Some applications of q-difference operator involving a family of meromorphic harmonic functions

In this paper, we establish certain new subclasses of meromorphic harmonic functions using the principles of q-derivative operator. We obtain new criteria of sense preserving and univalency. We also address other important aspects, such as distortion limits, preservation of convolution, and convexity limitations. Additionally, with the help of sufficiency criteria, we estimate sharp bounds of the real parts of the ratios of meromorphic harmonic functions to their sequences of partial sums.


Introduction and definitions
Univalent harmonic functions are a new research area that was initially developed by Clunie and Sheil-Small [15]; see also [40]. The significance of such functions is attributed to their usage in the analysis of minimal surfaces and in problems relevant to applied mathematics. Hengartner and Schober [18] introduced and analyzed some specific types of harmonic functions in the region D={z ∈ C : |z| > 1}. They proved that a harmonic complexvalued sense-preserving univalent mapping f defined in D and obeying f (∞) = ∞ must satisfy the following representation: f (z) = G 1 (z) + G 2 (z) + A log |z|, (1.1) where G 1 (z) = μ 1 z + ∞ n=1 a n z -n and G 2 (z) = μ 2 z + ∞ n=1 b n z -n with 0 ≤ |μ 2 | < |μ 1 | and A ∈ C. In 1999, Jahangiri and Silverman [26] gave adequate coefficient criteria for functions of type (1.1) to be univalent. They also provided necessary and sufficient coefficient criteria within certain constraints for functions to be harmonic and starlike. Using this idea, the authors of [24] contributed a certain family of harmonic closeto-convex functions involving the Alexander integral transform. In 2000, Jahangiri [22] and Murugusundaramoorthy [35,36] analyzed the families of meromorphic harmonic function in D. In [12,14] the authors used the technique developed by Zou and his coauthors in [55] to examine the natures of meromorphic harmonic starlike functions with respect to symmetrical conjugate points in the punctured disc D * ={z ∈ C : 0 < |z| < 1} = D\{0}. Particularly, in [14] a sharp approximation of the coefficients and a structural description of these functions are also determined. To understand the basics in a more clear way, we denote by H the family of harmonic functions f that can be represented in the series form where h and g are holomorphic functions in D * and D of the form and |a n | ≥ 1, |b n | ≥ 1 (n = 2, 3, . . .).
Also, let us denote by M H the set of complex-valued functions f ∈ H that are sense preserving and univalent in D * . Clearly, if g(z) ≡ 0 (z ∈ D), then M H matches with the collection M of holomorphic univalent normalized functions in D. The above foundational papers opened a new door for the researchers to add some input in this area of function theory. In this regard, we consider the collections of meromorphic harmonic starlike and meromorphic harmonic convex functions in D *

MS
where the notation ≺ shows the familiar subordination between the holomorphic functions, and Furthermore, many subfamilies of meromorphic harmonic functions have also been established by some well-known researchers; for example, see Bostanci [11], Bostanci and Öztürk [13], Öztürk and Bostanci [38], Wang et al. [54], Al-dweby and Darus [3], Al-Shaqsi and Darus [4], Ponnusamy and Rajasekaran [39], Ahuja and Jahangiri [2], Al-Zkeri and Al-Oboudi [5], Stephen et al. [53], and Khan et al. [32]. The investigation of q-calculus (q stands for quantum) fascinated and inspired many scholars due its use in various areas of the quantitative sciences. Jackson [20,21] was among the key contributors of all the scientists who introduced and developed the qcalculus theory. Just like q-calculus was used in other mathematical sciences, the formulations of this idea are commonly used to examine the existence of various structures of function theory. The first paper in which a link was established between certain geometric nature of the analytic functions and the q-derivative operator is due to the authors [19]. For the usage of q-calculus in function theory, a solid and comprehensive foundation is given by Srivastava [43]. After this development, many researchers introduced and studied some useful operators in q-analog with applications of convolution concepts. For example, Kanas and Răducanu [27] established the q-differential operator and then examined the behavior of this operator in function theory. For more applications of this operator, see [1,7,17]. This operator was generalized further for multivalent analytic functions by Arif et al. [8] and later studied in [30,41,51]. Analogous to q-differential operator Arif et al. [9] and Khan et al. [33] contributed the integral operators for analytic and multivalent functions, respectively. Similarly, in [6] the authors developed and analyzed operators in q-analog for meromorphic functions. Also, see the survey-type paper [44] on quantum calculus and its applications. In 2021, Srivastava, Arif, and Raza [46] introduced and studied a generalized convolution q-derivative operator for meromorphic harmonic functions. Using these operators, many researchers contributed some good papers in this direction in geometric function theory; see [16,23,25,28,29,31,37,45,50,52].
See also [10,48,49], and [47] for some recent applications of the q-difference operators in the theory of q-series and q-polynomials.
Similarly, we denote In this paper, we learn some nice properties for the currently established families including distortion limits, univalency criteria, partial-sum problems, sufficiency criteria, convexity conditions, and preserving convolutions.

Necessary and sufficient conditions
Proof If f (z) = 1 z , then we have h(z) = 1 z and g(z) = 0. This implies that Hence by the result of Lewy [34] the function f in D * is locally univalent and orientationpreserving. Now we show that f is univalent in D * . Let z 1 , z 2 ∈ D * with z 1 = z 2 . Then To It is easy to find that qD q H f (z) = -1 z and L -M > 0. This indicates that Hence f ∈ MS * H (q, L, M). Now let f ∈ H have be of the form (1.2), and let us choose n ≥ 1 such that a n = 0 or b n = 0. Also, by using Similarly, ρ n L-M ≥ n for n ≥ 1. Thus using (2.1) together with the above evidence, we get and therefore Therefore by Lewy's result [34] the function f in D * is sense-preserving and locally univalent. Moreover, if z 1, z 2 ∈ D * withz 1 = z 2 , then Hence by (2.4) we have This shows that f is univalent in D * , and thus f ∈ M H . Therefore f ∈ MS * H (q, L, M) if and only if there exists a holomorphic function u with u(0) = 0 and |u(z)| < 1 such that or, alternatively, To prove (2.5), it suffices to show that By substituting specific values of the parameters included in this result we obtain the following corollaries. Proof Taking the limit as q → 1-in the above corollary, we get the needed result.

Example 2.5 Let us choose the function
Then we easily get Thus T ∈ MS * H ϑ (q, L, M).
By using the above-mentioned theorem along with the notion of class MS c H (q, L, M) we can easily derive the following results.

Investigation of partial-sum problems
In this section, we investigate problems of partial sums of certain meromorphic harmonic functions belonging to MS * H (q, L, M). We produce some new findings that connect the meromorphic harmonic functions with their partial-sum sequences. Let f = h+ g with h and g given in (1.3). Then the partial-sum sequences of f are specified by

Now we find sharp lower bounds for
Re and Re  The findings above are best suited for the function
Proof Let us represent ∞ n=1 a n z n 1 z + t n=1 a n z n + ∞ n=1 b n z n .
Inequality (3.1) will be acquired if we are able to show that Re{ 1 (z)} > 0, and for this, we required to conclude that Alternatively, we have the following inequalities: From (2.1) we have that it suffices to guarantee that the left-hand side of (3.6) is bounded above by which is exactly equivalent to t n=1 and this is true because of (3.4). We observe that the function offers the best possible outcome. We see for z = re i π t that f (z) To examine (3.2), let us write 2 (z) = ∞ n=t+1 a n z n 1 z + ∞ n=1 a n z n + ∞ n=1 b n z n .

Inequality (3.7) is valid if the left-hand side of this inequality is bounded above by
and thus the proof is accomplished by using (2.1). and Re

9)
where I n is given by (3.3), and The equalities are achieved by considering the function Proof The proof for this specific outcome is similar to that of Theorem 3.1 and is thus omitted.  (3.12) and Re , (3.13) where I n is given by (3.3). The equalities are easily achieved by using (3.5).
Proof To establish (3.12), let us consider L-M ( ∞ n=t+1 a n z n + ∞ n=l+1 b n z n ) 1 z + t n=1 a n z n + l n=1 b n z n .
Therefore, to show inequality (3.12), it is sufficient to prove the inequality Now recalling the left-hand side of the above-mentioned inequality, by easy calculations we get .
Since we observe that from (2.1) that the denominator of the last inequality is positive. The right-hand side of the last inequality is also constrained by one if and only if the following inequality is maintained: |b n | ≤ 1. (3.14) Eventually, to verify inequality (3.12), it suffices to show that the left-hand side of (3.14) is bounded above by which is further analogous to and this is true due to (3.10). Now let us choose and X n , Y n ≥ 0 for n ∈ N are such that ∞ n=1 (X n + Y n ) = 1. Proof Let f be specified by (4.3). Then from (4.4) we get Thus f ∈ clcoMS * H ϑ (q, L, M). For the converse part, let f = h + g ∈ MS * H ϑ (q, L, M). Put Then utilizing (4.4) together with the hypothesis, we have which is the needed form (4.3). Thus the proof of Theorem 4.2 is completed.  For ∞ k=1 ξ k = 1, 0 ≤ ξ k < 1, the convex combination of f k is ξ k |b n,k | z n .
Then by Theorem 2.4 we can write

Conclusion
Utilizing the principles of quantum calculus, we have added some new subfamilies of meromorphic harmonic mappings linked to a circular domain. We learned also certain important problems for the newly specified function families, namely necessary and sufficient conditions, problems for partial sums, distortion limits, convexity conditions, and convolution preserving. For these families, other problems, such as topological properties, fundamental mean inequality, and their implications are open problems for the scholars to investigate. As pointed out in the survey-cum-expository review paper by Srivastava [44, p. 340], any attempt to produce the so-called (p, q)-variation of the q-results, which we have presented in this paper, will be trivial and inconsequential because the additional parameter p is obviously redundant or superfluous.