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Applications of a New -Difference Operator in Janowski-Type Meromorphic Convex Functions
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  introduced the -analogue of a Ruscheweyh differential operator and studied its various properties. More applications of this operator can be seen in the paper . Huda and Darus  utilized the -analogue of a Liu-Srivastava operator and defined an integral operator. In somewhat similar way, Mohammed and Darus  introduced a generalized operator along with investigating a class of functions relating to -hypergeometric functions. Later, Seoudy  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
In a similar way, for we get
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 ). 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
Proof. From (17), we have
Thus, we obtain that
From (20), we find that
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.
Since , we have
Theorem 5. If and is of the form (1), then where
Proof. If is in the class , then it satisfies
Since so is in the class with its representation which is given by
Now, by taking absolute on both sides with using the triangle inequality and using (39), we obtain
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
Therefore, by the principle of mathematical induction, we have
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.
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
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),
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)