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

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1 Acta Univ. Sapientiae, Mathematica, 10, DOI: /ausm 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 G. Murugusundaramoorthy Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Deemed to be University, India J. Sokó l University of Rzeszów, Faculty of Mathematics Natural Sciences, Pol 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 Mathematics Subject Classification: 30C45, 30C50 Key words phrases: analytic functions, bi-univalent, shell-like curve, Fibonacci numbers, starlike functions, convex functions 70 n=

2 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

3 7 H. Ö. Güney, G. Murugusundaramoorthy, J. Sokó l with where τ = 1 5/ 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/ 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/ 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/

4 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/ 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/ 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 t = = t + 1 t t + 1 t n=1 n=1 t + 1 t t 1 t t 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,, 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,,.

5 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 τ 5 z 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/ 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 is in the class P0. It follows that uz = c 1z + c c 1 z + c 3 c 1 c + c3 1 z

6 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 } z + c 3 c 1 c + c3 1 z } z + c 3 c 1 c + c3 1 z } 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 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 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 d 1 9 d 3 d 1 d + d3 1 w 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/ where z, w U g is given by.

7 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/ 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 wg w g w pw = 1 + τ w 1 τw τ w 18 where τ = 1 5/ 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

8 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/ where z, w U g is given by. Since α 1 + zf z zf z f + 1 α = αa z αa 3 z fz α 1 + 3αa z wg w wg w g + 1 α = αa w αa w gw Thus we have 1 + αa 3 w αa z αa αa 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 ] 4 c 1 c c 1 p + c3 1 8 p 3 z αa w αa 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 ] 4 d 1 d d 1 p + d3 1 8 p 3 w 3 +. It follows from 3 4 that αa = c 1τ, αa αa = 1 c c 1 τ + c 1 4 3τ, αa = d 1τ, αa 1 + αa 3 = 1 d d 1 τ + d 1 4 3τ. 8

9 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 α τ αa = 1 c + d τ 1 4 c 1 + d 1 τ c 1 + d 1 τ. 31 By putting 30 in 31, we have 1 + α [ 3ατ 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 αa = 1 c d τ. 34 a α a 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

10 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 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/ 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 Remark Taking γ = 0, we get SLG 0,Σ pz KL Σ pzthe class as given in Definition 6 satisfying the conditions given in 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].

11 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/ 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 [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 [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

12 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 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 So, we find 43 γa 3 43 γa = 1 c d τ. 56 Hence, we get 41 + γ a 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.

13 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 α [1 + α + 3ατ] τ d α 59 So we have where a 3 µa = hµ + hµ = c + hµ d 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.

14 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 τ 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, [] D. A. Brannan T. S. Taha, On some classes of bi-univalent functions, Stud. Univ. Babeş-Bolyai Math., , [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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