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Axioms
, Volume 12 (5) – Apr 26, 2023

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axioms Article Applications Laguerre Polynomials for Families of Bi-Univalent Functions Deﬁned with (p, q)-Wanas Operator 1 2 3, Abbas Kareem Wanas , Fethiye Müge Sakar and Alina Alb Lupas ¸ Department of Mathematics, College of Science, University of Al-Qadisiyah, Al Diwaniyah 58001, Iraq Department of Management, Faculty of Economics and Administrative Sciences, Dicle University, Diyarbakir 21280, Turkey Department of Mathematics and Computer Science, University of Oradea, 1 Universitatii Street, 410087 Oradea, Romania * Correspondence: dalb@uoradea.ro or alblupas@gmail.com Abstract: In current manuscript, using Laguerre polynomials and ( p q)-Wanas operator, we identify upper boundsja j andja j which are ﬁrst two Taylor-Maclaurin coefﬁcients for a speciﬁc bi-univalent 2 3 functions classes W (h, d, l, s, q, a, b, p, q; h) and K (x, r, s, q, a, b, p, q; h) which cover the convex S S and starlike functions. Also, we discuss Fekete-Szegö type inequality for deﬁned class. Keywords: bi-univalent function; Fekete-Szegö problem; coefﬁcient bound; Laguerre polynomial; ( p, q)-Wanas operator; subordination MSC: 30C45; 30C80 1. Introduction Denote by A function collections that have the style: Citation: Wanas, A.K.; Sakar, F.M.; f (z) = z + a z , z 2 D, (1) n=2 Alb Lupas, ¸ A. Applications Laguerre Polynomials for Families of holomorphic in D = fz : jzj < 1g in the complex plane C. Bi-Univalent Functions Deﬁned with Further, present by S the sub-set of A including of univalent functions in D fullﬁling ( p, q)-Wanas Operator. Axioms 2023, 1 1 (1). Taking account the Koebe theorem (see [1]), each f 2 S has an inverse f with 12, 430. https://doi.org/10.3390/ 4 1 1 the properties f ( f (z)) = z, for z 2 D and f ( f (w)) = w, with jwj < r ( f ), where axioms12050430 r ( f ) . If f is of the style (1), then Academic Editors: Georgia Irina Oros and Nhon Nguyen-Thanh 1 2 2 3 3 4 f (w) = w a w + 2a a w 5a 5a a + a w + , jwj < r ( f ). (2) 2 3 2 3 4 0 2 2 Received: 25 February 2023 Revised: 13 March 2023 When f and f are univalent functions, f 2 A is bi-univalent in D. The set of Accepted: 25 April 2023 bi-univalent functions can be expressed by S. The work on bi-univalent functions have Published: 26 April 2023 been brightened by Srivastava et al. [2] in recent years. The following functions can be exampliﬁed for functions in the set of bi-univalent. z 1 1 + z , log(1 z) and log . Copyright: © 2023 by the authors. 1 z 2 1 z Licensee MDPI, Basel, Switzerland. This article is an open access article Although Koebe function is not an element of bi-univalent set of functions, the S is not distributed under the terms and null set. conditions of the Creative Commons Later, such studies continued by Ali et al. [3], Bulut et al. [4], Srivastava et al. [5] and Attribution (CC BY) license (https:// others (see, for example, [6–18]). However, non decisive predictions of the ja j and ja j 2 3 creativecommons.org/licenses/by/ 4.0/). Axioms 2023, 12, 430. https://doi.org/10.3390/axioms12050430 https://www.mdpi.com/journal/axioms Axioms 2023, 12, 430 2 of 13 coefﬁcients in given by (1) were declared in different studies. Generalized inequalities on Taylor-Maclaurin coefﬁcients ja j (n 2 N; n = 3) for f 2 S has not been totally solved yet for several subfamilies of the S. a ma of the Fekete-Szegö function for f 2 S is well-known in the Geometric Function Theory. Its origin lies in the refutation of the Littlewood-Paley conjecture by Fekete-Szegö [19]. In that case, the coefﬁcients of odd (single-valued) univalent functions are bounded by unity. Functions have received much attention since then, especially in the investigation of many subclasses of the single-valued function family. This topic has become very interesting for Geometric Function Theorists (see for example [20–25]). The generator function for Laguerre polynomial L (t) is the polynomial answer f(t) of the differential equation ([26]) 00 0 tf + (1 + g t)f + nf = 0, where g > 1 and n is non-negative integers. The generating function of generator function for Laguerre polynomial L (t) is ex- pressed as below: tz 1z H t, z = L t z = , (3) ( ) ( ) g n g+1 1 z ( ) n=0 where t 2 R and z 2 D. The generator function for Laguerre polynomial can also be expressed given below: 2n + 1 + g t n + g g g g L (t) = L (t) L (t) (n 1), n+1 n1 n + 1 n + 1 with the initial terms t (g + 1)(g + 2) g g g L (t) = 1, L (t) = 1 + g t and L (t) = (g + 2)t + . (4) 0 1 2 2 2 Simply, when g = 0 the generator function for Laguerre polynomial leads to the simply Laguerre polynomial, L (t) = L (t). Let f and g be holomorphic in D, it is clear that f is subordinate to g, if there occurs a holomorphic function w in D such that w(0) = 0, and jw(z)j < 1, for z 2 D so that f (z) = g(w(z)). This subordination is indicated by f g. Moreover, if g is univalent in D, then we have the balance (see [27]), given by f (z) g(z) () f (D) g(D) and f (0) = g(0). The ( p, q)-derivative operator or ( p, q)-difference operator (0 < q < p 1), for a function f is stated by f ( pz) f (qz) D f (z) = (z 2 D = Dnf0g), p,q ( p q)z and D f (0) = f (0). p,q More information on the subject of ( p, q)-calculus are founded in [28–33]. For f 2 A, we conclude that n1 D f (z) = 1 + [n] a z , p,q å p,q n n=2 Axioms 2023, 12, 430 3 of 13 where the ( p, q)-bracket number or twin-basic [n] is showed by p,q n n p q n1 n2 n3 2 n2 n1 [n] = = p + p q + p q + + pq + q ( p 6= q), p,q p q which is a native generator number for q, namely is, we get (see [34,35]) 1 q lim [n] = [n] = . p,q q 1 q p!1 Obviously, the impression [n] is symmetric, namely, p,q [n] = [n] . p,q q, p s,q Wanas and Cotîrla ˇ [36] presented W : A ! A known as ( p q)-Wanas operator a,b, p,q showed by ¥ ¥ [Y (s, a, b)] [Y (s, a, b)] n p,q p,q s,q n n W f (z) = z + a z = z + a z , n n å å a,b, p,q [Y (s, a, b)] [Y (s, a, b)] 1 p,q 1 p,q n=2 n=2 where s s s s t+1 t t t+1 t t Y (s, a, b) = (1) (a + nb ), Y (s, a, b) = (1) (a + b ), n å 1 å t t t=1 t=1 and a 2 R, b 2 R with a + b > 0, n 1 2 N, s 2 N, q 2 N , 0 < q < p 1 and z 2 D. s,q Remark 1. The operator W is a generalized form of several operators given in previous a,b, p,q researches for some values of parameters which are mentioned below. s,q 1. For p = s = b = 1, q = n, <(n) > 1 and a 2 Cn Z , the operator W decreases to 0 a,b, p,q the q-Srivastava Attiya operator J [37]. q,a s,q 2. For p = s = b = 1, q = 1 and a > 1, the operator W decreases to the q-Bernardi a,b, p,q operator [38]. s,q 3. For p = s = a = b = 1 and q = 1, the operator W decreases to the q-Libera a,b, p,q operator [38]. s,q 4. For a = 0 and p = s = b = 1, the operator W decreases to the q-Sal ˘ agean ˘ operator [39]. a,b, p,q s,q 5. For q ! 1 and p = s = 1, the operator W decreases to the operator I was a,b a,b, p,q presented and studied by Swamy [40]. s,q 6. For q ! 1 , p = s = b = 1, q = n, <(n) > 1 and s 2 Cn Z , the operator W 0 a,b, p,q n n decreases to the operator J was presented by Srivastava and Attiya [41]. The operator J is a s well-known as Srivastava-Attiya operator by researchers. s,q 7. For q ! 1 , p = s = b = 1 and a > 1, the operator W , decreases to the operator a,b, p,q I was presented by Cho and Srivastava [42]. s,q q 8. For q ! 1 , p = s = a = b = 1, the operator W decreases to the operator I was a,b, p,q presented by Uralegaddi and Somanatha [43]. s,q 9. For q ! 1 , p = s = a = b = 1, q = x and x > 0, the operator W decreases to a,b, p,q x x the operator I was presented by Jung et al. [44]. The operator I is the Jung-Kim-Srivastava integral operator. s,q 10. For q ! 1 , p = s = b = 1, q = 1 and a > 1, the operator W decreases to the a,b, p,q Bernardi operator [45]. Axioms 2023, 12, 430 4 of 13 s,q 11. For q ! 1 , a = 0, p = s = b = 1 and q = 1, the operator W decreases to the a,b, p,q Alexander operator [46]. s,q 12. For q ! 1 , p = s = 1, a = 1 b and t 0, the operator W decreases to the a,b, p,q operator D was presented by Al-Oboudi [19]. s,q 13. For q ! 1 , p = s = 1, a = 0 and b = 1, the operator W decreases to the operator a,b, p,q S was presented by Sal ˘ agean ˘ [47]. 2. Main Results Firstly, We start to present the classes W (h, d, l, s, q, a, b, p, q; h) and K (x, r, s, q, S S a, b, p, q; h) given below: Deﬁnition 1. Suppose that 0 h 1, 0 l 1, 0 d 1 and h is analytic in D, h(0) = 1. f 2 S is in the class W (h, d, l, s, q, a, b, p, q; h) if it provides the subordinations: 0 1 2 0 13 h l 0 0 00 s,q s,q s,q z W f (z) z W f (z) z W f (z) a,b, p,q a,b, p,q a,b, p,q B C 6 B C7 (1 d) + d 1 + h(z) @ A 4 @ A5 s,q s,q s,q W f (z) W f (z) W f (z) a,b, p,q a,b, p,q a,b, p,q and 0 1 2 0 13 h l 0 0 00 s,q s,q s,q 1 1 1 w W f (w) w W f (w) w W f (w) a,b, p,q a,b, p,q a,b, p,q B C 6 B C7 (1 d) + d 1 + h(w), @ A 4 @ A5 s,q s,q 1 1 s,q W f (w) W f (w) 1 a,b, p,q a,b, p,q W f (w) a,b, p,q where f is given by (2). Deﬁnition 2. Suppose that 0 x 1, 0 r < 1 and h is analytic in D, h(0) = 1. f 2 S is in the class K (x, r, s, q, a, b, p, q; h) if it provides the subordinations: s,q z W f (z) a,b, p,q (1 x) s,q s,q (1 r)W f (z) + rz W f (z) a,b, p,q a,b, p,q 0 1 0 00 s,q s,q W f (z) + z W f (z) a,b, p,q a,b, p,q B C +x@ A h(z) 0 00 s,q s,q W f (z) + rz W f (z) a,b, p,q a,b, p,q and s,q w W f (w) a,b, p,q (1 x) s,q s,q 1 1 (1 r)W f (w) + rw W f (w) a,b, p,q a,b, p,q 0 1 0 00 s,q s,q 1 1 W f (w) + w W f (w) a,b, p,q a,b, p,q B C +x h(w), @ A 0 00 s,q s,q 1 1 W f (w) + rw W f (w) a,b, p,q a,b, p,q where f is given by (2). Theorem 1. Suppose that 0 h 1, 0 l 1 and 0 d 1. If f 2 S of the style (1) be an element of class W (h, d, l, s, q, a, b, p, q; h), with h(z) = 1 + e z + e z + , then S 1 2 (h + l(d + 1))[Y (s, a, b)] je j 2 1 je j p,q ja j = [Y (s, a, b)] W p,q Axioms 2023, 12, 430 5 of 13 and ( ( ) ( )) 2 2 e e je e e (2D + j)e 1 2 1 2 1 1 ja j min max , , max , , (5) 2 2 D D W D D D W D where (h+l(d+1))[Y (s,a,b)] p,q W = , [Y (s,a,b)] 1 p,q 2(h+l(2d+1))[Y (s,a,b)] 3 p,q D = , (6) [Y (s,a,b)] p,q 2q [h(h1)+l(d+1)(2h+(l1)(d+1))2(h+l(3d+1))][Y (s,a,b)] p,q j = . 2q 2[Y (s,a,b)] 1 p,q Proof. Assume that f 2 W (h, d, l, s, q, a, b, p, q; e ; e ). Then there consists two holomor- S 1 2 phic functions f, y : D ! D showed by 2 3 f(z) = r z + r z + r z + (z 2 D) (7) 1 2 3 and 2 3 y(w) = s w + s w + s w + (w 2 D), (8) 1 2 3 with f(0) = y(0) = 0, jf(z)j < 1, jy(w)j < 1, z, w 2 D so that 0 1 2 0 13 h l 0 0 00 s,q s,q s,q z W f (z) z W f (z) z W f (z) a,b, p,q a,b, p,q a,b, p,q B C 6 B C7 (1 d) + d 1 + @ A 4 @ A5 s,q s,q s,q W f (z) W f (z) W f (z) a,b, p,q a,b, p,q a,b, p,q = 1 + e f(z) + e f (z) + (9) 1 2 and 0 1 2 0 13 h l 0 0 00 s,q s,q s,q 1 1 1 w W f (w) w W f (w) w W f (w) a,b, p,q a,b, p,q a,b, p,q B C 6 B C7 (1 d) + d 1 + @ A 4 @ A5 s,q s,q 1 1 s,q W f (w) W f (w) 1 a,b, p,q a,b, p,q W f (w) a,b, p,q = 1 + e y(w) + e y (w) + . (10) 1 2 Uniﬁcation of (7), (8), (9) and (10), yield 0 1 2 0 13 h l 0 0 00 s,q s,q s,q z W f (z) z W f (z) z W f (z) a,b, p,q a,b, p,q a,b, p,q B C 6 B C7 (1 d) + d 1 + @ A 4 @ A5 s,q s,q s,q W f (z) W f (z) W f (z) a,b, p,q a,b, p,q a,b, p,q h i 2 2 = 1 + e r z + e r + e r z + (11) 1 1 1 2 2 and 0 1 2 0 13 h l 0 0 00 s,q 1 s,q 1 s,q 1 w W f (w) w W f (w) w W f (w) a,b, p,q a,b, p,q a,b, p,q B C 6 B C7 (1 d) + d 1 + @ A 4 @ A5 s,q s,q 1 1 s,q W f (w) W f (w) 1 W f (w) a,b, p,q a,b, p,q a,b, p,q h i 2 2 = 1 + e s w + e s + e s w + . (12) 1 1 1 2 2 It is clear that if f(z) < 1 and y(w) < 1, z, w 2 D, we obtain j j j j r 1 and s 1 (j 2 N). j j Axioms 2023, 12, 430 6 of 13 Taking into account (11) and (12), after simplifying, we ﬁnd that (h + l(d + 1))[Y (s, a, b)] p,q a = e r , (13) 2 1 1 [Y (s, a, b)] p,q 2(h + l(2d + 1))[Y (s, a, b)] p,q [Y (s, a, b)] p,q 2q [h(h 1) + l(d + 1)(2h + (l 1)(d + 1)) 2(h + l(3d + 1))][Y (s, a, b)] 2 p,q + a 2q 2[Y (s, a, b)] 1 p,q = e r + e r , (14) 1 2 2 (h + l(d + 1))[Y (s, a, b)] p,q a = e s (15) 1 1 [Y (s, a, b)] 1 p,q and 2(h + l(2d + 1))[Y (s, a, b)] 3 p,q 2a a 2 3 [Y (s, a, b)] 1 p,q 2q [h(h 1) + l(d + 1)(2h + (l 1)(d + 1)) 2(h + l(3d + 1))][Y (s, a, b)] p,q + a 2q 2[Y (s, a, b)] p,q = e s + e s . (16) 1 2 2 If we implement notation (6), then (13) and (14) becomes 2 2 Wa = e r , Da + ja = e r + e r . (17) 2 1 1 3 1 2 2 2 1 This gives D e je 2 1 a = r + r , (18) 3 2 e e W 1 1 and on using the given certain result ([48], p. 10): jr mr j maxf1,jmjg (19) for every m 2 C, we get D e je 2 1 ja j max 1, . (20) e e W 1 1 In the same way, (15) and (16) becomes 2 2 2 Wa = e s , D(2a a ) + ja = e s + e s . (21) 2 1 1 3 1 2 2 2 2 1 This gives D e (2D + j)e 2 1 a = s + s . (22) 3 2 e e W 1 1 Applying (19), we obtain D e (2D + j)e 2 1 ja j max 1, . (23) e e W 1 1 Inequality (5) follows from (20) and (23). If we take the generating function L (t) given by (3) common generalized Laguerre polynomials as h(z), then from the equalities given(4), we get e = 1 + g t and (g+1)(g+2) e = (g + 2)t + . We obtain following corollary from Theorem 1. 2 2 Axioms 2023, 12, 430 7 of 13 Corollary 1. If f 2 S given by style (1) is in the family W (h, d, l, s, q, a, b, p, q; H t, z ), ( ) S g then (h + l(d + 1))[Y (s, a, b)] j1 + g tj 2 j1 + g tj p,q ja j = [Y (s, a, b)] 1 p,q and ( ( ) (g+1)(g+2) t 2 (g + 2)t + 1 + g t j(1 + g t) 2 2 ja j min max , , D D W D ( )) (g+1)(g+2) t 2 (g + 2)t + 1 + g t (2D + j)(1 + g t) 2 2 max , , D D W D for all h, l, d so that 0 h 1, 0 l 1 and 0 d 1, where W, D, j are given by (6) and H (t, z) is given by (3). Theorem 2. Suppose that 0 x 1 and 0 r < 1. If f 2 S of the style (1) be an element of the class K (x, r, s, q, a, b, p, q; h), with h(z) = 1 + e z + e z + , then S 1 2 (x + 1)(1 r)[Y (s, a, b)] je j 2 1 je j p,q ja j = [Y (s, a, b)] U 1 p,q and ( ( ) ( )) 2 2 ce (2F + c)e e e e e 1 2 1 2 1 1 ja j min max , , max , , (24) 2 2 F F U F F F U F where (x+1)(1r)[Y (s,a,b)] p,q U = , [Y (s,a,b)] 1 p,q 2(2x+1)(1r)[Y (s,a,b)] 3 p,q (25) F = , [Y (s,a,b)] 1 p,q 2 2q (2x+1)(r 1)[Y (s,a,b)] 2 p,q c = . 2q [Y (s,a,b)] 1 p,q Proof. Assume that f 2 K (x , r, s, q, a, b, p, q; e ; e ). Then there consists two holomorphic S 1 2 functions f, y : D ! D such that 0 1 0 0 00 s,q s,q s,q z W f (z) W f (z) + z W f (z) a,b, p,q a,b, p,q a,b, p,q B C (1 x) + x @ A 0 0 00 s,q s,q s,q s,q (1 r)W f (z) + rz W f (z) W f (z) + rz W f (z) a,b, p,q a,b, p,q a,b, p,q a,b, p,q = 1 + e f(z) + e f (z) + (26) 1 2 and 0 1 0 0 00 s,q s,q s,q 1 1 1 w W f (w) W f (w) + w W f (w) a,b, p,q a,b, p,q a,b, p,q B C (1 x) + x@ A 0 0 00 s,q s,q s,q s,q 1 1 1 1 (1 r)W f (w) + rw W f (w) W f (w) + rw W f (w) a,b, p,q a,b, p,q a,b, p,q a,b, p,q = 1 + e y(w) + e y (w) + , (27) 1 2 Axioms 2023, 12, 430 8 of 13 where f and y given by the style (7) and (8). Uniﬁcation of (26) and (27), serve 0 1 0 0 00 s,q s,q s,q z W f (z) W f (z) + z W f (z) a,b, p,q a,b, p,q a,b, p,q B C (1 x) + x @ A 0 0 00 s,q s,q s,q s,q (1 r)W f (z) + rz W f (z) W f (z) + rz W f (z) a,b, p,q a,b, p,q a,b, p,q a,b, p,q h i 2 2 = 1 + e r z + e r + e r z + (28) 1 1 1 2 2 and 0 1 0 0 00 s,q s,q s,q 1 1 1 w W f (w) W f (w) + w W f (w) a,b, p,q a,b, p,q a,b, p,q B C (1 x) + x@ A 0 0 00 s,q s,q s,q s,q 1 1 1 1 (1 r)W f (w) + rw W f (w) W f (w) + rw W f (w) a,b, p,q a,b, p,q a,b, p,q a,b, p,q h i 2 2 = 1 + e s w + e s + e s w + . (29) 1 1 1 2 2 It is clear that if jf(z)j < 1 and jy(w)j < 1, z, w 2 D, we obtain r 1 and s 1 (j 2 N). j j Taking into account (28) and (29), after simplifying, we ﬁnd that (x + 1)(1 r)[Y (s, a, b)] p,q a = e r , (30) 2 1 1 [Y (s, a, b)] 1 p,q q 2 2q 2(2x + 1)(1 r)[Y (s, a, b)] (2x + 1)(r 1)[Y (s, a, b)] 3 2 p,q p,q 2 2 a + a = e r + e r , (31) 3 1 2 2 2 1 q 2q [Y (s, a, b)] [Y (s, a, b)] 1 p,q 1 p,q (x + 1)(1 r)[Y (s, a, b)] p,q a = e s (32) 2 1 1 [Y (s, a, b)] p,q and q 2 2q 2(2x + 1)(1 r)[Y (s, a, b)] (2x + 1)(r 1)[Y (s, a, b)] 3 2 p,q p,q 2 2 2a a + a 2 2 q 2q [Y (s, a, b)] [Y (s, a, b)] 1 1 p,q p,q = e s + e s . (33) 1 2 2 If we implement notation (25), then (30) and (31) becomes 2 2 Ua = e r , Fa + ca = e r + e r . (34) 2 1 1 3 1 2 2 2 1 This gives F e ce 2 1 a = r + r , (35) 3 2 e e U 1 1 and on using the given certain result ([48], p. 10): jr mr j maxf1,jmjg (36) for every m 2 C, we get F e ce 2 1 ja j max 1, . (37) e e 1 1 In the same way, (32) and (33) becomes 2 2 2 Ua = e s , F(2a a ) + ca = e s + e s . (38) 2 1 1 3 1 2 2 2 2 1 Axioms 2023, 12, 430 9 of 13 This gives F e (2F + c)e 2 1 a = s + s . (39) 3 2 e e U 1 1 Applying (36), we obtain F e (2F + c)e 2 1 ja j max 1, . (40) e e U 1 1 Inequality (24) follows from (37) and (40). If we take the generating function L (t) given by (3) common generalized Laguerre polynomials as h(z), then from the equalities given(4), we get e = 1 + g t and e = (g+1)(g+2) (g + 2)t + . We obtain following corollary from Theorem 2. 2 2 Corollary 2. If f 2 S of the style (1) be an element of the class K (x , r, s, q, a, b, p, q; H (t, z)), S g then (x + 1)(1 r)[Y (s, a, b)] j1 + g tj 2 j1 + g tj p,q ja j = [Y (s, a, b)] U p,q and ( ( ) (g+1)(g+2) t 2 1 + g t (g + 2)t + c 1 + g t ( ) 2 2 ja j min max , , F F U F ( )) (g+1)(g+2) t 2 1 + g t (g + 2)t + (2F + c)(1 + g t) 2 2 max , , F F U F for all x , r so that 0 x 1 and 0 r < 1, where U, F, c are introduced by (25) and H (t, z) is given by (3). We investigate the “Fekete-Szegö Inequalities” for the familiesW (h, d, l, s, q, a, b, p, q; h) and K (x, r, s, q, a, b, p, q; h) in next theorems. Theorem 3. If f 2 S of the style (1) be an element of family W (h, d, l, s, q, a, b, p, q; h), then je j e (zD j)e e (2D + j zD)e 2 2 2 1 1 1 a z a min max 1, + , max 1, , 2 2 D e W e W 1 1 for all z, h, l, d such that z 2 R, 0 h 1, 0 l 1 and 0 d 1, where W, D, j are given by (6) and e , e , a and a as deﬁned in Theorem 1. 1 2 2 3 Proof. We implement the impressions from the Theorem 1’s proof. From (17) and from (18), we get e e (zD j)e 2 1 2 1 2 a z a = r + + r 3 2 2 1 D e by using the certain result jr mr j maxf1,jmjg, we get je j e (zD j)e 2 1 1 ja z a j max 1, + . D e W In the same way, from (21) and from (22), we get e e (2D + j zD)e 1 2 1 2 2 a z a = s + s 3 2 2 1 D e W 1 Axioms 2023, 12, 430 10 of 13 and on using js ms j max 1,jmj , we get f g je j e (2D + j zD)e 1 2 1 ja z a j max 1, . 3 2 D e W Corollary 3. If f 2 S of the style (1) be an element of W (h, d, l, s, q, a, b, p, q; H (t, z)), then a z a ( ( ) (g+1)(g+2) (g + 2)t + j1 + g tj (zD j)(1 + g t) 2 2 min max 1, + , D 1 + g t W ( )) (g+1)(g+2) (g + 2)t + (2D + j zD)(1 + g t) 2 2 max 1, , 1 + g t for each z , h, l, d such that z 2 R, 0 h 1, 0 l 1 and 0 d 1, where W, D, j are given by (6) and H (t, z) is presented by (3). Theorem 4. If f 2 S of the style (1) is in the family K (x, r, s, q, a, b, p, q; h), then je j e (zF c)e e (2F + c zF)e 2 1 2 1 2 1 a z a min max 1, + , max 1, , 2 2 F e e U U 1 1 for all z, x, r such that z 2 R, 0 x 1 and 0 r < 1, where U, F, c are given by (25) and e , e , a and a as deﬁned in Theorem 2. 2 2 3 Proof. We implement the impressions from the Theorem 2’s proof. From (34) and from (35), we get e e (zF c)e 1 2 1 2 2 a z a = r + + r 3 2 2 1 F e by using the certain result jr mr j maxf1,jmjg, we get je j e (zF c)e 2 1 2 1 ja z a j max 1, + . F e U In the same way, from (38) and from (39), we get e e (2F + c zF)e 2 1 1 2 a z a = s + s 3 2 2 1 F e U and on using js ms j maxf1,jmjg, we get je j e (2F + c zF)e 2 1 2 1 ja z a j max 1, . F e U 1 Axioms 2023, 12, 430 11 of 13 Corollary 4. If f 2 S of the style (1) be an element of K (x, r, s, q, a, b, p, q; H t, z ), then ( ) S g a z a ( ( ) (g+1)(g+2) (g + 2)t + j1 + g tj (zF c)(1 + g t) 2 2 min max 1, + , F 1 + g t U ( )) (g+1)(g+2) (g + 2)t + (2F + c zF)(1 + g t) 2 2 max 1, , 1 + g t U for each z, x, r such that z 2 R, 0 x 1 and 0 r < 1, where U, F, c are given by (25) and H (t, z) is presented by (3). 3. Conclusions The main aim of this study was to constitute a new classes W (h, d, l, s, q, a, b, p, q; h) and K (x, r, s, q, a, b, p, q; h) of bi-univalent functions described through ( p q)-Wanas operator and also utilization of the generator function for Laguerre polynomial L (t), pre- sented by the equalities in (4) and the producing function H (t, z) given by (3). The initial Taylor-Maclaurin coefﬁcient estimates for functions of these freshly presented bi-univalent function classes W (h, d, l, s, q, a, b, p, q; h) and K (x, r, s, q, a, b, p, q; h) were produced S S and the well-known Fekete-Szegö inequalities were examined. Author Contributions: Conceptualization, A.K.W. and F.M.S.; methodology, A.K.W. and F.M.S.; software, A.A.L.; validation, A.A.L. and A.K.W.; formal analysis, A.K.W. and F.M.S.; investigation, A.K.W. and F.M.S.; resources, F.M.S.; data curation, A.K.W.; writing—original draft preparation, F.M.S.; writing—review and editing, A.A.L. and F.M.S.; visualization, A.K.W.; supervision, F.M.S.; project administration, F.M.S.; funding acquisition, A.A.L. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the University of Oradea. Data Availability Statement: Not applicable. Conﬂicts of Interest: The authors declare no conﬂict of interest. References 1. Duren, P.L. 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Axioms – Multidisciplinary Digital Publishing Institute

**Published: ** Apr 26, 2023

**Keywords: **bi-univalent function; Fekete-Szegö problem; coefficient bound; Laguerre polynomial; (*p*,*q*)-Wanas operator; subordination

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