Abstract
Let and be two nonvanishing holomorphic functions in the open unit disc with . For some holomorphic function we consider the class consisting of normalized holomorphic functions whose ratios and are subordinate to and , respectively. The majorization results are obtained for this class when is chosen either or or and
1. Introduction
In order to better explain the terminology included in our key observations, some of the essential relevant literature on geometric function theory needs to be provided and discussed here. We start with symbol which represents the class of holomorphic functions in the region of open unit disc and if is in then it satisfies the relationship Also, the family contains all univalent functions. Though function theory was started in 1851, in 1916, due to coefficient conjecture provided by Bieberbach [1], this field emerged as a good area of new research. This conjecture was proved by De-Branges [2] in 1985. Many good scholars of the period attempted to prove or disprove this conjecture between the years 1916 and 1985. As a result, they identified several subfamilies of a class of univalent functions linked to various image domains. The families of star-like and convex functions are the most basic, mostly studied, and beautiful geometric representations of these subfamilies, which are described as
In 1970, Roberston [3] established the idea of quasisubordination among holomorphic functions. Two functions are related to the relationship of quasisubordination, denoted mathematically by , if there exist functions such that is holomorphic in with , and satisfying the relationship
Also, by choosing and , we obtain the most useful concepts of geometric function theory known as subordination between analytic functions. In fact, if then, for , the subordination relationship has
By taking , the above definition of quasisubordination becomes the majorization between holomorphic functions and is written mathematically by , for . That is; if a function exists with in such a way that
This idea was introduced by MacGregor [4] in 1967. Numerous articles have been published in which this idea was used. The work of Altintas and Srivastava [5], Cho et al. [6], Goswami and Aouf [7], Goyal and Goswami [8, 9], Li et al. [10], Panigraht and El-Ashwah [11], Prajapat and Aouf [12], and the authors [13, 14] are worth mentioning on this topic.
The general form of the class was studied in 1992 by Ma and Minda [15] and was given by where is a regular function with positive real part and Also, the function maps onto a star-shaped region with respect to and is symmetric about the real axis. They addressed some specific results such as distortion, growth, and covering theorems. In recent years, several subfamilies of the set were studied as a special case of the class . For example, (i)if we take with , then the deduced family is described by the functions of the Janowski star-like family established in [16] and later studied in different directions in [17, 18](ii)the family with was developed in [19] by Sokól and Stankiewicz. The image of the function demonstrates that the image domain is bounded by the Bernoullis lemniscate right-half plan specified by (iii)by selecting the class leads to the family which was explored in [20] while has been produced in the article [21] and later studied in [22](iv)the family with is studied in [23] while and were recently examined by Raza and Bano [24], and Abdullah et.al [25], respectively
Now, let us take the nonvanishing analytic functions and in with Then, the families defined in this article consist of functions whose ratios and are subordinated to and , respectively, for some analytic function with as
We are now going to choose some particular functions instead of and . These choices are and by applying the above-mentioned concepts, we now consider the following classes:
In the present article, we discuss majorization problems for each of the above-defined classes , and
2. Main Results
To prove majorization results for the classes , and we need the following lemma.
Lemma 1. Let and for Then, satisfies the following inequalities:
Proof. If then for some Schwartz function Now, after some easy calculations, we have Let with ,A calculation shows that where Now, we can write Thus, we have Now, consider A calculation shows that the numbers, are the roots of equation (19) in Since is an even function, it is sufficient to consider . We observe that and . Now, we can write Hence, Similarly, one can easily show that Now, from well-known inequality for Schwartz function we obtain
Now, by applying (21), (22), and (23) in (14), we get (12).
Theorem 2. Let the functions and also suppose that in . Then, for , where is the smallest positive root of the equation
Proof. If then by the subordination relationship, we have Now, after simple calculations, we have Now, by using (21), (22), and (23) along with Lemma 1, we obtain From (4), we can write Differentiating the above equality on both sides, we get Also, the Schwartz function fulfils the below inequality: Now, applying (28) and (31) in (30), we have which by putting becomes the inequality where To determine , it is sufficient to choose or, equivalently, where Clearly, when , the function assumes its minimum value, namely, where Next, we have the following inequalities: There exists such that for all where is the smallest positive root of equation (25). Thus, the proof is completed.
Theorem 3. Let and also suppose that in . Then, for , where is the root of the equation:
Proof. If then by using (10) along with the subordination relationship, a holomorphic function in occurs with and in such a way that hold. Now, after simple calculations, we have Using (21), (22), and (23) along with Lemma 1, we obtain Also, with the use of (31) and (46) in (30), we easily get
Now, by the similar lines of Theorem 2 along with the virtue of (33), we easily obtain the required result.
Theorem 4. Let and also suppose that is majorized by in Then, for , where is the positive smallest root of the equation
Proof. Let Then, a holomorphic function in occurs with and so that Now, after simple calculations, we have Using (23), we obtain By virtue of (23) and Lemma 1, we obtain Now, using (31) and (53) in (30), we get
The required result follows directly using similar calculations as Theorem 2 along with the use of (33).
3. Conclusion
For some particular subfamilies of holomorphic functions which are connected with different shapes of image domains, we studied the problems of majorization. These problems can be examined for some other families such as for the families of meromorphic functions as well as for the families of harmonic functions.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare no conflict of interest.
Authors’ Contributions
All authors contributed equally to this research paper.
Acknowledgments
This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, under grant No. (DF-763-830-1441). Therefore, the authors gratefully acknowledge DSR technical and financial support.