CUBO, A Mathematical Journal Vol. 23, no. 02, pp. 299–312, August 2021 DOI: 10.4067/S0719-06462021000200299 Subclasses of λ-bi-pseudo-starlike functions with respect to symmetric points based on shell-like curves H. Özlem Güney1 G. Murugusundaramoorthy2 K. Vijaya2 1 Dicle University, Faculty of Science, Department of Mathematics, Diyarbakır, Turkey. ozlemg@dicle.edu.tr 2 School of Advanced Sciences, Vellore Institute of Technology, Vellore -632014, India. gmsmoorthy@yahoo.com; kvijaya@vit.ac.in ABSTRACT In this paper we define the subclass PSLλs,Σ(α,p̃(z)) of the class Σ of bi-univalent functions defined in the unit disk, called λ-bi-pseudo-starlike, with respect to symmetric points, related to shell-like curves connected with Fibonacci num- bers. We determine the initial Taylor-Maclaurin coefficients |a2| and |a3| for functions f ∈ PSLλs,Σ(α,p̃(z)). Further we determine the Fekete-Szegö result for the function class PSLλs,Σ(α,p̃(z)) and for the special cases α = 0, α = 1 and τ = −0.618 we state corollaries improving the initial Taylor- Maclaurin coefficients |a2| and |a3|. RESUMEN En este art́ıculo definimos la subclase PSLλs,Σ(α,p̃(z)) de la clase Σ de funciones bi-univalentes definidas en el disco uni- tario, llamadas λ-bi-pseudo-estrelladas, con respecto a pun- tos simétricos, relacionadas a curvas espirales en conexión con números de Fibonacci. Determinamos los coeficientes iniciales de Taylor-Maclaurin |a2| y |a3| para funciones f ∈ PSLλs,Σ(α,p̃(z)). Más aún determinamos el resultado de Fekete-Szegö para la clase de funciones PSLλs,Σ(α,p̃(z)) y para los casos especiales α = 0, α = 1 y τ = −0.618 enunciamos corolarios mejorando los coeficientes iniciales de Taylor-Maclaurin |a2| y |a3|. Keywords and Phrases: Analytic functions, bi-univalent, shell-like curve, Fibonacci numbers, starlike functions. 2020 AMS Mathematics Subject Classification: 30C45, 30C50. Accepted: 05 June, 2021 Received: 04 November, 2020 ©2021 H. Özlem Güney et al. This open access article is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/S0719-06462021000200299 https://orcid.org/0000-0002-3010-7795 https://orcid.org/0000-0001-8285-6619 https://orcid.org/0000-0002-3216-7038 300 H. Özlem Güney, G. Murugusundaramoorthy & K. Vijaya CUBO 23, 2 (2021) 1 Introduction Let A denote the class of functions f which are analytic in the open unit disk U = {z : z ∈ C and |z| < 1}. Also let S denote the class of functions in A which are univalent in U and normalized by the conditions f(0) = f′(0) − 1 = 0 and are of the form: f (z) = z + ∞∑ n=2 anz n. (1.1) The Koebe one quarter theorem [4] ensures that the image of U under every univalent function f ∈A contains a disk of radius 1 4 . Thus every univalent function f has an inverse f−1 satisfying f−1(f(z)) = z, (z ∈ U) and f(f−1(w)) = w (|w| < r0(f), r0(f) ≥ 14 ). A function f ∈A is said to be bi-univalent in U if both f and f−1 are univalent in U. Let Σ denote the class of bi-univalent functions defined in the unit disk U. Since f ∈ Σ has the Maclaurin series given by (1.1), a computation shows that its inverse g = f−1 has the expansion g(w) = f−1(w) = w −a2w2 + (2a22 −a3)w 3 + · · · . (1.2) We notice that the class Σ is not empty. For example, the functions z, z 1−z , − log(1 − z) and 1 2 log 1+z 1−z are members of Σ. However, the Koebe function is not a member of Σ. In fact, Srivastava et al. [15] have actually revived the study of analytic and bi-univalent functions in recent years, it was followed by such works as those by (see [2, 3, 9, 15, 16, 17]). An analytic function f is subordinate to an analytic function F in U, written as f ≺ F (z ∈ U), provided there is an analytic function ω defined on U with ω(0) = 0 and |ω(z)| < 1 satisfying f(z) = F(ω(z)). It follows from Schwarz Lemma that f(z) ≺ F(z) ⇐⇒ f(0) = F(0) and f(U) ⊂ F(U), z ∈ U (for details see [4, 8]). We recall important subclasses of S in geometric function theory such that if f ∈A and zf′(z) f(z) ≺ p(z) and 1 + zf′′(z) f′(z) ≺ p(z) where p(z) = 1+z 1−z , then we say that f is starlike and convex, respectively. These functions form known classes denoted by S∗ and C, respectively. Recently, in [14], Sokó l introduced the class SL of shell-like functions as the set of functions f ∈A which is described in the following definition: Definition 1.1. The function f ∈A belongs to the class SL if it satisfies the condition that zf′(z) f(z) ≺ p̃(z) with p̃(z) = 1 + τ2z2 1 − τz − τ2z2 , where τ = (1 − √ 5)/2 ≈−0.618. CUBO 23, 2 (2021) Subclasses of λ-bi-pseudo-starlike functions with respect to .... 301 It should be observed SL is a subclass of the starlike functions S∗. The function p̃ is not univalent in U, but it is univalent in the disc |z| < (3 − √ 5)/2 ≈ 0.38. For example, p̃(0) = p̃(−1/2τ) = 1 and p̃(e∓i arccos(1/4)) = √ 5/5, and it may also be noticed that 1 |τ| = |τ| 1 −|τ| , which shows that the number |τ| divides [0, 1] such that it fulfils the golden section. The image of the unit circle |z| = 1 under p̃ is a curve described by the equation given by (10x− √ 5)y2 = ( √ 5 − 2x)( √ 5x− 1)2, which is translated and revolved trisectrix of Maclaurin. The curve p̃(reit) is a closed curve without any loops for 0 < r ≤ r0 = (3 − √ 5)/2 ≈ 0.38. For r0 < r < 1, it has a loop, and for r = 1, it has a vertical asymptote. Since τ satisfies the equation τ2 = 1 + τ, this expression can be used to obtain higher powers τn as a linear function of lower powers, which in turn can be decomposed all the way down to a linear combination of τ and 1. The resulting recurrence relationships yield Fibonacci numbers un: τn = unτ + un−1. In [11] Raina and Sokó l showed that p̃(z) = 1 + τ2z2 1 − τz − τ2z2 = ( t + 1 t ) t 1 − t− t2 = 1 √ 5 ( t + 1 t )( 1 1 − (1 − τ)t − 1 1 − τt ) = ( t + 1 t ) ∞∑ n=1 unt n = 1 + ∞∑ n=1 (un−1 + un+1)τ nzn, (1.3) where un = (1 − τ)n − τn √ 5 , τ = 1 − √ 5 2 , t = τz (n = 1, 2, . . .). (1.4) This shows that the relevant connection of p̃ with the sequence of Fibonacci numbers un, such that u0 = 0, u1 = 1, un+2 = un + un+1 for n = 0, 1, 2, · · · . And they got p̃(z) = 1 + ∞∑ n=1 p̃nz n = 1 + (u0 + u2)τz + (u1 + u3)τ 2z2 + ∞∑ n=3 (un−3 + un−2 + un−1 + un)τ nzn = 1 + τz + 3τ2z2 + 4τ3z3 + 7τ4z4 + 11τ5z5 + · · · . (1.5) Let P(β), 0 ≤ β < 1, denote the class of analytic functions p in U with p(0) = 1 and Re{p(z)} > β. Especially, we will use P instead of P(0). 302 H. Özlem Güney, G. Murugusundaramoorthy & K. Vijaya CUBO 23, 2 (2021) Theorem 1.2. [6] The function p̃(z) = 1 + τ2z2 1 − τz − τ2z2 belongs to the class P(β) with β = √ 5/10 ≈ 0.2236. Now we give the following lemma which will use in proving. Lemma 1.3. [10] Let p ∈P with p(z) = 1 + c1z + c2z2 + · · · , then |cn| ≤ 2, for n ≥ 1. (1.6) 2 Bi-Univalent function class PSLλs,Σ(α,p̃(z)) In this section, we introduce a new subclass of Σ associated with shell-like functions connected with Fibonacci numbers and obtain the initial Taylor coefficients |a2| and |a3| for the function class by subordination. Firstly, let p(z) = 1 + p1z + p2z 2 + · · · , and p ≺ p̃. Then there exists an analytic function u such that |u(z)| < 1 in U and p(z) = p̃(u(z)). Therefore, the function h(z) = 1 + u(z) 1 −u(z) = 1 + c1z + c2z 2 + · · · (2.1) is in the class P. It follows that u(z) = c1z 2 + ( c2 − c21 2 ) z2 2 + ( c3 − c1c2 + c31 4 ) z3 2 + · · · (2.2) and p̃(u(z)) = 1 + p̃1c1z 2 + { 1 2 ( c2 − c21 2 ) p̃1 + c21 4 p̃2 } z2 + { 1 2 ( c3 − c1c2 + c31 4 ) p̃1 + 1 2 c1 ( c2 − c21 2 ) p̃2 + c31 8 p̃3 } z3 + · · · . (2.3) And similarly, there exists an analytic function v such that |v(w)| < 1 in U and p(w) = p̃(v(w)). Therefore, the function k(w) = 1 + v(w) 1 −v(w) = 1 + d1w + d2w 2 + · · · (2.4) is in the class P(0). It follows that v(w) = d1w 2 + ( d2 − d21 2 ) w2 2 + ( d3 −d1d2 + d31 4 ) w3 2 + · · · (2.5) and p̃(v(w)) = 1 + p̃1d1w 2 + { 1 2 ( d2 − d21 2 ) p̃1 + d21 4 p̃2 } w2 + { 1 2 ( d3 −d1d2 + d31 4 ) p̃1 + 1 2 d1 ( d2 − d21 2 ) p̃2 + d31 8 p̃3 } w3 + · · · . (2.6) CUBO 23, 2 (2021) Subclasses of λ-bi-pseudo-starlike functions with respect to .... 303 The class Lλ(α) of λ-pseudo-starlike functions of order α (0 ≤ α < 1) were introduced and investigated by Babalola [1] whose geometric conditions satisfy < ( z(f′(z))λ f(z) ) > α, λ > 0. He showed that all pseudo-starlike functions are Bazilevič of type ( 1 − 1 λ ) order α 1 λ and univalent in open unit disk U. If λ = 1, we have the class of starlike functions of order α, which in this context, are 1-pseudo-starlike functions of order α. A function f ∈ A is starlike with respect to symmetric points in U if for every r close to 1, r < 1 and every z0 on |z| = r the angular velocity of f(z) about f(−z0) is positive at z = z0 as z traverses the circle |z| = r in the positive direction. This class was introduced and studied by Sakaguchi [13] presented the class S∗s of functions starlike with respect to symmetric points. This class consists of functions f(z) ∈S satisfying the condition < ( 2zf′(z) f(z) −f(−z) ) > 0, z ∈ U. Motivated by S∗s , Wang et al. [18] introduced the class Ks of functions convex with respect to symmetric points, which consists of functions f(z) ∈S satisfying the condition < ( 2(zf′(z))′ (f(z) −f(−z))′ ) > 0, z ∈ U. It is clear that, if f(z) ∈Ks , then zf′(z) ∈S∗s . For such a function φ, Ravichandran [12] presented the following subclasses: A function f ∈ A is in the class S∗s (φ) if 2zf′(z) f(z) −f(−z) ≺ φ(z), z ∈ U, and in the class Ks(φ) if 2(zf′(z))′ (f(z) −f(−z))′ ≺ φ(z) z ∈ U. Motivated by aforementioned works [1, 13, 12, 18] and recent study of Sokól [14] (also see [11]), in this paper we define the following new subclass f ∈ PSLλs,Σ(p̃(z)) of Σ named as λ-bi-pseudo- starlike functions with respect to symmetric points, related to shell-like curves connected with Fibonacci numbers, and determine the initial Taylor-Maclaurin coefficients |a2| and |a3|. Further we determine the Fekete-Szegö result for the function class PSLλs,Σ(p̃(z)) and the special cases are stated as corollaries which are new and have not been studied so far. Definition 2.1. For 0 ≤ α ≤ 1; λ > 0; λ 6= 1 3 ,a function f ∈ Σ of the form (1.1) is said to be in the class PSLλs,Σ(α,p̃(z)) if the following subordination hold:( 2z(f′(z))λ f(z) −f(−z) )α ( 2[(z(f′(z)))′]λ [f(z) −f(−z)]′ )1−α ≺ p̃(z) = 1 + τ2z2 1 − τz − τ2z2 (2.7) and ( 2w(g′(w))λ g(w) −g(−w) )α ( 2[(w(g′(w)))′]λ [g(w) −g(−w)]′ )1−α ≺ p̃(w) = 1 + τ2w2 1 − τw − τ2w2 (2.8) where τ = (1 − √ 5)/2 ≈−0.618 where z,w ∈ U and g is given by (1.2). 304 H. Özlem Güney, G. Murugusundaramoorthy & K. Vijaya CUBO 23, 2 (2021) Specializing the parameter λ = 1 we have the following definitions, respectively: Definition 2.2. For 0 ≤ α ≤ 1, a function f ∈ Σ of the form (1.1) is said to be in the class PSL1s,Σ(α,p̃(z)) ≡MSLs,Σ(α,p̃(z)) if the following subordination hold:( 2zf′(z) f(z) −f(−z) )α ( 2(z(f′(z)))′ [f(z) −f(−z)]′ )1−α ≺ p̃(z) = 1 + τ2z2 1 − τz − τ2z2 (2.9) and ( 2wg′(w) g(w) −g(−w) )α ( 2(w(g′(w)))′ [g(w) −g(−w)]′ )1−α ≺ p̃(w) = 1 + τ2w2 1 − τw − τ2w2 (2.10) where τ = (1 − √ 5)/2 ≈−0.618 where z,w ∈ U and g is given by (1.2). Further by specializing the parameter α = 1 and α = 0 we state the following new classes SL∗s,Σ(p̃(z)) and KLs,Σ(p̃(z)) respectively. Definition 2.3. A function f ∈ Σ of the form (1.1) is said to be in the class PSL1s,Σ(1, p̃(z)) ≡ SL∗s,Σ(p̃(z)) if the following subordination hold: 2zf′(z) f(z) −f(−z) ≺ p̃(z) = 1 + τ2z2 1 − τz − τ2z2 (2.11) and 2wg′(w) g(w) −g(−w) ≺ p̃(w) = 1 + τ2w2 1 − τw − τ2w2 (2.12) where τ = (1 − √ 5)/2 ≈−0.618 where z,w ∈ U and g is given by (1.2). Definition 2.4. A function f ∈ Σ of the form (1.1) is said to be in the class PSL1s,Σ(0, p̃(z)) ≡ KLs,Σ(p̃(z)) if the following subordination hold: 2(z(f′(z)))′ [f(z) −f(−z)]′ ≺ p̃(z) = 1 + τ2z2 1 − τz − τ2z2 (2.13) and 2(w(g′(w)))′ [g(w) −g(−w)]′ ≺ p̃(w) = 1 + τ2w2 1 − τw − τ2w2 (2.14) where τ = (1 − √ 5)/2 ≈−0.618 where z,w ∈ U and g is given by (1.2). Definition 2.5. For λ > 0; λ 6= 1 3 , a function f ∈ Σ of the form (1.1) is said to be in the class PSLλs,Σ(p̃(z)) if the following subordination hold:( 2z(f′(z))λ f(z) −f(−z) ) ≺ p̃(z) = 1 + τ2z2 1 − τz − τ2z2 (2.15) and ( 2w(g′(w))λ g(w) −g(−w) ) ≺ p̃(w) = 1 + τ2w2 1 − τw − τ2w2 (2.16) where τ = (1 − √ 5)/2 ≈−0.618 where z,w ∈ U and g is given by (1.2). CUBO 23, 2 (2021) Subclasses of λ-bi-pseudo-starlike functions with respect to .... 305 Definition 2.6. For λ > 0; λ 6= 1 3 , a function f ∈ Σ of the form (1.1) is said to be in the class GSLλs,Σ(p̃(z)) if the following subordination hold:( 2[(z(f′(z)))′]λ [f(z) −f(−z)]′ ) ≺ p̃(z) = 1 + τ2z2 1 − τz − τ2z2 (2.17) and ( 2[(w(g′(w)))′]λ [g(w) −g(−w)]′ ) ≺ p̃(w) = 1 + τ2w2 1 − τw − τ2w2 (2.18) where τ = (1 − √ 5)/2 ≈−0.618 where z,w ∈ U and g is given by (1.2). In the following theorem we determine the initial Taylor coefficients |a2| and |a3| for the function class PSLλs,Σ(α,p̃(z)). Later we will reduce these bounds to other classes for special cases. Theorem 2.7. Let f given by (1.1) be in the class PSLλs,Σ(α,p̃(z)). Then |a2| ≤ |τ|√ 4λ2(α− 2)2 −{10λ2(α− 2)2 −λ− 2α + 3}τ . (2.19) and |a3| ≤ 2λ|τ| [ 2λ(α− 2)2 −{5λ(α− 2)2 + 4 − 3α}τ ] (3λ− 1)(3 − 2α) [4λ2(α− 2)2 −{10λ2(α− 2)2 −λ− 2α + 3}τ] (2.20) where 0 ≤ α ≤ 1; λ > 0 and λ 6= 1 3 . Proof. Let f ∈PSLλs,Σ(α,p̃(z)) and g = f−1. Considering (2.7) and (2.8), we have ( 2z(f′(z))λ f(z) −f(−z) )α ( 2[(z(f′(z)))′]λ [f(z) −f(−z)]′ )1−α = p̃(u(z)) (2.21) and ( 2w(g′(w))λ g(w) −g(−w) )α ( 2[(w(g′(w)))′]λ [g(w) −g(−w)]′ )1−α = p̃(v(w)) (2.22) for some Schwarz functions u and v where τ = (1− √ 5)/2 ≈−0.618 where z,w ∈ U and g is given by (1.2). Since ( 2z[f′(z)]λ f(z) −f(−z) )α ( 2[(z(f′(z)))′]λ [f(z) −f(−z)]′ )1−α = 1 − 2λ(α− 2)a2z + {[2λ2(α− 2)2 + 2λ(3α− 4)]a22 + (3λ− 1)(3 − 2α)a3}z 2 + · · · and ( 2w(g′(w))λ g(w) −g(−w) )α ( 2[(w(g′(w)))′]λ [g(w) −g(−w)]′ )1−α = 1 + 2λ(α−2)a2w +{[2λ2(α−2)2 + 2λ(5−3α) + 2(2α−3)]a22 + (3λ−1)(2α−3)a3}w 2 + · · · . 306 H. Özlem Güney, G. Murugusundaramoorthy & K. Vijaya CUBO 23, 2 (2021) Thus we have 1 − 2λ(α− 2)a2z + {[2λ2(α− 2)2 + 2λ(3α− 4)]a22 + (3λ− 1)(3 − 2α)a3}z 2 + · · · = 1 + p̃1c1z 2 + [ 1 2 ( c2 − c21 2 ) p̃1 + c21 4 p̃2 ] z2 + [ 1 2 ( c3 − c1c2 + c31 4 ) p̃1 + 1 2 c1 ( c2 − c21 2 ) p̃2 + c31 8 p̃3 ] z3 + · · · . (2.23) and 1 + 2λ(α− 2)a2w + {[2λ2(α− 2)2 + 2λ(5 − 3α) + 2(2α− 3)]a22 + (3λ− 1)(2α− 3)a3}w 2 = 1 + p̃1d1w 2 + [ 1 2 ( d2 − d21 2 ) p̃1 + d21 4 p̃2 ] w2 + [ 1 2 ( d3 −d1d2 + d31 4 ) p̃1 + 1 2 d1 ( d2 − d21 2 ) p̃2 + d31 8 p̃3 ] w3 + · · · . (2.24) It follows from (1.5), (2.23) and (2.24) that − 2λ(α− 2)a2 = c1τ 2 , (2.25) [2λ2(α− 2)2 + 2λ(3α− 4)]a22 + (3λ− 1)(3 − 2α)a3 = 1 2 ( c2 − c21 2 ) τ + 3 4 c21τ 2, (2.26) and 2λ(α− 2)a2 = d1τ 2 , (2.27) [2λ2(α− 2)2 + 2λ(5 − 3α) + 2(2α− 3)]a22 + (3λ− 1)(2α− 3)a3 = 1 2 ( d2 − d21 2 ) τ + 3 4 d21τ 2. (2.28) From (2.25) and (2.27), we have c1 = −d1, (2.29) and a22 = (c21 + d 2 1) 32λ2(α− 2)2 τ2. (2.30) Now, by summing (2.26) and (2.28), we obtain [ 4λ2(α− 2)2 + 2(λ + 2α− 3) ] a22 = 1 2 (c2 + d2)τ − 1 4 (c21 + d 2 1)τ + 3 4 (c21 + d 2 1)τ 2. (2.31) By putting (2.30) in (2.31), we have 2 [ 8λ2(α− 2)2 −{20λ2(α− 2)2 − 2(λ + 2α− 3)}τ ] a22 = (c2 + d2)τ 2. (2.32) Therefore, using Lemma 1.3 we obtain |a2| ≤ |τ|√ 4λ2(α− 2)2 −{10λ2(α− 2)2 −λ− 2α + 3}τ . (2.33) Now, so as to find the bound on |a3|, let’s subtract from (2.26) and (2.28). So, we find 2(3λ− 1)(3 − 2α)a3 − 2(3λ− 1)(3 − 2α)a22 = 1 2 (c2 −d2) τ. (2.34) CUBO 23, 2 (2021) Subclasses of λ-bi-pseudo-starlike functions with respect to .... 307 Hence, we get 2(3λ− 1)(3 − 2α)|a3| ≤ 2|τ| + 2(3λ− 1)(3 − 2α)|a2|2. (2.35) Then, in view of (2.33), we obtain |a3| ≤ 2λ|τ| [ 2λ(α− 2)2 −{5λ(α− 2)2 + 4 − 3α}τ ] (3λ− 1)(3 − 2α) [4λ2(α− 2)2 −{10λ2(α− 2)2 −λ− 2α + 3}τ] . (2.36) If we can take the parameter λ = 1 in the above theorem, we have the following the initial Taylor coefficients |a2| and |a3| for the function classes MSLs,Σ(α,p̃(z)). Corollary 2.8. Let f given by (1.1) be in the class MSLs,Σ(α,p̃(z)). Then |a2| ≤ |τ|√ 4(α− 2)2 − 2(5α2 − 21α + 21)τ (2.37) and |a3| ≤ |τ| [ 2(α− 2)2 −{5α2 − 23α + 24}τ ] (3 − 2α) [4(α− 2)2 −{10α2 − 42α + 42}τ] . (2.38) Further by taking α = 1 and α = 0 and τ = −0.618 in Corollary 2.8, we have the following im- proved initial Taylor coefficients |a2| and |a3| for the function classes SL∗s,Σ(p̃(z)) and KLs,Σ(p̃(z)) respectively. Corollary 2.9. Let f given by (1.1) be in the class SL∗s,Σ(p̃(z)). Then |a2| ≤ |τ| √ 4 − 10τ ' 0.19369 (2.39) and |a3| ≤ |τ|(1 − 3τ) 2 − 5τ ' 0.3465. (2.40) Corollary 2.10. Let f given by (1.1) be in the class KLs,Σ(p̃(z)). Then |a2| ≤ |τ| √ 16 − 42τ ' 0.0954 (2.41) and |a3| ≤ 4|τ|(1 − 3τ) 3(8 − 21τ) ' 0.17647. (2.42) Corollary 2.11. Let f given by (1.1) be in the class PSLλs,Σ(p̃(z)). Then |a2| ≤ |τ|√ 4λ2 −{10λ2 −λ + 1}τ (2.43) and |a3| ≤ 2λ|τ| [2λ−{5λ + 1}τ] (3λ− 1) [4λ2 −{10λ2 −λ + 1}τ] (2.44) where λ > 0 and λ 6= 1 3 . 308 H. Özlem Güney, G. Murugusundaramoorthy & K. Vijaya CUBO 23, 2 (2021) Corollary 2.12. Let f given by (1.1) be in the class GSLλs,Σ(p̃(z)). Then |a2| ≤ |τ|√ 16λ2 −{40λ2 −λ + 3}τ (2.45) and |a3| ≤ 2λ|τ| [8λ−{20λ + 4}τ] 3(3λ− 1) [16λ2 −{40λ2 −λ + 3}τ] (2.46) where λ > 0 and λ 6= 1 3 . 3 Fekete-Szegö inequality for the function class PSLλs,Σ(α,p̃(z)) Fekete and Szegö [7] introduced the generalized functional |a3−µa22|, where µ is some real number. Due to Zaprawa [19], in the following theorem we determine the Fekete-Szegö functional for f ∈ PSLλs,Σ(α,p̃(z)). Theorem 3.1. Let λ ∈ R with λ > 1 3 and let f given by (1.1) be in the class PSLλs,Σ(α,p̃(z)) and µ ∈ R. Then we have |a3 −µa22| ≤   |τ| 4(3λ− 1)(3 − 2α) , 0 ≤ |h(µ)| ≤ |τ| 4(3λ− 1)(3 − 2α) 4|h(µ)|, |h(µ)| ≥ |τ| 4(3λ− 1)(3 − 2α) where h(µ) = (1 −µ)τ2 4 [4λ2(α− 2)2 −{10λ2(α− 2)2 −λ− 2α + 3}τ] . (3.1) Proof. From (2.32) and (2.34) we obtain a3 −µa22 = (1 −µ)(c2 + d2)τ2 4 [4λ2(α− 2)2 −{10λ2(α− 2)2 −λ− 2α + 3}τ] + τ(c2 −d2) 4(3λ− 1)(3 − 2α) = ( (1 −µ)τ2 4 [4λ2(α− 2)2 −{10λ2(α− 2)2 −λ− 2α + 3}τ] + τ 4(3λ− 1)(3 − 2α) ) c2 + ( (1 −µ)τ2 4 [4λ2(α− 2)2 −{10λ2(α− 2)2 −λ− 2α + 3}τ] − τ 4(3λ− 1)(3 − 2α) ) d2. So we have a3 −µa22 = ( h(µ) + τ 4(3λ− 1)(3 − 2α) ) c2 + ( h(µ) − τ 4(3λ− 1)(3 − 2α) ) d2 (3.2) where h(µ) = (1 −µ)τ2 4 [4λ2(α− 2)2 −{10λ2(α− 2)2 −λ− 2α + 3}τ] . Then, by taking modulus of (3.2), we conclude that CUBO 23, 2 (2021) Subclasses of λ-bi-pseudo-starlike functions with respect to .... 309 |a3 −µa22| ≤   |τ| 4(3λ− 1)(3 − 2α) , 0 ≤ |h(µ)| ≤ |τ| 4(3λ− 1)(3 − 2α) 4|h(µ)|, |h(µ)| ≥ |τ| 4(3λ− 1)(3 − 2α) . Taking µ = 1, we have the following corollary. Corollary 3.2. If f ∈PSLλs,Σ(α,p̃(z)), then |a3 −a22| ≤ |τ| 4(3λ− 1)(3 − 2α) . (3.3) If we can take the parameter λ = 1 in Theorem 3.1, we can state the following: Corollary 3.3. Let f given by (1.1) be in the class MSLs,Σ(α,p̃(z)) and µ ∈ R. Then we have |a3 −µa22| ≤   |τ| 8(3 − 2α) , 0 ≤ |h(µ)| ≤ |τ| 8(3 − 2α) 4|h(µ)|, |h(µ)| ≥ |τ| 8(3 − 2α) where h(µ) = (1 −µ)τ2 4 [4(α− 2)2 −{10(α− 2)2 − 2α + 2}τ] . Further by fixing λ = 1 taking α = 1 and α = 0 in the above corollary, we have the following the Fekete-Szegö inequalities for the function classes SL∗s,Σ(p̃(z)) and KLs,Σ(p̃(z)), respectively. Corollary 3.4. Let f given by (1.1) be in the class SL∗s,Σ(p̃(z)) and µ ∈ R. Then we have |a3 −µa22| ≤   |τ| 24 , 0 ≤ |h(µ)| ≤ |τ| 24 4|h(µ)|, |h(µ)| ≥ |τ| 24 where h(µ) = (1 −µ)τ2 8 [2 − 5τ] . Corollary 3.5. Let f given by (1.1) be in the class KLs,Σ(p̃(z)) and µ ∈ R. Then we have |a3 −µa22| ≤   |τ| 8 , 0 ≤ |h(µ)| ≤ |τ| 8 4|h(µ)|, |h(µ)| ≥ |τ| 8 where h(µ) = (1 −µ)τ2 8 [8 − 21τ] . By assuming λ ∈ R; λ > 1 3 and taking α = 1 and α = 0 we have the following the Fekete-Szegö inequalities for the function classes PSLλs,Σ(p̃(z)) and GSL λ s,Σ(p̃(z)), respectively. 310 H. Özlem Güney, G. Murugusundaramoorthy & K. Vijaya CUBO 23, 2 (2021) Corollary 3.6. Let λ ∈ R with λ > 1 3 and let f given by (1.1) be in the class PSLλs,Σ(p̃(z)) and µ ∈ R. Then we have |a3 −µa22| ≤   |τ| 4(3λ− 1) , 0 ≤ |h(µ)| ≤ |τ| 4(3λ− 1) 4|h(µ)|, |h(µ)| ≥ |τ| 4(3λ− 1) where h(µ) = (1 −µ)τ2 4 [4λ2 −{10λ2 −λ + 1}τ] . Corollary 3.7. Let λ ∈ R with λ > 1 3 and let f given by (1.1) be in the class GSLλs,Σ(p̃(z)) and µ ∈ R. Then we have |a3 −µa22| ≤   |τ| 12(3λ− 1) , 0 ≤ |h(µ)| ≤ |τ| 12(3λ− 1) 4|h(µ)|, |h(µ)| ≥ |τ| 12(3λ− 1) where h(µ) = (1 −µ)τ2 4 [16λ2 −{40λ2 −λ + 3}τ] . Conclusions Our motivation is to get many interesting and fruitful usages of a wide variety of Fibonacci numbers in Geometric Function Theory. 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Introduction Bi-Univalent function class PSLs,(,(z)) Fekete-Szegö inequality for the function class PSLs,(,(z))