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: The motive behind this article is to apply the notions of q -derivative by introducing some new families of harmonic functions associated with the symmetric circular region. We develop a new criterion for sense preserving and hence the univalency in terms of q -differential operator. The necessary and sufﬁcient conditions are established for univalency for this newly deﬁned class. We also discuss some other interesting properties such as distortion limits, convolution preserving, and convexity conditions. Further, by using sufﬁcient inequality, we establish sharp bounds of the real parts of the ratios of harmonic functions to its sequences of partial sums. Some known consequences of the main results are also obtained by varying the parameters.


Introduction
To better understand the concepts used in our key findings, some of the basic relevant Geometric Function Theory literature needs to be included and studied here.To do this, we begin with notation H which indicates the harmonic functions class in D := D (1), where D (r) = {z ∈ C : |z| < r} .Also, let H 0 be denoted by the family of functions f = h + g ∈ H which have the series expansion: where h and g are holomorphic functions with the following series expansion: The series defined in (1) and (2) are convergent in the open unit disc D. Also, let's select the symbol S which contains all functions (say f ∈ S) that are univalent holomorphic in D and satisfy the relationship f (0) = f (0) − 1 = 0. Further, the notation S H denotes the family of complex-valued functions f ∈ H 0 which are sense preserving and univalent in D. Clearly, S H coincides with the set S of normalized univalent holomorphic functions if g(z) becomes zero in D. Clunie and Sheil-Small [1] and Sheil-Small [2] studied the class S H along with some of their subfamilies.Particularly, they explored and studied the families of starlike harmonic and convex harmonic functions in D, which are given as follows: and where D H f (z) = zh (z) − zg (z). ( Here the symbol "≺" represents the familiar subordination between holomorphic functions represented by " f 1 ≺ f 2 " and is defined as; two holomorphic functions f 1 : D → C and f 2 : D → C are connected by the subordination relationship, if a holomorphic function u occurs with the limitations u(0) = 0 and |u (z)| < 1, in such a way that f 1 (z) = f 2 (u (z)).Additionally, if f 2 is univalent, then we obtain: In 2015, Dziok [3] introduced a family S * H (L, M) , (L, M ∈ C with L = M) , of Janowski harmonic functions with the help of the concepts established in [4] and this class is defined by; where D H f (z) is given by (3).The families S * H (β with the restriction 0 ≤ β < 1, were examined by Jahangiri [5] and Ahuja [6].Moreover, for β = 0, we have the most basic families S * H := S * H (1, −1) and S c H := S c H (1, −1) of functions f ∈ S H which are starlike and convex in D (r) respectively, for any r ∈ [0, 1) .For more work on harmonic mappings, see [7][8][9][10].
Quantum calculus or just q-calculus is the classical calculus without the limit notion.This theory was originated by Jackson [11,12].The researchers were inspired by the learning of q-calculus because of its contemporary use in various claims; for example in differential equations, quantum theory, the theory of special functions, analytic number theory, combinatorics, operator theory, numerical analysis, and other related theories, see [13,14].Just as q-calculus has been used in other branches of Mathematics, the applications of this concept have been widely used to analyze the geometric nature of various analytical functions in Geometric functions theory.In this regard, Ismail et al. [15] published the first paper by exploring the geometry of q-starlike functions.This concept was further expanded by Agrawal and Sahoo [16] by introducing the q-starlike functions family with some order.Due to this advancement in functions theory, many researchers were inspired.They made significant contributions which gradually enhanced the attractiveness of this research area for potential researchers.For more literature on quantum calculus, see [17][18][19][20][21][22][23][24][25].
For the given q ∈ ]0, 1[, the q-analog derivative of f is defined as: Making use of (1) and (4) , one can get easily for n ∈ N where In 1956 Sakaguchi [26] established the family S * s of holomorphic univalent functions in D which are starlike with respect to symmetrical points; a holomorphic function f is said to be starlike with respect to symmetric points if Using this idea of Sakaguchi, Cho and Dziok [27] recently introduced the family where D H f (z) is given by (3).They investigated some useful properties such as coefficient estimates, subordination properties, distortion theorems, and integral representation for the functions belongs to the family HS * s (L, M).The family HS * s (L, M) generalizes various known families discussed earlier by many researchers, see [28][29][30][31].Motivated by the above work, we now define the following subfamilies of Janowski harmonic mappings involving q-derivative.Before definition and to prevent repetitions we will assume (except as otherwise stated) that Definition 1.Let HS * s (x, y, q, L, M) be the family of functions f ∈ S H such that where Additionally let's describe HS c s (x, y, q, L, M) := f ∈ S H : D q H f (z) ∈ HS * s (x, y, q, L, M) .
In this article, we obtain some interesting properties for the newly described classes including necessary and sufficient conditions, distortion limits, problems with partial sums, convolutions and convexity conditions.Several implications of the key results are also given.

Necessary and Sufficient Conditions
Theorem 1.Let f ∈ H be expressed in series expansion (1) and if the following inequality holds, then f ∈ HS * s (x, y, q, L, M) with Proof.For f (z) = z, we have h (z) = z and g (z) = 0. To prove that f is sense preserving and locally univalent, we consider Hence, due to the result of Lewy [9], the function f in D is thus orientation-preserving and locally univalent.Now we show that f is univalent in To prove that f ∈ HS * s (x, y, q, L, M) , we have to show that Hence, f ∈ HS * s (x, y, q, L, M) .Now let f ∈ H has the expansion form (1) and then let we presume that there is n ≥ 2 so that a n = 0 or b n = 0. Also by using and we have Similarly υ n L−M ≥ n for n ≥ 2. Therefore by using (8) along with the above facts, we obtain and so Now by the result proved by Lewy [9], the function f in D is thus orientation-preserving and locally univalent.Further, if z 1 , z 2 ∈ D with z 1 = z 2 , then Hence, by the virtue of (11) and (12) along with the triangle inequality, we have This yields that f is univalent in D and so f ∈ S H . Thus f ∈ HS * s (x, y, q, L, M) if and only if a holomorphic function u occurs with u (0) = 0 and |u (z or, alternatively Hence, it is enough to establish that where we have used (8) .Thus f ∈ S * H (x, y, q, L, M) .
Substituting particular values of the parameters used in this result, we achieve the following corollaries: Corollary 1.Let f ∈ H be expressed in the series expansion (1) and if the inequality Proof.The result follows by taking s = 1, t = −1 and q → 1 − in the above Theorem.
Corollary 2. Let f ∈ H be expressed in the series expansion (1) and if holds true with Proof.By putting s = 1, t = 0 and q → 1 − in the above Theorem, we obtain the required result.
Proof.To prove our result, it is enough to establish that each function f ∈ S * τ (x, y, q, L, M) satisfies the relation (8).Let f ∈ S * τ (x, y, q, L, M).Then it must satisfy (13 Setting z = r (r ∈ (0, 1)), we have It is clear that for r ∈ (0, 1) , the denominator of left hand side of (15) can not be zero.Further, it is positive for r = 0. Thus from (15) , we get It is also clear that the partial sums sequence {S n } connected with the series ∑ ∞ n=2 (σ n |a n | + υ n |b n |) is a non-decreasing sequence and with the use of (16) , it is bounded by (L − M).So {S n } is convergent and which yields assertion (8) .

Example 1. Consider the function
then we can easily obtain Hence, G ∈ S * τ (x, y, q, L, M) .
By using the above facts, the following two results are easily obtained.
Corollary 3. Let f ∈ H 0 be expressed in series expansion form (1) and if then f ∈ S c H (x, y, q, L, M) .
Proof.Let f ∈ H 0 be of the form (1) and set Then by using inequality (17) together with Theorem 1, we obtain F (z) ∈ HS * s (x, y, q, L, M) and hence by Alexandar type relation we get the required result.
Corollary 4. Let f ∈ τ 1 be the series expansion form (14). Then f ∈ S c τ (x, y, q, L, M) if and only if (17) is satisfied.
Proof.Using the relation (18) and Theorem 2, we easily get the required result.

Investigation of Partial Sums Problems
In this section, we study the partial sums of some harmonic functions that belong to the family S * H (x, y, q, L, M) .We develop certain new results which provide relationship between harmonic functions to its sequences of partial sums.
Let f = h + g with h and g are expressed in the form (2) .Then the sequences of partial sum of functions f are given by Here we determine sharp lower bounds of Theorem 3. Let f be of the form (1) .If f satisfies the condition (8 and and The above results are best possible for the given function where d t+1 is given by (22) .
Proof.Let us denote The inequality (19) will be obtained if we can prove Re{Φ 1 (z)} > 0 and for this we need to prove the below inequality: In other words By the virtue of (8) , it would be enough to ensure that the left hand side of (24) is bounded above by and this is equal to which is true due to relation (22).To prove that the function provides the best possible result.We note for z = re i π t that Now to prove (20) , let us write The inequality (25) is true only if the left side of this inequality is bounded above by and hence the proof is completed due to the use of (8) .
Theorem 4. Let f = h + g, where h and g are given by (2).If f satisfies the condition (8 where d n is given by (21) and Equalities are obtain by using the function Proof.The evidence of this specific result is analogous to that of Theorem 3, and is therefore exempted.
Theorem 5. Let f = h + g be of the form (1).If f fulfills the inequality (8 and where d n is given by (21).Equalities are obtained for the function given in (23) .
Proof.In order to show (30) , let us write Thus for the proof of inequality (19) it is enough to prove the following inequality: Now consider the left hand side of (32) and then by simple computation, we have Since, from the use of (8), we observe that the denominator of the last inequality is positive.Thus the right hand side of the last inequality is bounded above by one if and only if the following inequality hold Finally, to prove the inequality in (30) , it suffices to show that the left hand side of (33) is bounded by and the last inequality is true because of (28) .For the sharpness, let's consider which provides the sharp result.We note for z = re i π t that Similarly we obtain the assertion (31) .
Theorem 6.Let f = h + g with h and g are expressed by (2).If f satisfies the condition (8 and where d n is given by (28).These estimates are sharp for the function given by (29).
Proof.The proof is identical to that of Theorem 5 proof and is thus excluded.

Further Properties of a class
Proof.Let f = h + g ∈ S * τ (x, y, q, L, M) with h and g be the form (2). Using Theorem 2 and letting This proves (36) .On similar lines one can easily achieve (37).
Theorem 8.A function f ∈ S * τ (x, y, q, L, M) if and only if where In particular the points {h n } , {g n } are called the extreme points of the closed convex hull of the class S * τ (x, y, q, L, M) denoted by clcoS * τ (x, y, q, L, M) .
Proof.Let f be given by (38).Then, using (39) , we can easily attain which by Theorem 2 proves that f ∈ S * τ (x, y, q, L, M).Since for this function Thus f ∈ clcoS * τ (x, y, q, L, M) .Conversely, let f = h + g ∈ S * τ (x, y, q, L, M), where h and g are of the form (14) .Set Then by using (39) along with the given hypothesis, we have which is of the form (38).This proves Theorem 8.
Theorem 10.The class S * τ (x, y, q, L, M) is closed under convex combination.

Conclusions
Using the concepts of quantum calculus, we introduced some new subfamilies of Janowski harmonic mapping with symmetrical points.We studied some useful problems, including necessary and sufficient conditions, distortion limits, problems with partial sums, convolutions and convexity conditions for the newly defined classes of functions.For these classes, problems like Topological properties, integral mean inequalities, and their applications are open for the researchers to determine.Further, these problems can be studied for classes of meromorphic type harmonic functions as well.