Some Subclasses of Meromorphic Functions Associated with a Family of Integral Operators

Let denote the class of functions of the form

(1.1)

which are analytic in the punctured open unit disk

(1.2)

Let , where is given by (1.1) and is defined by

(1.3)

Then the Hadamard product (or convolution) of the functions and is defined by

(1.4)

Let denote the class of functions of the form

(1.5)

which are analytic and convex in and satisfy the condition

(1.6)

For two functions and , analytic in , we say that the function is subordinate to in , and write

(1.7)

if there exists a Schwarz function , which is analytic in with

(1.8)

such that

(1.9)

Indeed, it is known that

(1.10)

Furthermore, if the function is univalent in , then we have the following equivalence:

(1.11)

Analogous to the integral operator defined by Jung et al. [1], Lashin [2] introduced and investigated the following integral operator:

(1.12)

defined, in terms of the familiar Gamma function, by

(1.13)

By setting

(1.14)

we define a new function in terms of the Hadamard product (or convolution):

(1.15)

Then, motivated essentially by the operator , we now introduce the operator

(1.16)

which is defined as

(1.17)

where (and throughout this paper unless otherwise mentioned) the parameters and are constrained as follows:

(1.18)

We can easily find from (1.14), (1.15), and (1.17) that

(1.19)

where is the Pochhammer symbol defined by

(1.20)

Clearly, we know that .

It is readily verified from (1.19) that

(1.21)

(1.22)

By making use of the principle of subordination between analytic functions, we introduce the subclasses , , and of the class which are defined by

(1.23)

Indeed, the above mentioned function classes are generalizations of the general meromorphic starlike, meromorphic convex, meromorphic close-to-convex and meromorphic quasi-convex functions in analytic function theory (see, for details, [3–12]).

Next, by using the operator defined by (1.19), we define the following subclasses , , and of the class :

(1.24)

Obviously, we know that

(1.25)

(1.26)

In order to prove our main results, we need the following lemmas.

Lemma 1.1 (see [13]).

Let . Suppose also that is convex and univalent in with

(1.27)

If is analytic in with , then the subordination

(1.28)

implies that

(1.29)

Lemma 1.2 (see [14]).

Let be convex univalent in and let be analytic in with

(1.30)

If is analytic in and , then the subordination

(1.31)

implies that

(1.32)

The main purpose of the present paper is to investigate some inclusion relationships and integral-preserving properties of the subclasses

(1.33)

of meromorphic functions involving the operator . Several subordination and superordination results involving this operator are also derived.

We begin by presenting our first inclusion relationship given by Theorem 2.1.

Theorem 2.1.

Let and with

(2.1)

Then

(2.2)

Proof.

We first prove that

(2.3)

Let and suppose that

(2.4)

where is analytic in with Combining (1.21) and (2.4), we find that

(2.5)

Taking the logarithmical differentiation on both sides of (2.5) and multiplying the resulting equation by , we get

(2.6)

By virtue of (2.1), an application of Lemma 1.1 to (2.6) yields , that is . Thus, the assertion (2.3) of Theorem 2.1 holds.

To prove the second part of Theorem 2.1, we assume that and set

(2.7)

where is analytic in with . Combining (1.22), (2.1), and (2.7) and applying the similar method of proof of the first part, we get , that is Therefore, the second part of Theorem 2.1 also holds. The proof of Theorem 2.1 is evidently completed.

Theorem 2.2.

Let and with (2.1) holds. Then

(2.8)

Proof.

In view of (1.25) and Theorem 2.1, we find that

(2.9)

(2.10)

Combining (2.9) and (2.10), we deduce that the assertion of Theorem 2.2 holds.

Theorem 2.3.

Let , and with (2.1) holds. Then

(2.11)

Proof.

We begin by proving that

(2.12)

Let . Then, by definition, we know that

(2.13)

with , Moreover, by Theorem 2.1, we know that , which implies that

(2.14)

We now suppose that

(2.15)

where is analytic in with Combining (1.21) and (2.15), we find that

(2.16)

Differentiating both sides of (2.16) with respect to and multiplying the resulting equation by , we get

(2.17)

In view of (1.21), (2.14), and (2.17), we conclude that

(2.18)

By noting that (2.1) holds and

(2.19)

we know that

(2.20)

Thus, an application of Lemma 1.2 to (2.18) yields

(2.21)

that is , which implies that the assertion (2.12) of Theorem 2.3 holds.

By virtue of (1.22) and (2.1), making use of the similar arguments of the details above, we deduce that

(2.22)

The proof of Theorem 2.3 is thus completed.

Theorem 2.4.

Let , and with (2.1) holds. Then

(2.23)

Proof.

In view of (1.26) and Theorem 2.3, and by similarly applying the method of proof of Theorem 2.2, we conclude that the assertion of Theorem 2.4 holds.

In this section, we derive some integral-preserving properties involving two families of integral operators.

Theorem 3.1.

Let with and

(3.1)

Then the integral operator defined by

(3.2)

belongs to the class .

Proof.

Let . Then, from (3.2), we find that

(3.3)

By setting

(3.4)

we observe that is analytic in with . It follows from (3.3) and (3.4) that

(3.5)

Differentiating both sides of (3.5) with respect to logarithmically and multiplying the resulting equation by , we get

(3.6)

Since (3.1) holds, an application of Lemma 1.1 to (3.6) yields

(3.7)

which implies that the assertion of Theorem 3.1 holds.

Theorem 3.2.

Let with and (3.1) holds. Then the integral operator defined by (3.2) belongs to the class .

Proof.

By virtue of (1.25) and Theorem 3.1, we easily find that

(3.8)

The proof of Theorem 3.2 is evidently completed.

Theorem 3.3.

Let with and (3.1) holds. Then the integral operator defined by (3.2) belongs to the class .

Proof.

Let . Then, by definition, we know that there exists a function such that

(3.9)

Since , by Theorem 3.1, we easily find that , which implies that

(3.10)

We now set

(3.11)

where is analytic in with . From (3.3), and (3.11), we get

(3.12)

Combining (3.10), (3.11), and (3.12), we find that

(3.13)

By virtue of (1.21), (3.10), and (3.13), we deduce that

(3.14)

The remainder of the proof of Theorem 3.3 is much akin to that of Theorem 2.3. We, therefore, choose to omit the analogous details involved. We thus find that

(3.15)

which implies that . The proof of Theorem 3.3 is thus completed.

Theorem 3.4.

Let with and (3.1) holds. Then the integral operator defined by (3.2) belongs to the class .

Proof.

In view of (1.26) and Theorem 3.3, and by similarly applying the method of proof of Theorem 3.2, we deduce that the assertion of Theorem 3.4 holds.

Theorem 3.5.

Let with and

(3.16)

Then the function defined by

(3.17)

belongs to the class .

Proof.

Let and suppose that

(3.18)

Combining (3.17) and (3.18), we have

(3.19)

Now, in view of (3.17), (3.18), and (3.19), we get

(3.20)

Since (3.16) holds, an application of Lemma 1.1 to (3.20) yields

(3.21)

that is, . We thus complete the proof of Theorem 3.5.

Theorem 3.6.

Let with and (3.16) holds. Then the function defined by (3.17) belongs to the class .

Proof.

By virtue of (1.25) and Theorem 3.5, and by similarly applying the method of proof of Theorem 3.2, we conclude that the assertion of Theorem 3.6 holds.

Theorem 3.7.

Let with and (3.16) holds. Then the function defined by (3.17) belongs to the class .

Proof.

Let . Then, by definition, we know that there exists a function such that (3.9) holds. Since , by Theorem 3.5, we easily find that , which implies that

(3.22)

We now set

(3.23)

where is analytic in with . From (3.17) and (3.23), we get

(3.24)

Combining (3.22), (3.23), and (3.24), we find that

(3.25)

Furthermore, by virtue of (1.22), (3.22), and (3.25), we deduce that

(3.26)

The remainder of the proof of Theorem 3.7 is similar to that of Theorem 2.3. We, therefore, choose to omit the analogous details involved. We thus find that

(3.27)

which implies that . The proof of Theorem 3.7 is thus completed.

Theorem 3.8.

Let with and (3.16) holds. Then the function defined by (3.17) belongs to the class .

Proof.

By virtue of (1.26) and Theorem 3.7, and by similarly applying the method of proof of Theorem 3.2, we deduce that the assertion of Theorem 3.8 holds.

In this section, we derive some subordination and superordination results associated with the operator . By similarly applying the methods of proof of the results obtained by Cho et al. [15], we get the following subordination and superordination results. Here, we choose to omit the details involved. For some other recent sandwich-type results in analytic function theory, one can find in [16–30] and the references cited therein.

Corollary 4.1.

Let . If

(4.1)

where

(4.2)

then the subordination relationship

(4.3)

implies that

(4.4)

Furthermore, the function is the best dominant.

Corollary 4.2.

Let . If

(4.5)

where

(4.6)

then the subordination relationship

(4.7)

implies that

(4.8)

Furthermore, the function is the best dominant.

Denote by the set of all functions that are analytic and injective on , where

(4.9)

and such that for . If is subordinate to , then is superordinate to . We now derive the following superordination results.

Corollary 4.3.

Let . If

(4.10)

where is given by (4.2), also let the function be univalent in and , then the subordination relationship

(4.11)

implies that

(4.12)

Furthermore, the function is the best subordinant.

Corollary 4.4.

Let . If

(4.13)

where is given by (4.6), also let the function be univalent in and , then the subordination relationship

(4.14)

implies that

(4.15)

Furthermore, the function is the best subordinant.

Combining the above mentioned subordination and superordination results involving the operator , we get the following "sandwich-type results".

Corollary 4.5.

Let . If

(4.16)

where is given by (4.2), also let the function be univalent in and , then the subordination chain

(4.17)

implies that

(4.18)

Furthermore, the functions and are, respectively, the best subordinant and the best dominant.

Corollary 4.6.

Let . If

(4.19)

where is given by (4.6), also let the function be univalent in and , then the subordination chain

(4.20)

implies that

(4.21)

Furthermore, the functions and are, respectively, the best subordinant and the best dominant.