Abstract
The purpose of the present paper is to investigate some subordination and superordinationpreserving properties of certain nonlinear integral operators defined on the space of meromorphically pvalent functions in the punctured open unit disk. The sandwichtype theorems associated with these integral operators are established. Relevant connections of the various results presented here with those involving relatively simpler nonlinear integral operators are also indicated.
MSC: 30C45, 30C80.
Keywords:
differential subordination; differential superordination; meromorphic functions; integral operators; convex functions; closetoconvex functions; subordination (or Löwner) chain1 Introduction, definitions and preliminaries
Let
For
Let f and F be members of ℋ. The function f is said to be subordinate to F, or F is said to be superordinate to f, if there exists a function w analytic in
such that
In such a case, we write
Furthermore, if the function F is univalent in
Definition 1 (Miller and Mocanu [1])
Let
and let h be univalent in
then
for all
for all dominants q of (1.1) is said to be the best dominant.
Definition 2 (Miller and Mocanu [4])
Let
and let h be analytic in
then
for all
for all subordinants q of (1.2) is said to be the best subordinant.
Definition 3 (Miller and Mocanu [4])
We denote by
and are such that
We also denote the class
Let
which are analytic in the punctured open unit disk
For a function
Several members of the family of integral operators
Making use of the principle of subordination between analytic functions, Miller et al.[13] and, more recently, Owa and Srivastava [14] obtained some interesting subordinationpreserving properties for certain integral
operators. Moreover, Miller and Mocanu [4] considered differential superordinations as the dual concept of differential subordinations
(see also [15]). It should be remarked that in recent years several authors obtained many interesting
results involving various integral operators associated with differential subordination
and superordination (for example, see [5,1618]). In the present paper, we obtain the subordination and superordinationpreserving
properties of the general integral operator
The following lemmas will be required in our present investigation.
Lemma 1 (Miller and Mocanu [19])
Suppose that the function
satisfies the following condition:
for all realsand for alltwith
If the function
is analytic in
then
Lemma 2 (Miller and Mocanu [20])
Let
If
then the solution of the differential equation
is analytic in
Lemma 3 (Miller and Mocanu [1])
Let
and let
be analytic in
If the functionqis not subordinate to
for which
Let
If R is the univalent function defined in
then the open door function (see [1]) is defined by
Remark 1 The function
Lemma 4 (Totoi [21])
Let
If
and
and
where
A function
Lemma 5 (Miller and Mocanu [4])
Let
Also set
If
is a subordination chain and
then the following subordination condition:
implies that
Furthermore, if
has a univalent solution
Lemma 6 (Pommerenke [22])
The function
with
Suppose that
and
for some positive constants
2 Main results and their corollaries and consequences
We begin by proving a general subordination property involving the integral operator
Theorem 1Let
where
Then the subordination relation
implies that
where
is the best dominant.
Proof Let us define the functions F and G, respectively, by
We first show that if the function q is defined by
then
From the definition of (1), we obtain
We also have
It follows from (2.7) and (2.8) that
Now, by a simple calculation with (2.9), we obtain the following relationship:
Thus, from (2.1), we have
and by using Lemma 2, we conclude that the differential equation (2.10) has a solution
Put
where ρ is given by (2.2). From (2.1), (2.10) and (2.11), we obtain
We now proceed to show that
Indeed, from (2.11), we have
where
For ρ given by (2.2), we note that the coefficient of
That is, the function
We next prove that the subordination condition (2.3) implies that
for the functions F and G defined by (2.5). Without loss of generality, we can assume that G is analytic and univalent on
We now consider the function
We note that
and
Furthermore, since G is convex, by using the wellknown growth and distortion sharp inequalities for convex functions (see [23]), we can prove that the second condition of Lemma 6 is satisfied. Therefore, by virtue
of Lemma 6,
and
This implies that
We now suppose that F is not subordinate to G. Then, in view of Lemma 3, there exist points
Hence we have
by virtue of the subordination condition (2.3). This contradicts the above observation that
Therefore, the subordination condition (2.3) must imply the subordination given by
(2.15). Considering
We next prove a solution to a dual problem of Theorem 1 in the sense that the subordinations are replaced by superordinations.
Theorem 2Let
whereρis given by (2.2) and
where
implies that
Moreover, the function
is the best subordinant.
Proof Let the functions F and G be given by (2.5). We first note from (2.7) and (2.8) that
After a simple calculation, equation (2.17) yields the following relationship:
where the function q is given in (2.6). Then, by using the same method as in the proof of Theorem 1, we can prove that
that is, G defined by (2.5) is convex (univalent) in
We next prove that the superordination condition (2.16) implies that
For this purpose, we consider the function
Since G is convex and
If we combine Theorem 1 and Theorem 2, then we obtain the following sandwichtype theorem.
Theorem 3Let
whereρis given by (2.2) and the function
where
implies that
Moreover, the functions
are the best subordinant and the best dominant, respectively.
The assumption of Theorem 3, that is, the functions
need to be univalent in
Corollary 1Let
whereρis given by (2.2). Then the subordination relation
implies that
where
are the best subordinant and the best dominant, respectively.
Proof In order to prove Corollary 1, we have to show that the condition (2.20) implies
the univalence of
By noting that
By setting
Corollary 2Let
and the function
where
implies that
Moreover, the functions
are the best subordinant and the best dominant, respectively.
If we take
Corollary 3Let
and the function
where
implies that
Moreover, the functions
are the best subordinant and the best dominant, respectively.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors jointly worked on the results and they read and approved the final manuscript.
Acknowledgements
Dedicated to Professor Hari M Srivastava.
This research was supported by the Basic Science Research Program through the National Research Foundation of the Republic of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 20120002619).
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