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On certain subclasses of multivalent functions defined by multiplier transformations
Journal of Inequalities and Applications volume 2013, Article number: 441 (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.
Dedicated to Professor Hari M Srivastava
Let denote the class of functions of the form
which are analytic in the open unit disk . We write .
Suppose that f and g are analytic in . We say that the function f is subordinate to g in , and we write or (), if there exists an analytic function ω in with and for all such that in . If g is univalent in , then the following equivalence relationship holds true:
For functions f given by (1.1) and , the Hadamard product (or convolution) of f and g is defined by
For fixed parameters A, B (), let be the class of functions of the form
which are analytic in and satisfy the condition
The class was investigated in . We denote by the class of functions such that . Analogously, is the class of functions such that . It is easily seen that
the subclasses of , which are respectively, p-valently starlike of order ρ and p-valently convex of order ρ in . We also note that
where and are the subclasses of consisting of functions that are p-valently starlike and p-valently convex in , respectively.
In the present investigation, we shall make use of the Gauss hypergeometric function defined in by
where denotes the Pochhammer symbol (or shifted factorial) given by
We note that the series defined by (1.3) converges absolutely for all and hence represents an analytic function in [, Chapter 14].
Motivated by the multiplier transformation introduced in  on , we introduce an operator on by
The operator is related to the multiplier transformation studied in .
Corresponding to the function , we define a new function in terms of the Hadamard product by
We now introduce the operator by
If the function f is given by (1.1), then from (1.4) and (1.5) we deduce that
In view of (1.5), it follows that
In particular, we note that for ,
By using the operator , we introduce the subclass of as follows.
Definition For fixed parameters A, B (), , and , we say that a function is in the class if
It is readily seen that
For the sake of convenience, we write
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.
Let h be analytic and convex (univalent) in with . Suppose also that the function φ defined by (1.2) is analytic in . If
and ψ is the best dominant of (2.1).
Lemma 2 [, p.35]
Suppose that the function satisfies the condition
for , real and all . If the function φ, given by (1.2) is analytic in and
then in .
Lemma 3 
Let and satisfy
If , and satisfies
then in .
If with satisfies
then, for , we have
The value of β in (2.2) and the bound in (2.3) are best possible.
Lemma 4 
then () in , where
and . Further, for , we have
The value of β in (2.4) and the bound in (2.5) are best possible.
3 Inclusion relationships for the class
Unless otherwise mentioned, we assume throughout the sequel that
Theorem 1 We have
Further, for , we also have
and q is the best dominant of (3.1). Moreover, if (), then
where . The bound ρ is best possible.
Proof Let . Suppose that
and . Choosing the principal branch of g, we note that g is single-valued and analytic in . Taking the logarithmic differentiation in (3.4) and using identity (1.6) in the resulting equation, we get that
is of the form (1.2) and is analytic in . Again, carrying out logarithmic differentiation in (3.5) and using (1.6), we deduce that
Hence, by applying the result [, Corollary 3.2], we obtain
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.
Corollary 1 If () and satisfies
The result is best possible.
In the special case when , , () and , Theorem 1 gives the following.
Corollary 2 If and satisfies
The result is best possible.
Remarks 1. Putting and in Corollary 1, we find that for and ,
which in turn implies that
For , this result is contained in .
Setting , () and in Corollary 1 and in Corollary 2, we get the corresponding result obtained in .
Theorem 2 For , we have
The bound R is best possible.
we see that u is of the form (1.2), analytic and has a positive real part in . Taking the logarithmic differentiation in (3.8) and using identity (1.6), we deduce that
Now, by using the well-known  estimates
in (3.9), we obtain
which is certainly positive if , where R is given by (3.7).
To show that the bound R is best possible, we consider the function defined by
for , we complete the proof of Theorem 2. □
Remark For , , and , Theorem 2 yields the corresponding result contained in .
For a function , the generalized Bernardi-Libera-Livingston integral operator is defined by (cf., e.g., )
For convenience, we write , . It readily follows from (3.10) that .
Theorem 3 Let δ be a real number satisfying
If , then(3.12)
and q is the best dominant of (3.12). Consequently, the operator maps the class into itself.
If and , then(3.13)
The bound τ is best possible.
Proof From (1.5) and (3.10), it follows that
and . Choosing the principal branch of g, it follows that g is a single-valued and is analytic in . Taking the logarithmic differentiation in (3.15) and using identity (1.6) for , we deduce that the function
is analytic in and . Using identity (3.14) in (3.16), we obtain
Since , it is clear that in . So, by (3.17), we get
Again, by taking the logarithmic differentiation in (3.18) followed by the use of identity (1.6) in the resulting equation, we get
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). □
Remark Letting , , , () and in Theorem 3, we have the following implications [, Corollary 3.4 and Remark 3.2]:
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.
4 Properties involving the operator
Theorem 4 If satisfies
The result is best possible.
we note that φ is of the form (1.2) and is analytic in . On differentiating (4.3) and using identity (1.6) in the resulting equation, we deduce that
The proof of the remaining part of the theorem follows by using Lemma 1 and the techniques that proved Theorem 4 in .
With a view to stating a well-known result, we denote by the class of functions φ of the form (1.2) which are analytic in and satisfy the inequality
It is known  that if (; ), then
where . The bound is best possible. □
Theorem 5 If the functions (, ; ), then the function defined in by
Proof We have
Hence, by using (4.5), we deduce that
which, in view of Lemma 1 for
From (4.9), by using Theorem 4 for
we deduce that
where . If we put
then φ is of the form (1.2) analytic in , and a simple computation shows that
where . Thus, by using (4.8) in (4.11), we conclude that
Now, for all real , we have
by (4.7) and (4.10). Hence, by making use of Lemma 2, we get in . This completes the proof of Theorem 5. □
Theorem 6 If the functions (, ; ), then the function H defined in by
Proof From the definition of the function H, it is easily seen that
and the proof of the theorem is completed similarly to Theorem 5. □
Theorem 7 Let (). If the function defined in by (4.6) satisfies
Proof Using the fact that
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 .
Theorem 8 Let and . If satisfies
then , where
Further, for ,
The bound given by (4.13) and the estimate in (4.14) are best possible.
and choosing the principal branch in (4.15), we note that Θ is analytic in with . A simple computation shows that (4.12) is equivalent to
Now, by applying Lemma 1 (with , and ), we get
We further observe that
Hence, assertion (4.13) follows by using part (i) of Lemma 3. If we put
then we obtain (4.14) from part (ii) of Lemma 3.
To show that the estimates are best possible, we consider the function defined in by
From this, we obtain
and the sharpness follows from Lemma 4 (for ). □
Putting , and in Theorem 8, we get the following.
Corollary 3 If and satisfies
Further, for ,
The estimates are best possible.
Theorem 9 If and satisfies
then , where
For , we have
Further, , where is obtained from ϰ (given in (4.13)) for and upon replacing γ by . Moreover, for ,
The estimates are best possible.
Proof Since satisfies (4.16), by Theorem 4 (for and ) we obtain
Again, on writing (4.16) in the form
and using part (ii) of Lemma 3, we deduce that
which implies assertion (4.17). By using part (ii) of Lemma 3 with
we obtain (4.18). That and (4.19) now follow from Theorem 8 and (4.20).
To show the sharpness of the estimates, we consider the function f defined in by
Hence, by using identity (1.6), we get
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.
Corollary 4 If satisfies
and for ,
The result is sharp.
Remark For in Corollary 3 and Corollary 4, we get the corresponding results obtained in .
Theorem 10 If satisfies
for some real numbers β and γ such that , , , then .
Proof If we set
then ω is analytic in . Differentiating (4.22) logarithmically and using identity (1.6) in the resulting equation, we get
Now, we have
We claim that for . Otherwise, there exists a point such that . By using Jack’s lemma , we write , and , . Thus, from (4.23), it follows that
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 .
Corollary 5 Let and . If satisfies
for all , then .
Similarly, by setting , , , , and in Theorem 10, we obtain the following sufficient condition for convexity of multivalent functions.
Corollary 6 Let and . If satisfies
where is defined as in Corollary 5, then .
Janowski W: Some extremal problems for certain families of analytic functions I. Ann. Pol. Math. 1973, 28: 297–326.
Whittaker ET, Watson GN: A Course on Modern Analysis. 4th edition. Cambridge University Press, Cambridge; 1927. (reprinted)
Cho NE, Srivastava HM: Argument estimates of certain analytic functions defined by a class of multiplier transformations. Math. Comput. Model. 2003, 37: 39–49. 10.1016/S0895-7177(03)80004-3
Flett TM: The dual of an inequality of Hardy and Littlewood and some related inequalities. J. Math. Anal. Appl. 1972, 38: 746–765. 10.1016/0022-247X(72)90081-9
Sǎlǎgean GS: Subclasses of univalent functions. Lecture Notes in Math. 1013. In Complex Analysis - Fifth Romanian-Finnish Seminar, Part I Bucharest, 1981. Springer, Berlin; 1983:362–372.
Cho NE, Kim TH: Multiplier transformations and strongly close-to-convex functions. Bull. Korean Math. Soc. 2003, 40: 399–410.
Cho NE, Kim JA: Inclusion properties of certain subclasses of analytic functions defined by a multiplier transformation. Comput. Math. Appl. 2006, 52: 323–330. 10.1016/j.camwa.2006.08.022
Uralegaddi BA, Somanatha C: Certain classes of univalent functions. In Current Topics in Analytic Function Theory. World Scientific, River Edge; 1992:371–374.
Hallenbeck DJ, Ruscheweyh S: Subordination by convex functions. Proc. Am. Math. Soc. 1975, 52: 191–195. 10.1090/S0002-9939-1975-0374403-3
Miller SS, Mocanu PT: Differential Subordinations, Theory and Applications. Dekker, New York; 2000.
Ponnusamy, S, Singh, V: Convolution properties of some classes of analytic functions. Internal Report, SPIC Science Foundation (1990)
Miller SS, Mocanu PT: Univalent solutions of Briot-Bouquet differential subordinations. J. Differ. Equ. 1985, 56: 297–309. 10.1016/0022-0396(85)90082-8
Patel J, Cho NE, Srivastava HM: Certain classes of multivalent functions associated with a family of linear operators. Math. Comput. Model. 2006, 43: 320–338. 10.1016/j.mcm.2005.06.014
Miller SS, Mocanu PT, Reade MO: The order of starlikeness of α -convex functions. Mathematica 1978, 20(43):25–30.
Patel J, Cho NE: Some classes of analytic functions involving Noor integral operator. J. Math. Anal. Appl. 2005, 312: 564–575. 10.1016/j.jmaa.2005.03.047
MacGregor TH: Subordination of convex functions of order α . J. Lond. Math. Soc. 1975, 9(2):530–536.
Choi JH, Saigo M, Srivastava HM: Some inclusion properties of a certain family of integral operators. J. Math. Anal. Appl. 2002, 276: 432–445. 10.1016/S0022-247X(02)00500-0
Fukui S, Kim JA, Srivastava HM: On certain subclasses of univalent functions by some integral operators. Math. Jpn. 1999, 50: 359–370.
Stankiewicz J, Stankiewicz Z: Some applications of the Hadamard convolution in the theory of functions. Ann. Univ. Mariae Curie-Skl̄odowska, Sect. A 1986, 40: 251–256.
Lashin AY: Some convolution properties of analytic functions. Appl. Math. Lett. 2005, 18: 135–138. 10.1016/j.aml.2004.09.003
Jack IS: Functions starlike and convex of order α . J. Lond. Math. Soc. 1971, 3(2):469–474.
Owa S, Srivastava HM: Univalent and starlike generalized hypergeometric functions. Can. J. Math. 1987, 39: 1057–1077. 10.4153/CJM-1987-054-3
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).
The authors declare that they have no competing interests.
All authors jointly worked on the results and they read and approved the final manuscript.
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Patel, J., Cho, N.E. & Palit, A.K. On certain subclasses of multivalent functions defined by multiplier transformations. J Inequal Appl 2013, 441 (2013). https://doi.org/10.1186/1029-242X-2013-441
- analytic function
- multivalent function
- multiplier transformation
- integral operator