Abstract
By making use of differential operators, many distinct subclasses of analytic and meromorphic functions have already been defined and investigated from numerous perspectives. In this article, we investigated several majorization results for the class of meromorphic univalent functions of complex order, defined by differential operator. Moreover, we pointed out some new or known consequences of our results, which is in the form of corollaries.
1. Introduction and Definitions
Let represent the class of meromorphic functions in the form ofwhich are analytic in the punctured disc , where . For the two functions and belonging to the class , there exists a Schwartz function , which is analytic in with and , such that , and the function is subordinate to , written as . The following relationship holds if is univalent:
Because of its use in a variety of mathematical sciences, the study of calculus (quantum calculus) has fascinated and motivated many scholars. One of the primary contributors among all the mathematicians who introduced the concept of calculus theory was Jackson [1, 2]. The formulation of this concept is widely used to investigate the nature of different structures of function theory, such as calculus was used in other branches of mathematics. Though the authors of the first article [3] discussed the geometric nature of starlike functions, Srivastava [4] laid a solid foundation for the use of calculus in the context of function theory. Also, in [5], Srivastava provided a brief overview of basic or calculus operators and fractional calculus operators, as well as their applications in the geometric function theory of complex analysis. Later, the authors [6–8] investigated a number of useful properties for the newly defined linear differential operator, and Mehmood and Sokół [9] discussed the Ruscheweyh differential operator, while Srivastava et al. [10] introduced a generalized operator for meromorphic harmonic functions by using the idea of convolution.
Let . For any nonnegative integer , the -integer number is defined by
In general, we will denotefor a noninteger number . Also, the -number shifted factorial is defined by
Clearly,
Let and . Then, the shifted factorial is defined by
Let . Then, gamma function is as follows:
In a subset of , the derivative (or difference) operator of function is defined byprovided that exists. We can easily observe from the definition of (9) that .
Suppose that , then analog derivative of asor
In 1967, Mac Gregor [11] introduced the notion of majorization as follows.
Definition 1. Let complex-valued functions and be analytic in . We say that is majorized by in and writeif there exists a function , satisfying
Majorization (12) is closely related to the concept of quasi-subordination between analytic functions in . Several researchers have published articles on this topic; for example, Tang et al. [12] gave the concept of majorization for subclasses of starlike functions based on the sine and cosine functions, Arif et al. [13] discussed majorization for various new defined classes, Cho et al. [14] obtained coefficient estimates for majorization, and Tang and Deng [15] defined the majorization problem connected with Liu-Owa integral operator and exponential function. This concept is also defined for valent function by Altintas and Srivastava [16] and for complex order by Altintas et al. [17].
The basic goal of this article is to examine and explain the idea of majorization in the context of the meromorphic function. Many researchers have shown their interest in this site. Goyal and Goswami [18, 19] studied this concept for majorization for meromorphic function with the integral operator, Tang et al. [12] discussed it for meromorphic sin and cosine functions, Bulut et al., Tang et al., and Janani [20–22] explained this concept for meromorphic multivalent functions, Rasheed et al. [23] investigated a majorization problem for the class of meromorphic spiral-like functions related with a convolution operator, and Panigrahi and El-Ashwah [24] defined majorization for subclasses of multivalent meromorphic functions through iterations and combinations of the Liu–Srivastava operator and Cho–Kwon–Srivastava operator and much more. In addition, there are several other articles on this topic [18].
Here is the definition of our main function.
Definition 2. A function is said to be in the class of meromorphic functions of complex order in , ifNow, we are going to choose some particular functions instead of . These choices areand by applying the above-mentioned concepts, we now consider the following cases:In the present article, we discussed majorization problems for each of the above-defined classes of .
2. Majorization Problem for the Classes
We state the following analogue of the result given by Nehari [25] and Salvakumaran et al. [26].
Lemma 1 (see [27]). If the function is analytic and in , then
Theorem 1. Let the function and suppose if is majorized by in , i.e.,
Then, for ,where is the smallest positive root of the following equation:
Proof. Since , by using (19), we can find if andBy Lemma 1, there exists a bounded analytic function in andwith . From (24), we obtainLet Re with and . By simple calculation, we show thatWe easily see that the equation,has five roots in , that is, and . Because , we just need to consider . Also, noticing that , andThus, we haveFrom (24) and (28), we find thatSince is majorized by in , from (13), we haveBy applying derivative on the previous equation w.r.t as in [27] and then multiplying by , we haveNoting that is the Schwartz function, so in , for all , satisfies the analogue result given by [25] proved in Lemma 1.Now, using (29) and (32) in (31), we haveBy settingwe get the inequalityWe defineTo determine , it is sufficient to chooseequivalently,whereClearly, when , the above function assumes its minimum value, namely,whereNext, we obtain the following inequalities:There exists such that for all , where is the smallest positive root of (20). The proof of Theorem 1 is completed.
Theorem 2. Let the function and suppose if is majorized by in , i.e.,
Then, for ,where is the smallest positive root of the following equation:
Proof. Since , from (11) and the subordination relationship, we see thatwhere is as same as in (24). Similar to (28), we can verify thatwhere Re with and .
Combining (46a) and (47), it is easy to see thatBy virtue of (32) as well as (48) in (31), we immediately obtainIn succession, according to (34) and just as the proof of Theorem 1, we can deduce the required result (45). Hence, we have completed the proof of Theorem 2.
Theorem 3. Let the function and suppose if is majorized by in , i.e.,
Then, for ,where is the smallest positive root of the following equation:
Proof. Let . Then, from definition (16) in terms of the Schwartz function, we havewhich impliesNow, as Re with and we havewhich impliesNow, as in Theorem 2, we use (32), as well as (56) in (31), and we obtainLet us take and , ; we obtainWe defineTo determine , it is sufficient to chooseequivalently,whereThis clearly shows the result that, when , the above function assumes its minimum value, namely,whereNext, we obtain the following inequalities:there exists such that for all , where is the smallest positive root of (52). The proof of Theorem 3 is completed.
Theorem 4. Let the function and suppose if is majorized by in , i.e.,
Then, for ,where is the smallest positive root of the following equation:
Proof. Since , we readily obtained from definition (17) thatwhere is the well-known class of bounded analytic functions in such thatFrom (69) and (70) and making use of (71), we obtainNow, just like the above theorems, we use (32) as well as (72) in (31), and we obtainLet us take and , ; we obtainWe defineTo determine , it is sufficient to chooseequivalently,whereClearly, when , the above function assumes its minimum value, namely,whereNext, we obtained the following inequalities:there exists such that for all , where is the smallest positive root of (68). The proof of Theorem 4 is completed.
3. Conclusion
In this article, we investigated majorization and other results for such subclasses of meromorphic functions, such as the meromorphic univalent function of complex order associated with the differential operator. We also highlighted some special cases and new consequences of our main results. In order to conclude our current study, we attract the attention of interested readers to the potential of examining the fundamental or quantum (or ) extensions of the results obtained in this work. Applications of the q th majorization in the real world will be an interesting and encouraging future study for researchers.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors participated in every stage of the research, and all authors read and approved the final manuscript.
Acknowledgments
This work was supported by Universiti Kebangsaan Malaysia.