1. Preliminaries
The q-analysis is a generalization of the ordinary analysis that does not employ limit notation. Jackson presented the use and application of the q-calculus in [1]. The q-analogue of the derivative and integral operator were first defined in [2], along with some of their applications. The multifaceted uses of the q-derivative operator make it extremely important in the study of geometric function theory. Initially, Ismail et al. proposed the concept of q-extension of the class of q-starlike functions in [3]. Later, a number of mathematicians studied q-calculus in the context of geometric functions theory: the Ruscheweyh differential operator’s q-analogue was first presented in [4]; some applications for multivalent functions were examined in [5,6]; q-starlike functions associated with the generalized conic domain, by applying the convolution concept was investigated in [7]; an operator associated with q-hypergeometric function was investigated in [8], and so forth. Numerous authors have recently released a number of articles that examine a class of q-starlike functions from different perspectives (see [9,10,11,12,13,14]).
An extension of the q-calculus to the (p, q)-calculus, was taken into consideration by the researchers. The (p, q)-number, was first examined around the same time (1991) and subsequently on its own by [15,16,17,18]. Fibonacci oscillators were studied with the presentation of the (p, q)-number in [15]. The investigation of the (p, q)-number in [16] allows for the construction of a (p, q)-Harmonic oscillator. The (p, q)-number was investigated in [17] as a means of combining different types of q-oscillator algebras and in [18], the (p, q)-number was examined in order to determine the (p, q)-Stirling numbers.
Building on the aforementioned publications, since 1991, a large number of scientists have investigated (p, q)-calculus in a range of research domains. The results in [19] provided a syntax for embedding the q-series into a (p, q)-series. They also investigated (p, q)-hypergeometric series and found some results corresponding to (p, q)-extensions of the known q-identities. The q-identities are extended correspondingly to yield the (p, q)-series (see, e.g., [20]). We provide some basic definitions of the (p, q)-calculus concepts used in this paper. The (p, q)-bracket number is given by which is an extension of q-number (see [2]), that is . Note that is symmetric and if p = 1, then .
Let = be the open unit disk, and be the complex plane. Let represent the natural number set and be the real number set.
([21]). Let φ be a function defined on and . Then, the (p, q)-derivative of φ is defined by
and , provided exists.
We note that and . Therefore, as . For any constants and , it is obvious that . The product rules and quotient rules are satisfied by the (p, q)-derivative (see [22]). The exponential functions are used to define the (p, q)-analogues of many functions, including sine, cosine, and tangent, in the same way as their well-known Euler expressions. Duran et al. [23] examined the (p, q)-derivatives of these functions.
The set of functions , which are regular in and have the following form, is represented by :
(1)
with , and we denote a sub-set of containing univalent functions in by . If is of the form (1), then(2)
The Koebe theorem (see [24]) states that the inverse of each function in is given by
(3)
satisfying and . In , a member of given by (1) is called bi-univalent if and . The set of such functions in is represented by . , and are some of the functions in the family. Nevertheless, despite being in , , , and the Koebe function does not belong to . For a concise analysis and to discover some of the remarkable characteristics of the family , see [25,26,27,28] and the citation provided in these papers. Srivastava et al. [29] introduced several subclasses of the family , which are similar to the well-known subclasses of the family . As follow-ups, numerous authors have since examined a number of different subfamilies of (see, for instance, [30,31,32,33,34]).Several subclasses of the class were studied using the (p, q)-calculus. In [35], the (p, q)-derivative operator and the subordination principle were used to introduce the new generalized classes of (p, q)-starlike and (p, q)-convex functions. The Fekete-Szegö inequalities are also examined and the (p, q)-Bernardi integral operator for analytic functions is defined. The (p, q)-Bernardi integral operator was used to obtain some applications of the main results. Several studies have also introduced and investigated novel subclasses of the class related to the (p, q)-differential operator (see [36,37,38,39,40,41]).
Let and be polynomials with real coefficients. For, , the generalized bivariate Fibonacci polynomials (GBFP) are defined by the recurrence relation:
(4)
where , and The generating function of GBFP is (see [42])(5)
For specific selections of and , GBFP leads to various known polynomials (see [43]). Readers with an interest in GBFP can find a brief history and extensive information in [44] and its references. Interesting findings regarding coefficient estimates and Fekete–Szegö functional have been presented in [45,46,47] for members of certain subclasses of associated with GBFP.For brevity, we write hereafter that and . and are evident from (4).
By specializing k and l, many polynomial sequences can be inferred from GBFP. They are as follows: (i) the bivariate Fibonacci polynmials are obtained if and ; (ii) the Pell polynomials are achieved if and ; (iii) we derive the Jacobsthal polynomials if and ; and (iv) we arrive at the Fermat polynomials, if and , and so forth.
For , holomorphic in , is subordinate to , if there is a Schwarz function that is holomorphic in with and , such that . The notation or denotes this subordination. Specifically, when , we have
We present two new families of subordinate to GBFP as in (4), with the generating function as in (5). These families are motivated by the aforementioned trends on coefficient-related problems and the Fekete–Szegö functional [48] on certain subclasses of linked to GBFP. is as in (5), , an inverse function as in (3),, and are assumed throughout this paper, unless otherwise mentioned.
A function ϕ given by (1), which is said to be in the set , if
and
where
(6)
For specific choices of and , the family includes many new and existing subfamilies of . This is shown below:
1. let . Then is the collection of members of that satisfy
and
where is as mentioned in (6).
2. Assume that . Then and is the collection of elements of that satisfy
and
where is as mentioned in (6).
3. If and in the class , then we obtain a subset which is the collection of members of that satisfy
and
where is as mentioned in (6).
4. Let . Then is the set of elements of that satisfy
and
where is as mentioned in (6).
(i). , (ii). , and (iii).
A function ϕ given by (1) is said to a member of the set , , if
and
where is as mentioned in (6).
The family contains numerous new and existing subfamilies of for specific choices of and , as shown below:
Let . Then, is the collection of elements of that satisfy
and
where is as mentioned in (6).
Let . Then, is the family of members of that satisfy
and
where is as mentioned in (6).
If and in the class , then we get a subclass of functions satisfying
and
where is as mentioned in (6).
Suppose . Then, is the set of functions that satisfy
where is as mentioned in (6).
(ii).
Estimates for bounds on , , and for functions in the class are found in Section 2. The upper bounds for , , and for functions in the class are derived in Section 3. Along with pertinent connections to the published research, interesting results are also presented.
2. Results for the Class
First, for , the class as defined in Definition 2, we find the coefficient estimates.
Let . If ϕ , then
(7)
(8)
and for(9)
where(10)
(11)
(12)
and(13)
Let . Then, due to Definition 2, we obtain
(14)
and(15)
where(16)
are some holomorphic functions with and . It is known that(17)
From (14)–(16), it follows that
(18)
and(19)
In light of (4), Equations (18) and (19) can be expressed as follows.(20)
and(21)
Comparing (20) and (21), we have
(22)
(23)
(24)
and(25)
where P, Q, and R are as mentioned in (11), (12) and (13), respectively. From (22) and (24), we easily obtain(26)
and also(27)
By adding (23) and (25), we obtain(28)
The value of from (27) is substituted in (28), yielding(29)
Applying (17) to the coefficients and yields (7).We deduct (25) from (23) to get the bound on :
(30)
Then in view of (26) and (27), (30) becomesand applying (17) for the coefficients , , and we obtain (8).
From (29) and (30), for , we obtain
where
Clearly
which results in (9) and J as in (10). □
Let in the above theorem. Then, for any function the upper bounds of , and are given by (7), (8), and (9), respectively, with in (11), (12), and (13), respectively. are to be used in place of for J in (10).
Let in the above theorem. Then, for any function the upper bounds of and are given by (7), (8), and (9), respectively, where . are to be used in place of and R for J in (10).
From Theorem 1, taking and , we obtain
Let . If , then
and for
where
Here are a few examples of the above corollary’s special cases.
Letting in the class , we obtain a subclass of functions ϕ satisfying
and
where is as mentioned in (6).
The class is not an empty set, which is an important point to note. Consider the following example, where . The function = is in the class . Because, its inverse . So, we obtain that (see [46]).
Let . If , then
and for
where
Letting in the class , we obtain a subclass of functions ϕ satisfying
where is as mentioned in (6).
Let . If then
and for
where
in Corollary 4 yields Corollaries 3 and 7 of Yilmaz and Aktas [43].
Taking in the above theorem, we obtain
Let . If ϕ, then
and for
where
and P, Q, and R are as detailed in (11), (12), and (13), respectively.
3. Results for the Class
First, for the class , as defined in Definition 3, we determine the coefficient estimates.
Let . If ϕ, then
(31)
(32)
and for(33)
where(34)
(35)
(36)
and(37)
Let . Following that, due to Definition 3, we obtain
(38)
and(39)
where and are analytic functions as given in (16), with , and it is known that(40)
It follows from (38), (39) and (16) that
(41)
and
(42)
In light of (4), Equations (41) and (42) can be expressed as follows.
(43)
and
(44)
Comparing (43) and (44), we have
(45)
(46)
(47)
and(48)
where L, N, and M are as in (35), (36) and (37), respectively. From (45) and (47), we easily obtain(49)
and also(50)
The bound on , is obtained by adding (46) and (48).(51)
Substituting the value of from (50) in (51), we obtain(52)
Applying (40) for the coefficients and , we obtain (31).To obtain the bound on , we deduct (48) from (46):
(53)
Then, in view of (49) and (50), (53) becomesand applying (40) for the coefficients , , and we obtain (32).
From (52) and (53), for , we obtain
where
Clearly
from which we conclude (33) with as in (34). □
Let in the above theorem. Then, for any function the upper bounds of , and are given by (31), (32) and (33), respectively, with and . are to be used in place of for in (34).
Let in the above theorem. Then, for any function the upper bounds of , and are given by (31), (32) and (33), respectively, with . are to be used in place of for in (34).
From Theorem 2, taking and , we obtain
Let . If then
and for
where
Here are a few examples of the above corollary’s special cases.
Letting in the class , we obtain a subclass of functions ϕ satisfying
and
where is as mentioned in (6), .
Let . If then
and for
where
Letting , then is the subclass of functions ϕ satisfying
where is as mentioned in (6).
Let . If then
and for
in Corollary 11 yields Corollaries 2 and 6 of Yilmaz and Aktas [43].
Letting , then is the subclass of functions ϕ satisfying
where is as mentioned in .
The class is not an empty set, which is an important point to note. Consider the following example, where . The function = is in the class . Because, its inverse . So, we get that (see [46]).
Let . If , then
and for
Let . If ϕ, then
and for
where L, N, and M are as mentioned in (31), (32), and (33), respectively.
4. Conclusions
The current work defines two subfamilies of associated with GBFP and derives upper bounds of , and the Fekete–Szegö functional for functions in these subfamilies. We have been able to draw attention to several implications by altering the parameters in Theorem 1 and Theorem 2. Relevant connections to the current research are also discovered. The subfamilies this paper studies may encourage researchers to concentrate on the (p, q) -derivative operator. Future studies might look into extending obtained results to fractional derivatives, higher-order Hankel determinants, or Toeplitz determinants. The findings presented here give these advancements a strong foundation and emphasize the importance of geometric factors in the study of analytic function theory.
S.R.S. and B.A.F.: analysis, methodology, implementation, and original draft; D.B. and L.-I.C.: software and conceptualization, validation, resources, and editing. All authors have read and agreed to the published version of the manuscript.
Data are contained within the article.
The authors declare that they have no conflicts of interest.
Footnotes
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Abstract
Making use of generalized bivariate Fibonacci polynomials, we propose two families of regular functions of the type
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1 Department of Infomation Science and Engineering, Acharya Institute of Technology, Bengaluru 560 107, Karnataka, India;
2 Department of Mathematics, Faculty of Science, Al al-Bayt University, Mafraq, Jordan;
3 Department of Computing, Mathematics and Electronics, “1 Decembrie 1918” University of Alba Iulia, 510009 Alba Iulia, Romania
4 Department of Mathematics, Tehnical University of Cluj-Napoca, 400114 Cluj-Napoca, Romania