
Adriana Cătaş
Abstract
The purpose of this paper is to derive certain subordinations and
superordinations results involving a new differential operator. By
means of the new introduced operator, I m (λ, β, l)f (z), for certain
normalized analytic functions in the open unit disc, we establish differential sandwich-type theorems. These results extend corresponding previously known results.
2000 Mathematics Subject Classification: Primary 30C80, 30C45.
Key words and phrases: differential subordination, differential
superordination, dominant, differential operator, subordinant.
83
84
1
Adriana Cătaş
Introduction and definitions
Let H(U ) be the class of analytic functions in the open unit disc
U = {z ∈ C : |z| < 1}.
For a ∈ C and n ∈ N let H[a, n] be the subclass of H(U ) consisting of
functions of the form
f (z) = a + an z n + an+1 z n+1 + . . . .
Let
An = {f ∈ H(U ), f (z) = z + an+1 z n+1 + . . . }
with A1 := A.
With a view to recalling the principle of subordination between analytic
functions, let the functions f and g be analytic in U . Then we say that the
function f is subordinate to g, written symbolically as
f ≺g
or f (z) ≺ g(z),
z∈U
if there exists a Schwarz function w analytic in U such that f (z) = g(w(z)),
z ∈ U . In particular, if the function g is univalent in U , the above subordination is equivalent to f (0) = g(0) and f (U ) ⊂ g(U ).
Let p, h ∈ H(U ) and let ψ(r, s, t; z) : C3 × U → C.
If p and ψ(p(z), zp0 (z), z 2 p00 (z); z) are univalent and if p satisfies the
second order differential superordination
(1)
h(z) ≺ ψ(p(z), zp0 (z), z 2 p00 (z); z),
z∈U
On a Certain Differential Sandwich Theorem ...
85
then p is a solution of the differential superordination (1). If f is subordinate
to g, then g is superordinate to f .
An analytic function q is called a subordinant of the differential superordination, or more simply a subordinant if q ≺ p for all p satisfying (1).
A univalent subordinant qe that satisfies q ≺ qe for all subordinants q of (1)
is said to be the best subordinant. The best subordinant is unique up to a
rotation of U . Recently Miller and Mocanu [7] obtained conditions on h, q
and ψ for which the following implication holds:
h(z) ≺ ψ(p(z), zp0 (z), z 2 p00 (z); z) =⇒ q(z) ≺ p(z),
z ∈ U.
In order to prove our subordination and superordination results, we make
use of the following definition and lemmas.
Definition 1 [7] Denote by Q, the set of all functions f that are analytic
and injective on U − E(f ), where
E(f ) = {ζ ∈ ∂U : lim f (z) = ∞}
z→ζ
and are such that f 0 (ζ) 6= 0 for ζ ∈ ∂U − E(f ).
Lemma 1 [8] Let the function q be univalent in the unit disc U and θ and
φ be analytic in a domain D containing q(U ) with φ(w) 6= 0 when w ∈ q(U ).
Set
Q(z) = zq 0 (z)φ(q(z))
and h(z) = θ(q(z)) + Q(z).
Suppose that
(1) Q(z)½is starlike
¾ univalent in U and
zh0 (z)
(2) Re
> 0 for z ∈ U .
Q(z)
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Adriana Cătaş
If p is analytic with p(0) = q(0), p(U ) ⊆ D and
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 2 [4] Let q be convex univalent in the unit disc U and ν and ϕ be
analytic in ½
a domain ¾
D containing q(U ). Suppose that
ν 0 (q(z))
(1) Re
> 0 for z ∈ U and
ϕ(q(z))
(2) ψ(z) = zq 0 (z)ϕ(q(z)) is starlike univalent in U .
If p(z) ∈ H[q(0), 1] ∩ Q with p(U ) ⊆ D and ν(p(z)) + zp0 (z)ϕ(p(z)) is
univalent in U and
ν(q(z)) + zq 0 (z)ϕ(q(z)) ≺ ν(p(z)) + zp0 (z)ϕ(p(z))
then
q(z) ≺ p(z)
and q is the best subordinant.
2
Main results
Definition 2 Let the function f be in the class An . For m, β ∈ N0 =
{0, 1, 2, . . . }, λ ≥ 0, l ≥ 0, we define the following differential operator
¸m
∞ ·
X
1 + λ(k − 1) + l
m
(2)
I (λ, β, l)f (z) := z +
C(β, k)ak z k
1
+
l
k=n+1
On a Certain Differential Sandwich Theorem ...
where
µ
C(β, k) :=
and
87
¶
k+β−1
(β + 1)k−1
=
β
(k − 1)!

 1,
n=0
(a)n :=
 a(a + 1) . . . (a + n − 1), n ∈ N = N − {0}
0
is Pochhamer symbol.
Using simple computation one obtains the next result.
Proposition 1 For m, β ∈ N0 , λ ≥ 0, l ≥ 0
(3) (l+1)I m+1 (λ, β, l)f (z) = (1−λ+l)I m (λ, β, l)f (z)+λz(I m (λ, β, l)f (z))0
and
(4)
z(I m (λ, β, l)f (z))0 = (1 + β)I m (λ, β + 1, l)f (z) − βI m (λ, β, l)f (z).
Remark 1 Special cases of this operator includes the Ruscheweyh derivative operator I 0 (1, β, 0)f (z) ≡ Dβ defined in [9], the Sălăgean derivative
operator I m (1, 0, 0)f (z) ≡ Dm , studied in [10], the generalized Sălăgean
operator I m (λ, 0, 0) ≡ Dλm introduced by Al-Oboudi in [1], the generalized
Ruscheweyh derivative operator I 1 (λ, β, 0)f (z) ≡ Dλ,β introduced in [6], the
m
operator I m (λ, β, 0) ≡ Dλ,β
introduced by K. Al-Shaqsi and M. Darus in [3]
and finally the operator I m (λ, 0, l) ≡ I1 (m, λ, l) introduced in [5].
The main object of the present paper is to find sufficient conditions for
certain normalized analytic functions f to satisfy
q1 (z) ≺
I m+1 (λ, β, l)f (z)
≺ q2 (z),
I m (λ, β, l)f (z)
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Adriana Cătaş
where m, β ∈ N0 , λ ≥ 0 and q1 , q2 are given univalent functions in U . Also,
we obtain the number of known results as their special cases.
Theorem 1 Let m, β ∈ N0 , λ > 0 and q be convex univalent in U with
q(0) = 1. Further, assume that
½
¾
2(δ + α)q(z)
zq 00 (z)
(5)
Re
+1+ 0
> 0.
δ
q (z)
Let
(6)
ψ(m, λ, β, δ, α; z) =
+
δ[1 − λ(1 + β) + l] I m+1 (λ, β, l)f (z)
· m
+
λ
I (λ, β, l)f (z)
δλ(β + 1)(β + 2) I m (λ, β + 2, l)f (z)
·
+
l+1
I m (λ, β, l)f (z)
δ(1 + β)[1 − λ(β + 2) + l] I m (λ, β + 1, l)
·
+
l+1
I m (λ, β, l)
·
µ
¶¸ µ m+1
¶2
I
(λ, β, l)f (z)
l+1
+ α+δ 1−
.
λ
I m (λ, β, l)f (z)
+
If f ∈ An satisfies
ψ(m, λ, β, δ, α; z) ≺ δzq 0 (z) + (δ + α)(q(z))2
(7)
then
I m+1 (λ, β, l)f (z)
≺ q(z)
I m (λ, β, l)f (z)
(8)
and q is the best dominant.
Proof.
Define the function p(z) by
(9)
p(z) =
I m+1 (λ, β, l)f (z)
,
I m (λ, β, l)f (z)
z ∈ U.
On a Certain Differential Sandwich Theorem ...
89
Then the function p(z) is analytic in U and p(0) = 1.
Therefore, by making use of (3) and (4) we have
δ[1 − λ(1 + β) + l] I m+1 (λ, β, l)f (z)
· m
+
λ
I (λ, β, l)f (z)
(10)
δλ(β + 1)(β + 2) I m (λ, β + 2, l)f (z)
+
·
+
l+1
I m (λ, β, l)f (z)
δ(1 + β)[1 − λ(β + 2) + l] I m (λ, β + 1, l)
·
+
l+1
I m (λ, β, l)
·
µ
¶¸ µ m+1
¶2
l+1
I
(λ, β, l)f (z)
+ α+δ 1−
=
λ
I m (λ, β, l)f (z)
+
= δzp0 (z) + (δ + α)(p(z))2 .
By using (10) in (7) we get
δzp0 (z) + (δ + α)(p(z))2 ≺ δzq 0 (z) + (δ + α)(q(z))2 .
By setting θ(w) = (δ + α)w2 and φ(w) = δ are analytic in C \ {0} and
that φ(w) 6= 0. Hence the result follows by an application of Lemma 1.
Remark 2 Similar results were obtained earlier in [6] for the operator defined in [2].
Let
q(z) =
1 + Az
,
1 + Bz
−1 ≤ B < A ≤ 1
in Theorem 1. One obtains the following result.
Corollary 1 Let m, β ∈ N0 , λ > 0. Assume that (5) holds. If f ∈ An ,
then, differential subordination
(11)
δ(A − B)z
ψ(m, λ, β, δ, α; z) ≺
+ (δ + α)
(1 + Bz)2
µ
1 + Az
1 + Bz
¶2
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Adriana Cătaş
implies
1 + Az
I m+1 (λ, β, l)f (z)
≺
m
I (λ, β, l)f (z)
1 + Bz
and
1 + Az
is the best dominant.
1 + Bz
Corollary 2 Let m, β ∈ N0 , λ > 0. Assume that (5) holds. If f ∈ An ,
then differential subordination
2δz
ψ(m, λ, β, δ, α; z) ≺
+ (δ + α)
(1 − z)2
(12)
µ
1+z
1−z
¶2
implies
1+z
I m+1 (λ, β, l)f (z)
≺
m
I (λ, β, l)f (z)
1−z
and
1+z
is the best dominant.
1−z
Corollary 3 Let m, β ∈ N0 , λ > 0, 0 < µ ≤ 1. Assume that (5) holds. If
f ∈ An , then differential subordination
(13)
2δµz
ψ(m, λ, β, δ, α; z) ≺
(1 − z)2
µ
1+z
1−z
¶µ−1
µ
+ (α + δ)
implies
I m+1 (λ, β, l)f (z)
≺
I m (λ, β, l)f (z)
µ
and
1+z
1−z
¶µ
is the best dominant.
µ
1+z
1−z
¶µ
1+z
1−z
¶2µ
On a Certain Differential Sandwich Theorem ...
91
Theorem 2 Let q be convex univalent in U with q(0) = 1. Assume that
½
¾
2(δ + α)q(z)q 0 (z)
(14)
Re
> 0.
δ
I m+1 (λ, β, l)f (z)
∈ H[q(0), 1] ∩ Q.
I m (λ, β, l)f (z)
If function ψ(m, λ, β, δ, α; z), given by (6), is univalent in U and
Let f ∈ A,
(15)
(δ + α)(q(z))2 + δzq 0 (z) ≺ ψ(m, λ, β, δ, α; z)
then
q(z) ≺
I m+1 (λ, β, l)f (z)
I m (λ, β, l)f (z)
and q is the best subordinant.
Proof.
Theorem 2 follows by using the same technique to prove Theorem 1 and
by an application of Lemma 2.
By using Theorem 2 we obtain the following corollaries.
Corollary 4 Let q(z) =
1 + Az
, −1 ≤ B < A ≤ 1, f ∈ A and
1 + Bz
I m+1 (λ, β, l)f (z)
∈ H[q(0), 1] ∩ Q.
I m (λ, β, l)f (z)
Assume that (14) holds. If
µ
(16)
then
(δ + α)
1 + Az
1 + Bz
¶2
+
δ(A − B)z
≺ ψ(m, λ, β, δ, α; z)
(1 + Bz)2
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Adriana Cătaş
1 + Az
I m+1 (λ, β, l)f (z)
≺ m
1 + Bz
I (λ, β, l)f (z)
and
1 + Az
is the best subordinant.
1 + Bz
Corollary 5 Let q(z) =
1+z
, f ∈ A and
1−z
I m+1 (λ, β, l)f (z)
∈ H[q(0), 1] ∩ Q.
I m (λ, β, l)f (z)
Assume that (14) holds. If
2δz
+ (δ + α)
(1 − z)2
(17)
µ
1+z
1−z
¶2
≺ ψ(m, λ, β, δ, α; z)
then
1+z
I m+1 (λ, β, l)f (z)
≺ m
1−z
I (λ, β, l)f (z)
1+z
is the best subordinant.
1−z
µ
¶µ
1+z
Corollary 6 Let q(z) =
, 0 < µ ≤ 1, f ∈ A and
1−z
and
I m+1 (λ, β, l)f (z)
∈ H[q(0), 1] ∩ Q.
I m (λ, β, l)f (z)
Assume that (14) holds. If
µ
¶µ−1
µ
¶2µ
2δµz
1+z
1+z
(18)
+ (α + δ)
≺ ψ(m, λ, β, δ, α; z)
(1 − z)2 1 − z
1−z
then
µ
µ
and
1+z
1−z
¶µ
1+z
1−z
¶µ
≺
I m+1 (λ, β, l)f (z)
I m (λ, β, l)f (z)
is the best subordinant.
On a Certain Differential Sandwich Theorem ...
93
Combining the results of differential subordination and superordination
we state the following Sandwich Theorems.
Theorem 3 Let q1 and q2 be convex univalent in U and satisfy (14) and
(5) respectively.
I m+1 (λ, β, l)f (z)
If f ∈ A, m
∈ H[q(0), 1] ∩ Q and ψ(m, λ, β, δ, α; z) given
I (λ, β, l)f (z)
in (6) is univalent in U and
(19)
δzq10 (z) + (δ + α)(q1 (z))2 ≺ ψ(m, λ, β, δ, α; z) ≺
≺ δzq20 (z) + (δ + α)(q2 (z))2 ,
then
q1 (z) ≺
I m+1 (λ, β, l)f (z)
≺ q2 (z)
I m (λ, β, l)f (z)
and q1 and q2 are the best subordinant and best dominant respectively.
1 + A1 z
1 + A2 z
, q2 (z) =
, where −1 ≤ B2 < B1 < A1 ≤
1 + B1 z
1 + B2 z
A2 ≤ 1 we have the following corollary.
For q1 (z) =
Corollary 7 If f ∈ A,
I m+1 (λ, β, l)f (z)
∈ H[q(0), 1] ∩ Q and
I m (λ, β, l)f (z)
δ(A1 − B1 )z
+ (δ + α)
(1 + B1 z)2
µ
1 + A1 z
1 + B1 z
¶2
δ(A2 − B2 )z
≺
+ (δ + α)
(1 + B2 z)2
≺ ψ(m, λ, β, δ, α; z) ≺
µ
1 + A2 z
1 + B2 z
¶2
,
then
I m+1 (λ, β, l)f (z)
1 + A2 z
1 + A1 z
≺ m
≺
1 + B1 z
I (λ, β, l)f (z)
1 + B2 z
1 + A2 z
1 + A1 z
and
are the best subordinant and the best dominant
Hence
1 + B1 z
1 + B2 z
respectively.
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Adriana Cătaş
References
[1] Al-Oboudi, F.M.: On univalent functions defined by a generalized
Sălăgean operator. Int. J. Math. Math. Sci. 27, 1429-1436, (2004).
[2] Al-Shaqsi, K., Darus, M.: Differential subordination with generalized
derivative operator. (submitted).
[3] Al-Shaqsi, K., Darus, M.: On univalent functions with respect to ksymmetric points defined by a generalized Ruscheweyh derivatives operator. (submitted).
[4] Bulboacă, T.:
Classes of first order differential superordinations.
Demonstratio Math. 35(2), 287-292, (2002).
[5] Cătaş, A.: On certain class of p-valent functions defined by a new multiplier transformations. Proceedings Book of the International Symposium G.F.T.A., Istanbul Kultur University, Turkey, 241-250, (2007).
[6] Darus, M., Al-Shaqsi, K.: Differential sandwich theorems with generalized derivative operator. Int. J. of Computational and Mathematical
Sciences. Vol 2, No. 2 Spring, 75-78, (2008).
[7] Miller, S.S., Mocanu, P.T.: Subordinants of differential superordinations. Complex Variables, Theory and Applications. 48(10), 815-826,
(2003).
[8] Miller, S.S., Mocanu, P.T.: Differential Subordinations: Theory and
Applications. Pure and Applied Mathematics. No. 225, Marcel Dekker,
New York, (2000).
On a Certain Differential Sandwich Theorem ...
95
[9] Ruscheweyh, S.: New criteria for univalent functions. Proc. Amer.
Math. Soc. 49, 109-115, (1975).
[10] Sălăgean, G. Şt.: Subclasses of univalent functions. Lecture Note in
Math. (Springer-Verlag), 1013, 362-372, (1983).
Adriana Cătaş
Faculty of Sciences, University of Oradea
Department of Mathematics and Computer Science
1 University Street, 410087 Oradea, Romania
E-mail: acatas@gmail.com