Subclasses of bi-univalent functions related to shell-like curves connected with Fibonacci numbers

Acta Univ. Sapientiae, Mathematica, 10, 1018 70-84 DOI: 10.478/ausm-018-0006 Subclasses of bi-univalent functions related to shell-like curves connected with Fibonacci numbers H. Özlem Güney Dicle University, Faculty of Science, Department of Mathematics, Turkey email: ozlemg@dicle.edu.tr G. Murugusundaramoorthy Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Deemed to be University, India email: gmsmoorthy@yahoo.com J. Sokó l University of Rzeszów, Faculty of Mathematics Natural Sciences, Pol email: jsokol@ur.edu.pl Abstract. In this paper, we introduce investigate new subclasses of bi-univalent functions related to shell-like curves connected with Fibonacci numbers. Furthermore, we find estimates of first two coefficients of functions in these classes. Also, we determine Fekete-Szegö inequalities for these function classes. 1 Introduction Let U = {z : z < 1} denote the unit disc on the complex plane. The class of all analytic functions of the form f z = z + a n z n 1 010 Mathematics Subject Classification: 30C45, 30C50 Key words phrases: analytic functions, bi-univalent, shell-like curve, Fibonacci numbers, starlike functions, convex functions 70 n=

Subclasses of bi-univalent functions related to shell-like curves 71 in the open unit disc U with normalization f0 = f 0 1 = 0 is denoted by A the class S A is the class which consists of univalent functions in U. The Koebe one quarter theorem [3] ensures that the image of U under every univalent function f A contains a disk of radius 1 4. Thus every univalent function f A has an inverse f 1 satisfying f 1 fz = z, z U ff 1 w = w w < r 0 f, r 0 f 1 4 A function f A is said to be bi-univalent in U if both f 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 Maclaurian series given by 1, a computation shows that its inverse g = f 1 has the expansion gw = f 1 w = w a w + a a 3w 3 +. One can see a short history examples of functions in the class Σ in [1]. Several authors have introduced investigated subclasses of bi-univalent functions obtained bounds for the initial coefficients see [1,, 8, 1, 13, 14]. 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 ω0 = 0 ωz < 1 satisfying fz = Fωz. It follows from Schwarz Lemma that. fz Fz f0 = F0 fu FU, z U for details see [3], [7]. We recall important subclasses of S in geometric function theory such that if f A zf z fz pz 1 + zf z f z pz where pz = 1+z 1 z, then we say that f is starlike convex, respectively. These functions form known classes denoted by S C, respectively. Recently,in [11], 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 The function f A belongs to the class SL if it satisfies the condition that zf z pz fz

7 H. Ö. Güney, G. Murugusundaramoorthy, J. Sokó l with where τ = 1 5/ 0.618. pz = 1 + τ z 1 τz τ z, It should be observed SL is a subclass of the starlike functions S. Later, Dziok et al. in [4] [5] defined introduced the class KSL SLM α of convex α convex functions related to a shell-like curve connected with Fibonacci numbers, respectively. These classes can be given in the following definitions. Definition The function f A belongs to the class KSL of convex shell-like functions if it satisfies the condition that where τ = 1 5/ 0.618. 1 + zf z f z pz = 1 + τ z 1 τz τ z, Definition 3 The function f A belongs to the class SLM α, 0 α 1 if it satisfies the condition that α 1 + zf z f + 1 α zf z pz = 1 + τ z z fz 1 τz τ z, where τ = 1 5/ 0.618. The class SLM α is related to the class KSL only through the function p SLM α KSL for all α 1. It is easy to see that KSL = SLM 1. Besides, let s define the class SLG γ of so-called gamma-starlike functions related to a shell-like curve connected with Fibonacci numbers as follows. Definition 4 The function f A belongs to the class SLG γ, γ 0, if it satisfies the condition that zf z γ 1 + zf z 1 γ fz f pz = 1 + τ z z 1 τz τ z, where τ = 1 5/ 0.618.

Subclasses of bi-univalent functions related to shell-like curves 73 The function p is not univalent in U, but it is univalent in the disc z < 3 5/ 0.38. For example, p0 = p 1/τ = 1 pe i arccos1/4 = 5/5, it may also be noticed that 1 = 1, which shows that the number divides [0, 1] such that it fulfils the golden section. The image of the unit circle z = 1 under p is a curve described by the equation given by 10x 5 y = 5 x 5x 1, which is translated revolved trisectrix of Maclaurin. The curve pre it is a closed curve without any loops for 0 < r r 0 = 3 5/ 0.38. For r 0 < r < 1, it has a loop, for r = 1, it has a vertical asymptote. Since τ satisfies the equation τ = 1 + τ, this expression can be used to obtain higher powers τ n as a linear function of lower powers, which in turn can be decomposed all the way down to a linear combination of τ 1. The resulting recurrence relationships yield Fibonacci numbers u n : τ n = u n τ + u n 1. In [10], taking τz = t, Raina Sokó l showed that pz = 1 + τ z 1 τz τ z = = 1 t + 1 5 t = = t + 1 t t + 1 t n=1 n=1 t + 1 t t 1 t t 1 1 1 τt 1 1 τt 1 τ n τ n t n 5 u n t n = 1 + u n 1 + u n+1 τ n z n, n=1 3 where u n = 1 τn τ n 5, τ = 1 5 n = 1,,.... 4 This shows that the relevant connection of p with the sequence of Fibonacci numbers u n, such that u 0 = 0, u 1 = 1, u n+ = u n + u n+1 for n = 0, 1,,.

74 H. Ö. Güney, G. Murugusundaramoorthy, J. Sokó l And they got pz = 1 + p n z n = 1 + u 0 + u τz + u 1 + u 3 τ z + n=1 u n 3 + u n + u n 1 + u n τ n z n n=3 = 1 + τz + 3τ z + 4τ 3 z 3 + 7τ 4 z 4 + 11τ 5 z 5 +. 5 Let Pβ, 0 β < 1, denote the class of analytic functions p in U with p0 = 1 Re{pz} > β. Especially, we will use P instead of P0. Theorem 1 [5] The function pz = 1+τ z 1 τz τ z β = 5/10 0.36. Now we give the following lemma which will use in proving. belongs to the class Pβ with Lemma 1 [9] Let p P with pz = 1 + c 1 z + c z +, then c n, for n 1. 6 In this present work, we introduce two subclasses of Σ associated with shelllike functions connected with Fibonacci numbers obtain the initial Taylor coefficients a a 3 for these function classes. Also, we give bounds for the Fekete-Szegö functional a 3 µa for each subclass. Bi-univalent function class SLM α,σ pz In this section, we introduce a new subclass of Σ associated with shell-like functions connected with Fibonacci numbers obtain the initial Taylor coefficients a a 3 for the function class by subordination. Firstly, let pz = 1 + p 1 z + p z +, p p. Then there exists an analytic function u such that uz < 1 in U pz = puz. Therefore, the function hz = 1 + uz 1 uz = 1 + c 1z + c z +... 7 is in the class P0. It follows that uz = c 1z + c c 1 z + c 3 c 1 c + c3 1 z 3 4 + 8

Subclasses of bi-univalent functions related to shell-like curves 75 puz = 1 + p 1 { c1 z + + p { c1 z + + p 3 { c1 z + = 1 + p 1c 1 z + { 1 + } c c 1 z + c 3 c 1 c + c3 1 z 3 4 + } z + c 3 c 1 c + c3 1 z 3 4 + } z + c 3 c 1 c + c3 1 z 3 3 4 + + } c c 1 p 1 + c 1 4 p z c c 1 c c 1 { 1 c 3 c 1 c + c3 1 4 p 1 + 1 c 1 c c 1 } p + c3 1 8 p 3 z 3 +. And similarly, there exists an analytic function v such that vw < 1 in U pw = pvw. Therefore, the function kw = 1 + vw 1 vw = 1 + d 1w + d w +... 10 is in the class P0. It follows that vw = d 1w + d d 1 w + { 1 + pvw = 1 + p 1d 1 w { 1 + d 3 d 1 d + d3 1 4 d d 1 p 1 + 1 d 1 9 d 3 d 1 d + d3 1 w 3 4 + 11 p 1 + d 1 d d 1 4 p } w p + d3 1 8 p 3 } 1 w 3 +. Definition 5 For 0 α 1,a function f Σ of the form 1 is said to be in the class SLM α,σ pz if the following subordination hold: α 1 + zf z zf z f + 1 α pz = 1 + τ z z fz 1 τz τ z 13 α 1 + wg w g + 1 α w wg w pw = gw 1 + τ w 1 τw τ w 14 where τ = 1 5/ 0.618 where z, w U g is given by.

76 H. Ö. Güney, G. Murugusundaramoorthy, J. Sokó l Specializing the parameter α = 0 α = 1 we have the following, respectively: Definition 6 A function f Σ of the form 1 is said to be in the class SL Σ pz if the following subordination hold: zf z fz pz = 1 + τ z 1 τz τ z 15 wg w gw pw = 1 + τ w 1 τw τ w 16 where τ = 1 5/ 0.618 where z, w U g is given by. Definition 7 A function f Σ of the form 1 is said to be in the class KL Σ pz if the following subordination hold: 1 + zf z f z pz = 1 + τ z 1 τz τ z 17 1 + wg w g w pw = 1 + τ w 1 τw τ w 18 where τ = 1 5/ 0.618 where z, w U g is given by. In the following theorem we determine the initial Taylor coefficients a a 3 for the function class SLM α,σ pz. Later we will reduce these bounds to other classes for special cases. Theorem Let f given by 1 be in the class SLM α,σ pz. Then a 1 + α 1 + α + 3ατ 19 a 3 [ 1 + α 3α + 9α + 4τ ] 1 + α1 + α [1 + α + 3ατ]. 0 Proof. Let f SLM α,σ pz g = f 1. Considering 13 14, we have α 1 + zf z zf z f + 1 α = puz 1 z fz

Subclasses of bi-univalent functions related to shell-like curves 77 α 1 + wg w wg w g + 1 α = pvw w gw where τ = 1 5/ 0.618 where z, w U g is given by. Since α 1 + zf z zf z f + 1 α = 1 + 1 + αa z + 1 + αa 3 z fz α 1 + 3αa z + 1 + wg w wg w g + 1 α = 1 1 + αa w + 3 + 5αa w gw Thus we have 1 + αa 3 w + 1 + 1 + αa z + 1 + αa 3 1 + 3αa z + = 1 + p [ ] 1c 1 z 1 + c c 1 p 1 + c 1 4 p z [ 1 + c 3 c 1 c + c3 1 p 1 + 1 ] 4 c 1 c c 1 p + c3 1 8 p 3 z 3 +. 1 1 + αa w + 3 + 5αa 1 + αa 3w +, = 1 + p [ ] 1d 1 w 1 + d d 1 p 1 + d 1 4 p w [ 1 + d 3 d 1 d + d3 1 p 1 + 1 ] 4 d 1 d d 1 p + d3 1 8 p 3 w 3 +. It follows from 3 4 that 3 4 1 + αa = c 1τ, 5 1 + αa 3 1 + 3αa = 1 c c 1 τ + c 1 4 3τ, 6 1 + αa = d 1τ, 7 3 + 5αa 1 + αa 3 = 1 d d 1 τ + d 1 4 3τ. 8

78 H. Ö. Güney, G. Murugusundaramoorthy, J. Sokó l From 5 7, we have Now, by summing 6 8, we obtain c 1 = d 1, 9 a = c 1 + d 1 41 + α τ. 30 1 + αa = 1 c + d τ 1 4 c 1 + d 1 τ + 3 4 c 1 + d 1 τ. 31 By putting 30 in 31, we have 1 + α [ 3ατ + 1 + α] a = 1 c + d τ. 3 Therefore, using Lemma 1 we obtain a 1 + α 1 + α + 3ατ. 33 Now, so as to find the bound on a 3, let s subtract from 6 8. So, we find Hence, we get Then, in view of 33, we obtain 41 + αa 3 41 + αa = 1 c d τ. 34 a 3 41 + α a 3 + 41 + α a. 35 [ 1 + α 3α + 9α + 4τ ] 1 + α1 + α [1 + α + 3ατ]. 36 If we can take the parameter α = 0 α = 1 in the above theorem, we have the following the initial Taylor coefficients a a 3 for the function classes SL Σ pz KSL Σ pz, respectively. Corollary 1 Let f given by 1 be in the class SL Σ pz. Then a 1 τ 37 a 3 1 4τ 1 τ. 38

Subclasses of bi-univalent functions related to shell-like curves 79 Corollary Let f given by 1 be in the class KSL Σ pz. Then a 4 10τ 39 a 3 1 4τ 3 5τ. 40 3 Bi-univalent function class SLG γ,σ pz In this section, we define a new class SLG γ,σ pz of γ bi-starlike functions associated with Shell-like domain. Definition 8 For γ 0, we let a function f Σ given by 1 is said to be in the class SLG γ,σ pz, if the following conditions are satisfied: zf z γ 1 + zf z 1 γ fz f pz = 1 + τ z z 1 τz τ z 41 wg w γ 1 + wg w 1 γ gw g pw = 1 + τ w w 1 τz τ w, 4 where τ = 1 5/ 0.618 where z, w U g is given by. Remark 1 Taking γ = 1, we get SLG 1,Σ pz SL Σ pz the class as given in Definition 5 satisfying the conditions given in 15 16. Remark Taking γ = 0, we get SLG 0,Σ pz KL Σ pzthe class as given in Definition 6 satisfying the conditions given in 17 18. Theorem 3 Let f given by 1 be in the class SLG γ,σ pz. Then a γ 5γ 1γ + 0τ a 3 [ γ 5γ 9γ + 3τ ] 3 γ [ γ 5γ 1γ + 0τ].

80 H. Ö. Güney, G. Murugusundaramoorthy, J. Sokó l Proof. Let f SLG γ,σ pz g = f 1 given by Considering 41 4, we have zf z γ 1 + zf z 1 γ fz f puz 43 z wg w γ 1 + wg w 1 γ gw g pvw 44 w where τ = 1 5/ 0.618 where z, w U g is given by. Since, zf z fz + γ 1 + zf z 1 γ f = 1 + γa z z 3 γa 3 + 1 [γ 34 3γ]a z + puz 45 wg w gw + γ 1 + wg w 1 γ g = 1 γa w w [81 γ + 1 γγ + 5]a 3 γa 3 w + pvw. 46 Equating the coefficients in45 46, with 9 1 respectively we get, γa = c 1τ, 47 3 γa 3 + 1 [γ 34 3γ]a = 1 c c 1 τ + c 1 4 3τ, 48 γa = d 1τ, 49 3 γa 3 + [81 γ + 1 γγ + 5]a = 1 d d 1 τ + d 1 4 3τ. 50 From 47 49, we have a = c 1τ γ = d 1τ γ, 51

Subclasses of bi-univalent functions related to shell-like curves 81 which implies c 1 = d 1 5 Now, by summing 48 50, we obtain a = c 1 + d 1 τ 8 γ. 53 γ 3γ + 4a = 1 c + d τ 1 4 c 1 + d 1 τ + 3 4 c 1 + d 1 τ. 54 Proceeding similarly as in the earlier proof of Theorem, using Lemma 1 we obtain a γ 5γ 1γ + 0τ. 55 Now, so as to find the bound on a 3, let s subtract from 48 50. So, we find 43 γa 3 43 γa = 1 c d τ. 56 Hence, we get 41 + γ a 3 + 41 + γ a. 57 Then, in view of 55, we obtain a 3 [ γ 5γ 9γ + 3τ ] 3 γ [ γ 5γ 1γ + 0τ]. 58 Remark 3 By taking γ = 1 γ = 0 in the above theorem, we have the initial Taylor coefficients a a 3 for the function classes SL Σ pz KSL Σ pz, as stated in Corollary 1 Corollary respectively. 4 Fekete-Szegö inequalities for the function classes SLM α,σ pz SLG γ,σ pz Fekete Szegö [6] introduced the generalized functional a 3 µa, where µ is some real number. Due to Zaprawa [15], in the following theorem we determine the Fekete-Szegö functional for f SLM α,σ pz.

8 H. Ö. Güney, G. Murugusundaramoorthy, J. Sokó l Theorem 4 Let f given by 1 be in the class SLM α,σ pz µ R. Then we have a 3 µa 1+α, 1+α[1+α +3ατ] µ 1 1+α. 1 µ τ 1+α[1+α +3ατ], Proof. From 3 34we obtain 1+α[1+α +3ατ] µ 1 1+α a 3 µa = 1 µ τ c + d 41 + α [1 + α + 3ατ] + τc d 81 + α 1 µτ = 41 + α [1 + α + 3ατ] + τ c 81 + α 1 µτ + 41 + α [1 + α + 3ατ] τ d. 81 + α 59 So we have where a 3 µa = hµ + hµ = c + hµ d 60 81 + α 81 + α 1 µτ 41 + α [1 + α + 3ατ]. 61 Then, by taking modulus of 60, we conclude that { a 3 µa 1+α, 0 hµ 81+α 4 hµ, hµ 81+α Taking µ = 1, we have the following corollary.. Corollary 3 If f SLM α,σ pz, then a 3 a 1 + α. 6 If we can take the parameter α = 0 α = 1 in the above theorem, we have the following the Fekete-Szegö inequalities for the function classes SL Σ pz KSL Σ pz, respectively.

Subclasses of bi-univalent functions related to shell-like curves 83 Corollary 4 Let f given by 1 be in the class SL Σ pz µ R. Then we have { 1 τ a 3 µa, µ 1. 1 τ µ 1 1 µ τ 1 τ, Corollary 5 Let f given by 1 be in the class KSL Σ pz µ R. Then we have { 5τ a 3 µa 6, µ 1 3. 5τ µ 1 1 µ τ 5τ, In the following theorem, we find the Fekete-Szegö functional for f SLG γ,σ pz. Theorem 5 Let f given by 1 be in the class SLG γ,σ pz µ R. Then we have a 3 µa 3 γ, µ 1 γ + 5γ +1γ 0τ 43 γ. 1 µ τ γ +[ 5γ +1γ 0τ], Taking µ = 1, we have the following corollary. Corollary 6 If f SLG γ,σ pz, then 3 µ 1 γ + 5γ +1γ 0τ 43 γ a 3 a 3 γ. 63 By taking γ = 1 γ = 0 in the above theorem, we have the Fekete-Szegö inequality for the function classes SL Σ pz KSL Σ pz, as stated in Corollary 4 Corollary 5, respectively. References [1] D. A. Brannan, J. Clunie W. E. Kirwan, Coefficient estimates for a class of star-like functions, Canad. J. Math., 1970, 476 485. [] D. A. Brannan T. S. Taha, On some classes of bi-univalent functions, Stud. Univ. Babeş-Bolyai Math., 31 1986, 70 77. [3] P. L. Duren, Univalent Functions. In: Grundlehren der Mathematischen Wissenschaften, B 59, New York, Berlin, Heidelberg Tokyo, Springer-Verlag, 1983.

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