Applications of a New -Difference Operator in Janowski-Type Meromorphic Convex Functions

Abstract

The main aim of the present article is the introduction of a new differential operator in -analogue for meromorphic multivalent functions which are analytic in punctured open unit disc. A subclass of meromorphic multivalent convex functions is defined using this new differential operator in -analogue. Furthermore, we discuss a number of useful geometric properties for the functions belonging to this class such as sufficiency criteria, coefficient estimates, distortion theorem, growth theorem, radius of starlikeness, and radius of convexity. Also, algebraic property of closure is discussed of functions belonging to this class. Integral representation problem is also proved for these functions.

1. Introduction and Definitions

Let denote the family of all meromorphic -valent functions that are analytic in the punctured disc and obeying the normalization

Also, let denote the well-known family of meromorphic -valent convex functions of order and defined as

For the -difference operator or -derivative of a function is defined by

It can easily be seen that for , where stands for the set of natural numbers and , where

For any nonnegative integer , the -number shift factorial is defined by

Also, the -generalized Pochhammer symbol for is given by

In (3), if , then this operator becomes the conventional derivative in the classical calculus, so the limits can be generalized by introducing the parameter , with condition , and all such concepts, which have been developed thus, are known as quantum calculus (-calculus). Many physical phenomena are better explained using this generalized operator, and as a result, this field attracted a lot of the researchers due to its various applications in the branches of mathematics and physics (see details in [1, 2]). Jackson [3, 4] was the pioneer of this field, who gave some applications of -calculus and introduced the -analogues of derivative and integral. Aral and Gupta [1, 2, 5] defined an operator, which is known as -Baskakov Durrmeyer operator by using -beta functions. The generalization of complex operators known as -Picard and -Gauss-Weierstrass singular integral operators was discussed by Aral and Anastassiu in [6–8]. Later, Kanas and Rducanu [9] introduced the -analogue of a Ruscheweyh differential operator and studied its various properties. More applications of this operator can be seen in the paper [10]. Huda and Darus [11] utilized the -analogue of a Liu-Srivastava operator and defined an integral operator. In somewhat similar way, Mohammed and Darus [12] introduced a generalized operator along with investigating a class of functions relating to -hypergeometric functions. Later, Seoudy [13] estimated coefficient bounds for some -starlike and -convex functions of complex order. Recently, Arif and Ahmad defined a new -differential operator for meromorphic multivalent functions and investigated classes related to -meromorphic starlike and convex functions in their articles [14, 15].

In this article, we introduce a new -differential operator for meromorphic functions and use this operator to define and study some properties of a new family of meromorphic multivalent functions associated with circular domain.

We now define the differential operator by where

Using (1), we can easily obtain

We take

In a similar way, for we get

From (8) and (11), we get the following identity:

We now define a subfamily of by using the operator as follows.

Definition 1. Forandwe defineto be in the classif it satisfiesHere, the relation symbol “” is used for the subordinations.

We see that for particular values of , and , we get some of the well-known classes few of which are listed below: (1)For and , we get the class of meromorphic multivalent convex functions associated with Janowski functions denoted by (2)For and , we get , the class of meromorphic multivalent convex functions in -analogue(3)For , and , we get the class of meromorphic multivalent convex functions denoted by (4)For , and , we get , the class of meromorphic convex functions

It can easily be verified that a function will be in the class , if and only if

The following lemma is used in our main results.

Lemma 2 (see [16]). Let be analytic in and have the form and is analytic and convex in with series representation

So if , then , for

2. Main Results and Their Consequences

In this section, we start with sufficiency criteria for this newly defined class and then, we give the coefficient estimates for the functions belonging to this class. The following lemma is proved which will be used in this section.

Lemma 3. Suppose that the sequence is defined by

Then,

Proof. From (17), we have

Thus, we obtain that

From (20), we find that

Thus,

In conjunction with (17), we complete the proof of Lemma 3.

Theorem 4. Ifis of the form (1), then it will be in the classif and only if the inequalityis satisfied.

Proof. For , we need to prove the inequality (14). For this, consider By using (8) and with the help of (3) and (11), Now, if we use the inequality (23), then and this completes the direct part of the proof.

Conversely, let and be of the form (1); then, from (14), we have for ,

Since , we have

Now if the values of are chosen on the real axis, then is real. Using some calculations in the inequality (28) and letting through real values, we finally get (23).

Theorem 5. If and is of the form (1), then where

Proof. If is in the class , then it satisfies

Now, let

Since so is in the class with its representation which is given by

Now,

But

Now, using Lemma 2, we get now putting the series expansions of and in (34), simplifying and comparing the coefficients of on both sides which implies that

Now, by taking absolute on both sides with using the triangle inequality and using (39), we obtain

Using the notation (31) and (32) implies that

Now, we define the sequence as follows:

In order to prove that we use the principle of mathematical induction. It is easy to verify that

Thus, assuming that we find from (44) and (48) that

Therefore, by the principle of mathematical induction, we have

By means of Lemma 3 and (45), we know that

Combining (50) and (51), we readily get the coefficient estimates (30).

3. Closure Theorems

Let the functions be defined by

Theorem 6. Let the functions defined by (52) be in the class . Then, the function where

Proof. From (53), we have

By Theorem 4, we have

Hence, by Theorem 4,

Theorem 7. The class is closed under convex combination.

Proof. Let the function given by (52) be in the class . It is enough to show that is in the class Since for by Theorem 4, we have Hence, by Theorem 4,

Theorem 8. Let the function given by (52) belong to ; then, their weighted mean is also in the class , where is defined by

Proof. From (59), one can easily write

To prove , we consider

Hence, by Theorem 4,

4. Distortion Theorem

In the next two results, we shall discuss the growth and distortion theorems for our newly defined class of functions.

Theorem 9. Ifis in the classand has the form (1), then for, we havewhere

Proof. As for , we have for and Similarly, Now, if , then by (23), But we know that Hence, which implies that Now, by putting this value in (65) and (66), we get the required proof.

Theorem 10. Let and have the form (1). Then, for , where

Proof. From the help of (3) and (4), we can write Since implies that for and , hence Similarly, Since is in the class , so by (23), we have the inequality from which it can be deduced that but it can easily be seen that which implies Now, using this inequality in (74) and (75), we obtain the required proof.

5. Integral Representation

Theorem 11. Let the functiongiven by (1) be in the classThen, the functionrepresented byis in the class

Proof. From (1),

Then,

Consider since

Therefore, by Theorem 4,

6. Radius Problems

The following results are about the radii of convexity and starlikeness for the functions of the class

Theorem 12. If, thenfor, where

Proof. Let . To prove , we only need to show

Using (1) along with some simple computation yields

As is in the class , so we have from (23)

Equivalently,

Now, inequality (86) will be true, if the following holds: which implies that and so

We get the required condition.

Theorem 13. Let. Then,for, where

Proof. We know that , if and only if

Using (1) and with some simplification, we get

Now, from (23), we can easily obtain that

For inequality (94) to hold, it will be enough if which implies that and hence,

Thus, we obtain the required result.

7. Conclusion

The applications of -calculus have been the focus point in the recent times in various branches of mathematics. This article introduces a new operator in -analogue for meromorphic multivalent functions. Then, a new subclass of multivalent convex functions is defined and studied for some of its geometric properties like sufficient conditions, coefficient estimates, and distortion. Also, problems of closure and integral representation are discussed in detail. Many other classes can be defined using this operator which will open a lot of new opportunities for research in this and related fields.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

All authors jointly worked on the results, and they read and approved the final manuscript.