- Open Access
On certain subclasses of multivalent functions defined by multiplier transformations
© Patel et al.; licensee Springer. 2013
- Received: 3 January 2013
- Accepted: 1 August 2013
- Published: 21 September 2013
The purpose of the present paper is to introduce and investigate various properties of a certain class of multivalent functions in the open unit disk defined by a multiplier transformation. In particular, we obtain some inclusion relationships and integral preserving properties of this class of functions. Relevant connections of the results presented in this paper with various known results are also pointed out.
- analytic function
- multivalent function
- multiplier transformation
- integral operator
Dedicated to Professor Hari M Srivastava
which are analytic in the open unit disk . We write .
where and are the subclasses of consisting of functions that are p-valently starlike and p-valently convex in , respectively.
We note that the series defined by (1.3) converges absolutely for all and hence represents an analytic function in [, Chapter 14].
The operator is related to the multiplier transformation studied in .
By using the operator , we introduce the subclass of as follows.
The object of the present paper is to investigate some inclusion properties of the class . Integral-preserving and convolution properties in connection with the operator are also considered. Relevant connections of the results presented here with those obtained in the earlier works are pointed out.
We denote by ℋ the class of all analytic functions in and by ℬ the class of functions such that and for .
We shall need the following lemmas to prove our results.
and ψ is the best dominant of (2.1).
Lemma 2 [, p.35]
then in .
Lemma 3 
- (i)If , and satisfies
- (ii)If with satisfies
The value of β in (2.2) and the bound in (2.3) are best possible.
Lemma 4 
The value of β in (2.4) and the bound in (2.5) are best possible.
where . The bound ρ is best possible.
where q is the best dominant of (3.1) and Q is given by (3.2).
The proof of the remaining part can now be deduced along the same lines as in [, Theorem 1]. The bound ρ in (3.3) is best possible as q is the best dominant of (3.1). This evidently completes the proof of the theorem. □
Setting , and in Theorem 1, we get the following corollary.
The result is best possible.
In the special case when , , () and , Theorem 1 gives the following.
The result is best possible.
Setting , () and in Corollary 1 and in Corollary 2, we get the corresponding result obtained in .
The bound R is best possible.
which is certainly positive if , where R is given by (3.7).
for , we complete the proof of Theorem 2. □
Remark For , , and , Theorem 2 yields the corresponding result contained in .
For convenience, we write , . It readily follows from (3.10) that .
- (i)If , then(3.12)
- (ii)If and , then(3.13)
The bound τ is best possible.
The proof of the remaining part is the same as that of [, Theorem 1], and we choose to omit the details. The result is best possible as q is the best dominant of (3.12). □
where and . The containment relations are best possible, and they improve the corresponding work in  for suitable values of the parameters p, η and δ.
The result is best possible.
The proof of the remaining part of the theorem follows by using Lemma 1 and the techniques that proved Theorem 4 in .
where . The bound is best possible. □
by (4.7) and (4.10). Hence, by making use of Lemma 2, we get in . This completes the proof of Theorem 5. □
and the proof of the theorem is completed similarly to Theorem 5. □
and by following the same lines of proof as in Theorem 5, we get the required result. □
Remark Putting , and in Theorems 5, 6 and 7, respectively, we obtain the corresponding results contained in .
The bound given by (4.13) and the estimate in (4.14) are best possible.
then we obtain (4.14) from part (ii) of Lemma 3.
and the sharpness follows from Lemma 4 (for ). □
Putting , and in Theorem 8, we get the following.
The estimates are best possible.
The estimates are best possible.
we obtain (4.18). That and (4.19) now follow from Theorem 8 and (4.20).
and the sharpness follows from Lemma 4. The fact that is sharp follows from (4.20) and the sharpness of Theorem 8. □
Putting , and in Theorem 9, we have the following.
The result is sharp.
Remark For in Corollary 3 and Corollary 4, we get the corresponding results obtained in .
for some real numbers β and γ such that , , , then .
This contradicts the hypothesis (4.21) and hence for . This proves the theorem. □
Taking , , , , and in Theorem 10, we get the following interesting criterion for starlikeness for multivalent functions, which improves the corresponding work in  for .
for all , then .
Similarly, by setting , , , , and in Theorem 10, we obtain the following sufficient condition for convexity of multivalent functions.
where is defined as in Corollary 5, then .
This work was supported by a Research Grant of Pukyong National University (2013 year) and the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2011-0007037).
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