Abstract
The authors introduce new classes of analytic functions in the open unit disc which are defined by using multiplier transformations. The properties of these classes will be studied by using techniques involving the Briot-Bouquet differential subordinations. Also an integral transform is established.
1. Introduction and Definitions
Let be the class of analytic functions in the open unit discand let be the subclass of consisting of functions of the form . Let denote the class of functions normalized bywhich are analytic in the open unit disc. In particular, we set If a function belongs to the class it has the form For two functions given by (1.4) and for given bythe Hadamard product (or convolution) is defined, as usual, by If and are analytic in , we say that is subordinate to , written symbolically asif there exists a Schwarz function in which is analytic in with and such that , .
We consider the following multiplier transformations.
Definition 1.1 (See [1]). Let . For , , , , define the multiplier transformations on by the following infinite series:
It follows from (1.8) that
Remark 1.2 (See [1]). For , , , , , , the operator was introduced and studied by Al-Oboudi [2] which is reduced to the Sălăgean differential operator [3] for . The operator was studied recently by Cho and Srivastava [4] and by Cho and Kim [5]. The operator was studied by Uralegaddi and Somanatha [6], the operator was introduced by Acu and Owa [7] and the operator was investigated recently by Sivaprasad Kumar et al. [8].
If is given by (1.2), then we havewhere In particular, we set In order to prove our main results, we will make use of the following lemmas.
Lemma 1.3 (See [9]). For real or complex numbers , , and , the following hold:
Lemma 1.4 (See [10]). Let , and let be convex in , with If the function , then
Lemma 1.5 (See [11]). Let be a positive measure on the unit interval . Let be a function analytic in , for each and integrable in , for each and for almost all . Suppose also that is real for real and Ifthen
Lemma 1.6 (See [12]). Let be univalent in the unit disc and let and be analytic in a domain with ,
when .
Set
Suppose that
(1) is starlike in and(2) for .
If is analytic in ,
with , ,
andthen and is the best dominant.
Lemma 1.7 (See [12, Theorem 3.3d]). Let , with and let satisfy eitherwhen , orwhen . If satisfiesthenwhere is the univalent solution of the differential equationIn addition, the function , is the best -dominant and the function is given bywhereand the univalent function is given by
Now we define new classes of analytic functions by using the multiplier transformations defined by (1.8) as follows.
2. Main Results
Definition 2.1. Let , , , . A function is said to be in the class if it satisfies the following subordination:
Remark 2.2. We note thatwhere and denote the subclasses of functions in which are, respectively, starlike of order and convex of order in . Also we have the classstudied by Patel [13].
Let be analytic in and . We introduce the following definition.
Definition 2.3. A function is said to be in the class if it satisfies the following subordination:
Remark 2.4. We note that the classes were investigated recently by Sivaprasad Kumar et al. [8].
Theorem 2.5. Let , , ,
and
(i) Then
Further, for
the following
hold:
whereand is the best dominant of (2.7).
(ii) Furthermore, in addition to (2.5), one consider
the inequalitywhere ,
thenwhere
The result is the best possible.
Proof. Settingwe note that is analytic in and
Using the identityin definition of and carrying out logarithmic differentiation
in the resulting equation, one obtains
Since ,
we get
By applying Lemma 1.4, we obtain that
Hence we have shown the inclusion (2.6). Also, making
use of Lemma 1.7 with and , is the best dominant of (2.7) and is defined by (2.8). This proves part (i) of
Theorem 2.5.
To establish (2.10), we need to show
thatThe proof of the assertion
(2.18) will be deduced on the same lines as in [14] making use of
Lemma 1.5. If we set , ,
then by using (1.13), (1.14), and
(1.15), we find from (2.8) thatwhereBy using (1.13), the above
equality yieldswhereis a positive measure on the
closed interval .
For ,
we note that , is real for and andTherefore, by using Lemma 1.5,
one obtainswhich, upon letting ,
yieldsNow, the assertion (2.18)
follows by using Lemma 1.5. The result is the best possible and is the best dominant of (2.7). This completes
the proof of Theorem 2.5.
Taking , , , , , in Theorem 2.5, we get the following result due to MacGregor [15].
Corollary 2.6. For , one obtainswhere The result is the best possible.
Theorem 2.7. Let be univalent in with , , and let be starlike in . Let be defined byThen
Proof. Settingwe note that is analytic in .
By a simple computation, we observe from (2.30) that
Making use of the identity (2.14), one obtains from
(2.31)
By the hypothesis of Theorem 2.7 that belongs to the class and in view of (2.32), we have
If we letwhereand since is starlike, our theorem is an immediate
consequence of Lemma 1.6.
Theorem 2.8. Let be univalent in , and let
be a complex number. Suppose that
(1)
and(2) is starlike in U.
Let the function be defined byand the functionthen implies .
Proof. From the definition of andif we letthen we note that is analytic in . Using (2.38) and (2.39), one obtains Differentiating this equality, we obtain For , we have from (2.41) If we letwhereand since is starlike in , our theorem is an immediate consequence of Lemma 1.6.
Theorem 2.9. Let . Then belongs to the class if and only if defined bybelongs to the class .
Proof. From the definition of , we have By convoluting (2.46) with the functionand using a convolution propertyone obtainsthat is, By using identity (2.14), we get Also, we obtain From (2.51) and (2.52), we get By the hypothesis of Theorem 2.9 thatand using (2.53), the desired result follows at once.