Abstract
In this article, we introduce a new subclass of analytic functions utilizing the idea of Mittag-Leffler type Poisson distribution associated with the Janowski functions. Further, we discuss some important geometric properties like necessary and sufficient condition, convex combination, growth and distortion bounds, Fekete-Szegö inequality, and partial sums for this newly defined class.
1. Introduction, Definitions, and Motivation
Let represent the collections of holomorphic (analytic) functions defined in the open unit disc:such that the Taylor series expansion of is given by
By convention, stands for a subclass of class comprising of univalent functions of the form (2) in the open unit disc . Let represent the class of all functions that are holomorphic in with the conditionand has the series representation
Next, we recall the definition of subordination, for two functions , we say is subordinated to and is symbolically written asif there exists an analytic function with the propertiessuch that
Further if , then the above condition becomes
Now, recall the definition of convolution, let given by (2) and given bythen their convolution denoted by is given by
The most important and well-known family of analytic functions is the class of starlike functions denoted by and is defined as
Next, for , Janowski [1] generalized the class as follows.
Definition 1. A function with property that is placed in the class if and only if
Janowski also proved that for a function , a function belongs to if the following relation holds
Also, function of form (2) belongs to the class if
Kanas et al. (see [2, 3]; see also [4, 5]) were the first to define the conic domain as follows:
Moreover, for fixed , represents the conic region bounded successively by the imaginary axis . For , it is a parabola, and for , it is the right-hand branch of the hyperbola, and for , it represents an ellipse.
For these conic regions, the following functions play the role of extremal functions:whereand is chosen such that . Here is Legendre’s complete elliptic integral of first kind and , that is, is the complementary integral of . Assume that
Then, in [6], it has been shown that, for (16), one can havewherewith
Definition 2. A function of the form (2) is said to be in class , if and only if
Noor and Malik [7] combined the concepts of the Janowski functions and the conic regions and gave the following definition.
Definition 3. A function is said to be in the class if and only if
Geometrically, takes all values in domain , which is defined as followsthe domain represents conic-type regions, which was introduced and studied by Noor and Malik [7] and is further generalized by the many authors, see for example [8] and the references cited therein.
Definition 4 [7]. A function is said to be in the class if and only if
The generalized exponential series:is one special-type function with single parameter , was introduced by Mittag-Leffer (see [9]), and is therefore known as the Mittag-Leffler function. Another function with two parameters and having similar properties to those of Mittag-Leffler function is given byand was introduced by Wiman [10, 11] Agrawal [12], and by the many other (see for example [13–16]). It can be seen that the series converges for all finite values of if
During the last years, the interest in Mittag-Leffler type functions has considerably increased due to their vast potential of applications in applied problems such as fluid flow, electric networks, probability, and statistical distribution theory. For a detailed account of properties, generalizations and applications of functions (27) and (28), one may refer to [17–19] and [20].
Geometric properties including starlikeness, convexity, and close-to-convexity for the Mittag-Leffler function were recently investigated by Bansal and Prajapat in [21]. Differential subordination results associated with generalized Mittag-Leffler function were also obtained in [22].
A variable is said to be Poisson distributed if it takes the values with probabilities respectively, where is called the parameter. Thus,
It is easy to see that (30) is a mass probability function because
The power series given bywhich coefficients are probabilities of Poisson distribution is introduced by Porwal [23]. We can see that by ratio test the radius of convergence of is infinity. Porwal [23] also defined and introduced the following series:
The works of Porwal [23] motivate researchers to introduced a new probability distribution if it assumes the positive values and its mass function is given by (30) (see for example [24–26]).
It was Porwal and Dixit [24] who studied and connected the Poisson distribution and the well-known Mittag-Leffer function systematically. They called it the Mittag-Leffer type Poisson distribution and prevailed moments. The Mittag-Leffer type Poisson distribution is given by (see [24])where is a normalized function of class , since
The probability mass function for the Mittag-Leffer type Poisson distribution series is given bywhere is given by (28). It is worthy to note that the Mittag-Leffer type Poisson distribution is a generalization of Poisson distribution. Furtheremore, Bajpai [27] also studied and obtain some necessary and sufficient conditions for this distribution series.
Very recently, using the Mittag-Leffer type Poisson distribution series, Alessa et al. [28] defined the convolution operator aswhere
Using this convolution operator, they defined and studied a new subclass of analytic function systematically. They obtained certain coefficient estimates, neighborhood results, partial sums, and convexity and compactness properties for their defined functions class.
In recent years, binomial distribution series, Pascal distribution series, Poisson distribution series, and Mittag-Leffer type Poisson distribution series play important role in the geometric function theory of complex analysis. The sufficient ways were innovated for certain subclasses of starlike and convex functions involving these special functions (see for example [26, 29–32]). Motivated by the abovementioned works and from the work of Alessa et al. [28], in this article, by mean of certain convolution operator for Mittag-Leffer type Poisson distribution, we shall define a new subclass of starlike functions involving both the conic-type regions and the Janowski functions. We then obtain some interesting properties for this newly defined function class including for example necessary and sufficient condition, convex combination, growth and distortion bounds, Fekete-Szegö inequality, and partial sums. We now define a subclass of Janowski-type starlike functions involving the conic domains by mean of certain convolution operator for Mittag-Leffer type Poisson distribution as follows.
Definition 5. For , a function is in class ifwhere
For the proofs of our key findings, we need the following lemma.
Lemma 6 [33]. Let have the series expansion of form (4), then
2. Main Results
Theorem 7. Let and is of the form (2), then
The result is sharp for the function given in (51).
Proof. Suppose that inequality (42) holds true, then it is enough to show thatFor this, considerAs we have settherefore, after some straightforward simplifications, we haveBy using (42), the above inequality is bounded above by 1, and hence, the proof is completed.☐
Example 8. For the functionsuch thatwe haveHence, and the result is sharp.
Corollary 9. Let the function of the form (2) be in the class Then,The result is sharp for the function given by
Theorem 10. The class is closed under convex combination.
Proof. Let such thatIt is enough to show thatAsNow, by Theorem 7, we haveHence,which completes the proof.☐
Theorem 11. Let , then for
The result is sharp for the function given in (51) for .
Proof. Let . Using Theorem 7, we can deduce the following inequity:Similarly,☐
Theorem 12. Let , then for
The result is sharp for the function given in (51) for .
Proof. The proof is quite similar to Theorem 11, so left for reader.☐
Now, we evaluate a kind of Hankel determinant problem, which is also known as the Fekete-Szegö functional.
Theorem 13. If is of the form (2) and belongs to , thenwhere and are defined by (19) and (20), respectively.
Proof. To prove inequality (61), we letthen from (26), we haveThus, ifthen by simple computation, we getNow, from (63), there exists an analytic function such thatis analytic andin open unit disc . Also, we havewhereAfter comparing the (69) and (70), we getNow, by making use of (71) and (72), in conjunction with Lemma, we havewhich is the required result.☐
3. Partial Sums
In this section, we will examine the ratio of a function of form (2) to its sequence of partial sumswhen the coefficients of are sufficiently small to satisfy condition (42). We will determine sharp lower bounds for
Theorem 14. If of form (2) satisfies condition (42), thenwhere
The result is sharp for the function given in (51).
Proof. It is easy to verify thatThus, in order to prove the inequality (76), we setWe now setThen, we find after some suitable simplification thatThus, clearly, we find thatBy applying the trigonometric inequalities with , we arrived at the following inequality:We can now see thatif and only ifwhich implies thatFinally, to prove the inequality in (76), it suffices to show that the left-hand side of (87) is bounded above by the following sum:which is equivalent toIn virtue of (89), the proof of inequality in (76) is now completed.
Next, in order to prove the inequality (77), we setwhereThis last inequality in (91) is equivalent toFinally, we can see that the left-hand side of the inequality in (92) is bounded above by the following sum:so we have completed the proof of the assertion (77).☐
We next turn to ratios involving derivatives.
Theorem 15. If of the form (2) satisfies condition (42), thenwhere is given by (78). The result is sharp for the function given in (51).
Proof. The proof of Theorem 15 is similar to that of Theorem 14; we here choose to omit the analogous details.☐
4. Concluding Remarks and Observations
In our present work, by making use of the idea of Mittag-Leffler type Poisson distribution, we have defined and studied certain new subclasses of starlike functions involving the Janowski functions. Further, we have discussed some important geometric properties like necessary and sufficient condition, convex combination, growth and distortion bounds, Fekete-Szegö inequality, and partial sums for this newly defined functions class.
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.