Differential Sandwich Theorems for Analytic Functions Defined by an Extended Multiplier Transformation ()
1. Introduction
Let
be the class of functions analytic in the open unit disk
. Let
be the subclass of
consisting of functions of the form:
(1.1)
For simplicity, let
. Also, let
be the subclass of
consisting of functions of the form:
(1.2)
If
we say that
is subordinate to
written
if there exists Schwarz function
which (by definition) is analytic in
with
and
for all
such that
Furthermore, if the function
is univalent in
then we have the following equivalence , (cf., e.g. [1,2]; see also [3]):
We denote this subordination by
![](https://www.scirp.org/html/5-5300192\5690e5d8-fbfc-48dc-8828-039be074a266.jpg)
and ![](https://www.scirp.org/html/5-5300192\6961811c-2f8f-405c-96af-31500ae0726b.jpg)
Let
and let
. If p and
are univalent and if p satisfies the second-order superordination
, (1.3)
then
is a solution of the differential superordination (1.3). Note that if
is subordinate to
, then
is superordinate to
. An analytic function
is called a subordinant if
for all
satisfying (1.3). A univalent subordinant
that satisfies
for all subordinants of (1.3) is called the best subordinant. Recently Miller and Mocanu [4] obtained conditions on the functions
and
for which the following implication holds:
(1.4)
Using the results of Miller and Mocanu [4], Bulboaca [5] considered certain classes of first-order differential superordinations as well as superordination-preserving integral operators [6]. Ali et al. [7] have used the results of Bulboaca [5] and obtained sufficient conditions for normalized analytic functions
to satisfy:
![](https://www.scirp.org/html/5-5300192\f2801ca5-65e3-4f74-954a-8b09370cf952.jpg)
where
and
are given univalent functions in
. Also, Tuneski [8] obtained a sufficient condition for starlikeness of
in terms of the quantity
.
Recently Shanmugam et al. [9] obtained sufficient conditions for a normalized analytic functions
to satisfy
![](https://www.scirp.org/html/5-5300192\03ac5295-3317-4b8d-8f7b-0003da71341c.jpg)
and
![](https://www.scirp.org/html/5-5300192\ef49c8c7-fef9-47fe-962b-fb27579b42bf.jpg)
Many essentially equivalent definitions of multiplier transformation have been given in literature (see [10-12]. In [13] Catas defined the operator
as follows:
Definition 1.1. [13] Let the function
. For
where
The extended multiplier transformation
on
is defined by the following infinite series:
(1.5)
It follows form (1.5) that ![](https://www.scirp.org/html/5-5300192\5b6a43dc-d02e-429d-86bd-8e9d11d0e9ea.jpg)
(1.6)
and
(1.7)
for all integers
and
. We note that:
1)
(see [14]);
2)
(see [15]);
3)
(see [10,11]);
4)
(see [12]).
Also if
, then we can write
![](https://www.scirp.org/html/5-5300192\850cd6ec-d674-4ce8-9f7d-6b801d3bc28a.jpg)
where
![](https://www.scirp.org/html/5-5300192\c057baa6-e570-4269-8f2a-57b41bf57e55.jpg)
In this paper, we obtain sufficient conditions for the normalized analytic function
defined by using an extended multiplier transformation
to satisfy:
![](https://www.scirp.org/html/5-5300192\b46764ba-6363-4c4e-9708-8ca3dbec5934.jpg)
and
![](https://www.scirp.org/html/5-5300192\bb23b95f-7f7e-4ba6-afc8-f1857343da22.jpg)
and
and
are given univalent functions in
.
2. Definitions and Preliminaries
In order to prove our results, we shall make use of the following known results.
Definition 2.1. [4]
Denote by
the set of all functions
that are analytic and injective on
where
![](https://www.scirp.org/html/5-5300192\6215d1e9-6f45-47a5-a4a7-912f8f0b2d20.jpg)
and are such that
for ![](https://www.scirp.org/html/5-5300192\a8822829-5991-48df-9cf1-5c9235fb41ec.jpg)
Lemma 2.1. [4]
Let the function
be univalent in the open unit disc
and
and
be analytic in a domain
containing
with
when
. Set
. (2.1)
Suppose that 1)
is starlike univalent in
2)
for
.
If
is analytic with
and
(2.2)
then
and
is the best dominant. Taking
and
in lemma 1, Shanmugam et al. [9] obtained the following lemma.
Lemma 2.2. [2]
Let
be univalent in
with
Let
;
further assume that
![](https://www.scirp.org/html/5-5300192\8ff2525d-f294-43bd-8bd0-d4a22d92a6c2.jpg)
If
is analytic in
, and
![](https://www.scirp.org/html/5-5300192\37ab7369-06a0-479a-ba7f-8ff777e51dd6.jpg)
then
and
is the best dominant.
Lemma 2.3. [5]
Let the function
be univalent in the open unit disc
and
and
be analytic in a domain
containing
Suppose that 1)
for
and 2)
is starlike univalent in
.
If
with
,
, is univalent in
and
(2.3)
then
and
is the best subordinant.
Taking
and
in Lemma 2.3, Shanmugam et al. [9] obtained the following lemma.
Lemma 2.4. [2]
Let
be convex univalent in
,
Let
,
and
If
is univalent in
and
then
and
is the best subordinant.
3. Applications to an Extended Multiplier Transformation and Sandwich Theorems
Theorem 3.1.
Let
be convex univalent in
with
Further, assume that
(3.1)
If
,
for
and
(3.2)
then
![](https://www.scirp.org/html/5-5300192\8c8fe27f-58d9-4d84-a9aa-15b0fa4c6287.jpg)
and
is the best dominant.
Proof. Define a function
by
(3.3)
Then the function
is analytic in
and
. Therefore, differentiating (3.3) logarithmically with respect to
and using the identity (1.6) in the resulting equation, we have
![](https://www.scirp.org/html/5-5300192\7ff8ae87-a652-4e9c-a5fa-e4c3eb8e5bd9.jpg)
that is,
![](https://www.scirp.org/html/5-5300192\d79bda98-317c-42ed-8860-cdc5e3990594.jpg)
and therefore, the theorem follows by applying Lemma 2.2.
Putting
![](https://www.scirp.org/html/5-5300192\4c4742f9-9529-46c1-a939-6a55a899ce48.jpg)
in Theorem 3.1, we have the following corollary.
Corollary 3.1.
If
and
satisfy
![](https://www.scirp.org/html/5-5300192\3819c8bc-31d6-461c-8eb3-d90236ee2287.jpg)
then
![](https://www.scirp.org/html/5-5300192\c3452cb0-39e5-4f53-b5a5-c44481dd8230.jpg)
Putting
and
in Corollary 3.1, we have
Corollary 3.2.
If
and
satisfy
![](https://www.scirp.org/html/5-5300192\00504bdb-4871-4999-99f4-8b5d4716b20d.jpg)
then
![](https://www.scirp.org/html/5-5300192\1a7016f3-3296-4109-958f-e18b46c0bc28.jpg)
Taking
in Theorem 1, we have
Corollary 3.3.
Let
be convex univalent in
with
. Further, assume that (3.1) holds. If
, and
![](https://www.scirp.org/html/5-5300192\1792acb0-3269-4d4f-93a0-86dd2b122de0.jpg)
then
![](https://www.scirp.org/html/5-5300192\1ae05d54-aef9-4277-9a1f-a365a76e23ca.jpg)
and
is the best dominant.
Taking
in Theorem 3.1, we have
Corollary 3.4.
Let
be convex univalent in
with
. Further, assume that (3.1) holds. If
, and
![](https://www.scirp.org/html/5-5300192\eed7e5be-b70d-4bc3-b532-cfd2175e9c1d.jpg)
then
![](https://www.scirp.org/html/5-5300192\56f8860a-d018-42ea-bf10-80cc4ef0586d.jpg)
and
is the best dominant.
Taking
in Theorem 3.1, we have
Corollary 3.5.
Let
be convex univalent in
with
. Further, assume that (3.1) holds. If
, and
![](https://www.scirp.org/html/5-5300192\03d3d72c-a85b-43bc-af26-98fb6de74697.jpg)
then
![](https://www.scirp.org/html/5-5300192\68bcb08e-1053-4f59-a5df-2fc18c4b251f.jpg)
and
is the best dominant.
Taking
in Theorem 1, we have
Corollary 3.6.
Let
be convex univalent in
with
. Further, assume that (3.1) holds. If
, and
![](https://www.scirp.org/html/5-5300192\8483ad49-4c3c-469d-b4a4-496654d829d3.jpg)
then
![](https://www.scirp.org/html/5-5300192\cafc4397-3409-4efb-bfe2-542977392f99.jpg)
and
is the best dominant.
Now, by appealing to Lemma 2.4 it can be easily prove the following theorem.
Theorem 3.2.
Let
be convex univalent in
. Let
with ![](https://www.scirp.org/html/5-5300192\689759f2-036b-4b52-8412-fc13630db5d4.jpg)
If
, ![](https://www.scirp.org/html/5-5300192\93b86153-6f78-45cd-a3d2-b7c16f92254d.jpg)
![](https://www.scirp.org/html/5-5300192\290a59e3-2ed1-438e-8d7b-a3f6064c7200.jpg)
is univalent in
, and
![](https://www.scirp.org/html/5-5300192\0ee08ce9-e4cd-4a08-99d0-305eb3f47589.jpg)
then
![](https://www.scirp.org/html/5-5300192\b5006f4e-d435-40da-b9d5-02b914230752.jpg)
and
is the best subordinant.
Taking
, in Theorem 3.2, we have
Corollary 3.7.
Let
be convex univalent in
. Let
with ![](https://www.scirp.org/html/5-5300192\77e06570-073b-4aa7-ae3f-b6da9c714041.jpg)
If
, ![](https://www.scirp.org/html/5-5300192\8a92b80c-320e-41dc-b1d8-363181cf75d3.jpg)
![](https://www.scirp.org/html/5-5300192\6be76dee-9421-425c-a960-13859286f425.jpg)
is univalent in
, and
then
![](https://www.scirp.org/html/5-5300192\cf68caee-6aec-4a18-877a-432bc091d891.jpg)
and
is the best subordinant.
Taking
in Theorem 3.2, we have
Corollary 3.8.
Let
be convex univalent in
. Let
with ![](https://www.scirp.org/html/5-5300192\ed1aa334-2d32-49ad-9c05-78ad2c31dbdc.jpg)
If
, ![](https://www.scirp.org/html/5-5300192\41afc637-70e2-4770-b4d2-df5706357f07.jpg)
![](https://www.scirp.org/html/5-5300192\46eced3c-7ede-4154-be89-6ae553393e27.jpg)
is univalent in
, and
then
![](https://www.scirp.org/html/5-5300192\5e569fee-a189-44be-b12c-234ee89ccf61.jpg)
and
is the best subordinant.
Taking
in Theorem 3.2, we have
Corollary 3.9.
Let
be convex univalent in
. Let
with ![](https://www.scirp.org/html/5-5300192\66a125a9-d95d-407d-970e-c0ffa1db3dbd.jpg)
If
, ![](https://www.scirp.org/html/5-5300192\4fef0d27-880c-4b58-b282-9e1a90fbb663.jpg)
![](https://www.scirp.org/html/5-5300192\f2559149-e760-4701-9744-0eb5309621f5.jpg)
is univalent in
, and
![](https://www.scirp.org/html/5-5300192\f4d3ff54-212c-4634-aca9-3392484f59e0.jpg)
then
![](https://www.scirp.org/html/5-5300192\996b8840-775c-4000-b455-acd1f1f5f93e.jpg)
and
is the best subordinant.
Taking
in Theorem 3.2, we have
Corollary 3.10.
Let
be convex univalent in
. Let
with ![](https://www.scirp.org/html/5-5300192\437904f4-df69-4fe9-89b0-f75ddb033421.jpg)
If
, ![](https://www.scirp.org/html/5-5300192\35e38357-4d42-4f36-8933-3b6e68092902.jpg)
![](https://www.scirp.org/html/5-5300192\dc836b9f-d82f-4904-8dbe-2af5803d07fa.jpg)
is univalent in
, and
![](https://www.scirp.org/html/5-5300192\915d6084-b029-4923-b78c-a3de4a2e8f2b.jpg)
then
![](https://www.scirp.org/html/5-5300192\b9904e40-6765-4df8-9465-c2be6d251267.jpg)
and
is the best subordinant.
Combining Theorems 3.1 and 3.2, we get the following sandwich theorem.
Theorem 3.3.
Let
be convex univalent in
,
with
be univalent in
and satisfies (3.1). If ![](https://www.scirp.org/html/5-5300192\199cd2dd-891f-4243-889d-8bd0bd3d58cb.jpg)
![](https://www.scirp.org/html/5-5300192\8cd17cca-b485-4ff0-93a7-e0ef2875aeb8.jpg)
![](https://www.scirp.org/html/5-5300192\2340a393-ee3b-456c-a1e8-1efb6d0d6b2a.jpg)
is univalent in
, and
![](https://www.scirp.org/html/5-5300192\792b68ca-5718-4848-bc8b-6b435f89e9b8.jpg)
Then
![](https://www.scirp.org/html/5-5300192\095e464a-c0ad-40fe-9a60-60d9973f0a9e.jpg)
and
and
are respectively, the best subordinant and the best dominant.
4. Remarks
Combining: 1) Corollary 3.3 and Corollary 3.7; 2) Corollary 3.4 and Corollary 3.8; 3) Corollary 3.5 and Corollary 3.9; 4) Corollary 3.6 and Corollary 3.10, we obtain similar sandwich theorems for the corresponding operators.
Theorem 3.4.
Let
be convex univalent in
,
. Further, assume that (3.1) holds.
If
satisfies
![](https://www.scirp.org/html/5-5300192\cf2e20df-145e-46c8-a156-3f4a5db32f4f.jpg)
then
![](https://www.scirp.org/html/5-5300192\f13ee68b-46e9-4c7a-9fe3-bce6e282c53c.jpg)
and q is the best dominant.
Proof. Define the function
by
.
Then, simple computations show that
![](https://www.scirp.org/html/5-5300192\7c36d964-59ed-4e68-b958-ff0a34f74f6e.jpg)
Applying Lemma 2, the theorem follows.
Taking
in Theorem 3.4, we have the following corollary.
Corollary 3.11.
Let
be convex univalent in
,
. Further, assume that (3.1) holds. If
satisfies
![](https://www.scirp.org/html/5-5300192\b4366121-8471-4df5-bf58-46d9ca2fa192.jpg)
then
![](https://www.scirp.org/html/5-5300192\8010faa4-e77b-45d4-ac05-b80330ab0497.jpg)
and
is the best dominant.
Taking
in Theorem 3.4, we have
Corollary 3.12.
Let
be convex univalent in
,
. Further, assume that (3.1) holds. If
satisfies
![](https://www.scirp.org/html/5-5300192\5f907ccf-ebd6-4b4f-a221-313dfd360aa1.jpg)
then
![](https://www.scirp.org/html/5-5300192\06fba51b-1227-4078-91f0-d318194e20e9.jpg)
and
is the best dominant.
Taking
in Theorem 3.4, we have
Corollary 3.13.
Let
be convex univalent in
,
. Further, assume that (3.1) holds. If
satisfies
![](https://www.scirp.org/html/5-5300192\4eb811b1-450e-476e-8be4-d3769cd264c2.jpg)
then
![](https://www.scirp.org/html/5-5300192\d7520fb0-abdd-4fa0-b639-b7a07e15d49b.jpg)
and
is the best dominant.
Taking
in Theorem 3.4, we have
Corollary 3.14.
Let
be convex univalent in
,
. Further, assume that (3.1) holds. If
satisfies
![](https://www.scirp.org/html/5-5300192\0d0844f6-3270-4d45-b0cc-03b8693abf12.jpg)
then
![](https://www.scirp.org/html/5-5300192\6430ab0c-b125-4786-bcf5-1e5221579ac1.jpg)
and
is the best dominant.
Theorem 3.5.
Let
be convex univalent in
. Let
with ![](https://www.scirp.org/html/5-5300192\2f2cf11f-2dee-444e-9f88-c483bfb5909b.jpg)
If
, ![](https://www.scirp.org/html/5-5300192\12c8b50c-0d95-4d29-a4b2-07947568ba75.jpg)
![](https://www.scirp.org/html/5-5300192\4c4dce87-b207-43a6-8f69-43af33fc0e3d.jpg)
is univalent in
and
![](https://www.scirp.org/html/5-5300192\157ed301-3038-479b-91d6-b3af216d47f6.jpg)
then
![](https://www.scirp.org/html/5-5300192\93f24803-73ae-4f91-8d3b-8fe850d0727e.jpg)
and
is the best subordinant.
Proof. The proof follows by applying Lemma 3.4.
Combining Theorems 3.4 and 3.5, we get the following sandwich theorem.
Theorem 3.6.
Let
be convex univalent in
,
with
be univalent in
and satisfies (3.1). If
, ![](https://www.scirp.org/html/5-5300192\c61e3bfc-5b48-4b59-a2f5-2a08b5531925.jpg)
![](https://www.scirp.org/html/5-5300192\255bf641-48a0-46d3-80dc-188db2b0161f.jpg)
is univalent in
and
![](https://www.scirp.org/html/5-5300192\a11698ea-5ce6-4c83-992e-956573ab8f0b.jpg)
then
![](https://www.scirp.org/html/5-5300192\0f002fc6-5e47-4785-a109-b375853058d3.jpg)
and
and
are respectively the best subordinant and the best dominant.