Certain classes of bi-univalent functions related to Shell-like curves connected with Fibonacci numbers
aa r X i v : . [ m a t h . C V ] O c t CERTAIN CLASSES OF BI-UNIVALENT FUNCTIONS RELATED TOSHELL-LIKE CURVES CONNECTED WITH FIBONACCI NUMBERS
N. MAGESH, V. K. BALAJI AND C. ABIRAMI
Abstract.
Recently, in their pioneering work on the subject of bi-univalent functions,Srivastava et al. [23] actually revived the study of the coefficient problems involvingbi-univalent functions. Inspired by the pioneering work of Srivastava et al. [23], therehas been triggering interest to study the coefficient problems for the different subclassesof bi-univalent functions. Motivated largely by Ali et al. [1], Srivastava et al. [23] andG¨uney et al. [10] in this paper, we consider certain classes of bi-univalent functionsrelated to shell-like curves connected with Fibonacci numbers to obtain the estimatesof second, third Taylor-Maclaurin coefficients and Fekete - Szeg¨o inequalities. Further,certain special cases are also indicated. Some interesting remarks of the results presentedhere are also discussed. Introduction and definitions
Let R = ( −∞ , ∞ ) be the set of real numbers, C be the set of complex numbers and N := { , , , . . . } = N \ { } be the set of positive integers. Let A denote the class of functions of the form f ( z ) = z + ∞ X n =2 a n z n (1.1)which are analytic in the open unit disk D = { z : z ∈ C and | z | < } . Further, by S weshall denote the class of all functions in A which are univalent in D . Let P denote the class of functions of the form p ( z ) = 1 + p z + p z + p z + . . . ( z ∈ D )which are analytic with ℜ { p ( z ) } > . Here p ( z ) is called as Caratheodory functions [8].It is well known that the following correspondence between the class P and the class ofSchwarz functions w exists: p ∈ P if and only if p ( z ) = 1 + w ( z ) / 1 − w ( z ) . Let P ( β ) , ≤ β < , denote the class of analytic functions p in D with p (0) = 1 and ℜ { p ( z ) } > β. For analytic functions f and g in D , f is said to be subordinate to g if there exists ananalytic function w such that w (0) = 0 , | w ( z ) | < f ( z ) = g ( w ( z )) ( z ∈ D ) . This subordination will be denoted here by f ≺ g ( z ∈ D )or, conventionally, by f ( z ) ≺ g ( z ) ( z ∈ D ) . Mathematics Subject Classification.
Primary 30C45; Secondary 30C50.
Key words and phrases.
Univalent functions, bi-univalent functions, shell-like function, convex shell-like function, pseudo starlike function, Bazilevi´c function.
In particular, when g is univalent in D ,f ≺ g ( z ∈ D ) ⇔ f (0) = g (0) and f ( D ) ⊂ g ( D ) . Some of the important and well-investigated subclasses of the univalent function class S include (for example) the class S ∗ ( α ) of starlike functions of order α (0 ≦ α <
1) in D and the class K ( α ) of convex functions of order α (0 ≦ α <
1) in D , the class S ∗ ( ϕ ) ofMa-Minda starlike functions and the class K ( ϕ ) of Ma-Minda convex functions ( ϕ is ananalytic function with positive real part in D , ϕ (0) = 1 , ϕ ′ (0) > ϕ maps D onto aregion starlike with respect to 1 and symmetric with respect to the real axis) (see [8]).It is well known that every function f ∈ S has an inverse f − , defined by f − ( f ( z )) = z ( z ∈ D )and f ( f − ( w )) = w ( | w | < r ( f ); r ( f ) ≧
14 ) , where f − ( w ) = w − a w + (2 a − a ) w − (5 a − a a + a ) w + . . . . A function f ∈ A is said to be bi-univalent in D if both f ( z ) and f − ( z ) are univalent in D . Let Σ denote the class of bi-univalent functions in D given by (1.1). Recently, in theirpioneering work on the subject of bi-univalent functions, Srivastava et al. [23] actuallyrevived the study of the coefficient problems involving bi-univalent functions. Varioussubclasses of the bi-univalent function class Σ were introduced and non-sharp estimateson the first two coefficients | a | and | a | in the Taylor-Maclaurin series expansion (1.1)were found in several recent investigations (see, for example, [1, 2, 3, 4, 5, 9, 11, 17, 18,19, 20, 21, 22, 24, 25, 26] and references therein). The afore-cited all these papers on thesubject were actually motivated by the pioneering work of Srivastava et al. [23]. However,the problem to find the coefficient bounds on | a n | ( n = 3 , , . . . ) for functions f ∈ Σ isstill an open problem.The classes SL (˜ p ) and KSL (˜ p ) of shell-like functions and convex shell-like functions arerespectively, characterized by zf ′ / f ( z ) ≺ ˜ p ( z ) or 1 + z f ′′ / f ′ ( z ) ≺ ˜ p ( z ) , where ˜ p ( z ) =(1 + τ z ) / (1 − τ z − τ z ) , τ = (1 − √
5) / 2 ≈ − . . The classes SL (˜ p ) and KSL (˜ p )were introduced and studied by Sok´o l [16] and Dziok et al. [6] respectively (see also [7, 15]).The function ˜ p is not univalent in D , but it is univalent in the disc | z | < (3 − √
5) / 2 ≈ . . For example, ˜ p (0) = ˜ p ( − τ ) = 1 and ˜ p ( e ∓ arccos (1 / √ | τ | = | τ | / 1 − | τ | which shows that the number | τ | divides [0 , | z | = 1 under ˜ p is acurve described by the equation given by (cid:0) x − √ (cid:1) y = (cid:0) √ − x (cid:1) (cid:0) √ x − (cid:1) , whichis translated and revolved trisectrix of Maclaurin. The curve ˜ p ( re it ) is a closed curvewithout any loops for 0 < r ≤ r = (3 − √
5) / 2 ≈ . . For r < r < , it has a loopand for r = 1, it has a vertical asymptote. Since τ satisfies the equation τ = 1 + τ , thisexpression 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 and 1 . The resulting recurrence relationships yield Fibonacci numbers u n τ n = u n τ + u n − . HELL-LIKE CURVES CONNECTED WITH FIBONACCI NUMBERS 3
Recently Raina and Sok´o l [15], taking τ z = t, showed that˜ p ( z ) = 1 + τ z − τ z − τ z = 1 + ∞ X n =2 ( u n − + u n +1 ) τ n z n , (1.2)where u n = (1 − τ ) n − τ √ , τ = 1 − √ , ; n = 1 , , . . . . (1.3)This shows that the relevant connection of ˜ p with the sequence of Fibonacci numbers u n ,such that u = 0 , u = 1 , u n +2 = u n + u n +1 for n = 0 , , , , . . . . Hence˜ p ( z ) = 1 + τ z + 3 τ z + 4 τ z + 7 τ z + 11 τ z + . . . . (1.4)We note that the function ˜ p belongs to the class P ( β ) with β = √ ≈ . Definition 1.1.
A function f ∈ Σ of the form f ( z ) = z + ∞ X n =2 a n z n , belongs to the class WSL Σ ( γ, λ, α, ˜ p ) , γ ∈ C \{ } , α ≥ λ ≥ , if the followingconditions are satisfied:1+ 1 γ (cid:18) (1 − α + 2 λ ) f ( z ) z + ( α − λ ) f ′ ( z ) + λzf ′′ ( z ) − (cid:19) ≺ g p ( z ) = 1 + τ z − τ z − τ z , z ∈ D (1.5)and for g ( w ) = f − ( w )1+ 1 γ (cid:18) (1 − α + 2 λ ) g ( w ) w + ( α − λ ) g ′ ( w ) + λwg ′′ ( w ) − (cid:19) ≺ ] p ( w ) = 1 + τ w − τ w − τ w , w ∈ D , (1.6)where τ = 1 − √ ≈ − . . It is interesting to note that the special values of α, γ and λ lead the class WSL Σ ( γ, λ, α, ˜ p )to various subclasses, we illustrate the following subclasses:(1) For α = 1+2 λ, we get the class WSL Σ ( γ, λ, λ, ˜ p ) ≡ F SL Σ ( γ, λ, ˜ p ) . A function f ∈ Σ of the form f ( z ) = z + ∞ X n =2 a n z n , is said to be in F SL Σ ( γ, λ, ˜ p ) , if the following conditions1 + 1 γ ( f ′ ( z ) + λzf ′′ ( z ) − ≺ g p ( z ) = 1 + τ z − τ z − τ z , z ∈ D HELL-LIKE CURVES CONNECTED WITH FIBONACCI NUMBERS 4 and for g ( w ) = f − ( w )1 + 1 γ ( g ′ ( w ) + λwg ′′ ( w ) − ≺ ] p ( w ) = 1 + τ w − τ w − τ w , w ∈ D , hold, where τ = 1 − √ ≈ − . . (2) For λ = 0 , we obtain the class WSL Σ ( γ, , α, ˜ p ) ≡ BSL Σ ( γ, α, ˜ p ) . A function f ∈ Σ of the form f ( z ) = z + ∞ X n =2 a n z n , is said to be in BSL Σ ( γ, α, ˜ p ) , if the following conditions1 + 1 γ (cid:18) (1 − α ) f ( z ) z + αf ′ ( z ) − (cid:19) ≺ g p ( z ) = 1 + τ z − τ z − τ z , z ∈ D and for g ( w ) = f − ( w )1 + 1 γ (cid:18) (1 − α ) g ( w ) w + αg ′ ( w ) − (cid:19) ≺ ] p ( w ) = 1 + τ w − τ w − τ w , w ∈ D hold, where τ = 1 − √ ≈ − . . (3) For λ = 0 and α = 1 , we have the class WSL Σ ( γ, , , ˜ p ) ≡ HSL Σ ( γ, ˜ p ) . A function f ∈ Σ of the form f ( z ) = z + ∞ X n =2 a n z n , is said to be in HSL Σ ( γ, ˜ p ) , if the following conditions1 + 1 γ ( f ′ ( z ) − ≺ g p ( z ) = 1 + τ z − τ z − τ z , z ∈ D and for g ( w ) = f − ( w )1 + 1 γ ( g ′ ( w ) − ≺ ] p ( w ) = 1 + τ w − τ w − τ w , w ∈ D hold, where τ = 1 − √ ≈ − . . Definition 1.2.
A function f ∈ Σ of the form f ( z ) = z + ∞ X n =2 a n z n , belongs to the class RSL Σ ( γ, λ, ˜ p ) , γ ∈ C \{ } and λ ≥ , if the following conditions aresatisfied: 1 + 1 γ (cid:18) z − λ f ′ ( z )( f ( z )) − λ − (cid:19) ≺ g p ( z ) = 1 + τ z − τ z − τ z , z ∈ D (1.7)and for g ( w ) = f − ( w )1 + 1 γ (cid:18) w − λ g ′ ( w )( g ( w )) − λ − (cid:19) ≺ ] p ( w ) = 1 + τ w − τ w − τ w , w ∈ D , (1.8) HELL-LIKE CURVES CONNECTED WITH FIBONACCI NUMBERS 5 where τ = 1 − √ ≈ − . . (1) For λ = 0 , we have the class RSL Σ ( γ, , ˜ p ) ≡ SL Σ ( γ, ˜ p ) . A function f ∈ Σ of theform f ( z ) = z + ∞ X n =2 a n z n , is said to be in SL Σ ( γ, ˜ p ) , if the following conditions1 + 1 γ (cid:18) zf ′ ( z ) f ( z ) − (cid:19) ≺ g p ( z ) = 1 + τ z − τ z − τ z , z ∈ D and for g ( w ) = f − ( w )1 + 1 γ (cid:18) wg ′ ( w ) g ( w ) − (cid:19) ≺ ] p ( w ) = 1 + τ w − τ w − τ w , w ∈ D hold, where τ = 1 − √ ≈ − . . Remark . For γ = 1 the class SL Σ (1 , ˜ p ) ≡ SL Σ (˜ p ) was introduced and studiedG¨uney et al. [10].(2) For λ = 1 , we have the class RSL Σ ( γ, , ˜ p ) ≡ HSL Σ ( γ, ˜ p ) . Definition 1.4.
A function f ∈ Σ of the form f ( z ) = z + ∞ X n =2 a n z n , belongs to the class SLB Σ ( λ ; ˜ p ) , λ ≥ , if the following conditions are satisfied: z [ f ′ ( z )] λ f ( z ) ≺ g p ( z ) = 1 + τ z − τ z − τ z , z ∈ D (1.9)and for g ( w ) = f − ( w ) w [ g ′ ( w )] λ g ( w ) ≺ ] p ( w ) = 1 + τ w − τ w − τ w , w ∈ D , (1.10)where τ = 1 − √ ≈ − . . (1) For λ = 1 , we have the class SLB Σ (1; ˜ p ) ≡ SL Σ (˜ p ) . A function f ∈ Σ of the form f ( z ) = z + ∞ X n =2 a n z n , is said to be in SL Σ (˜ p ) , if the following conditions zf ′ ( z ) f ( z ) ≺ g p ( z ) = 1 + τ z − τ z − τ z , z ∈ D and for g ( w ) = f − ( w ) wg ′ ( w ) g ( w ) ≺ ] p ( w ) = 1 + τ w − τ w − τ w , w ∈ D , HELL-LIKE CURVES CONNECTED WITH FIBONACCI NUMBERS 6 hold, where τ = 1 − √ ≈ − . . Definition 1.5.
A function f ∈ Σ of the form f ( z ) = z + ∞ X n =2 a n z n , belongs to the class PSL Σ ( λ ; ˜ p ) , ≤ λ ≤ , if the following conditions are satisfied: zf ′ ( z ) + λz f ′′ ( z )(1 − λ ) f ( z ) + λzf ′ ( z ) ≺ g p ( z ) = 1 + τ z − τ z − τ z , z ∈ D (1.11)and for g ( w ) = f − ( w ) wf ′ ( w ) + λw g ′′ ( w )(1 − λ ) g ( w ) + λwg ′ ( w ) ≺ ] p ( w ) = 1 + τ w − τ w − τ w , w ∈ D , (1.12)where τ = 1 − √ ≈ − . . (1) For λ = 0 , we have the class PSL Σ (0; ˜ p ) ≡ SL Σ (˜ p ) . A function f ∈ Σ of the form f ( z ) = z + ∞ X n =2 a n z n , is said to be in SL Σ (˜ p ) , if the following conditions zf ′ ( z ) f ( z ) ≺ g p ( z ) = 1 + τ z − τ z − τ z , z ∈ D and for g ( w ) = f − ( w ) wg ′ ( w ) g ( w ) ≺ ] p ( w ) = 1 + τ w − τ w − τ w , w ∈ D , hold, where τ = 1 − √ ≈ − . . (2) For λ = 1 , we have the class PSL Σ (1; ˜ p ) ≡ KSL Σ (˜ p ) . A function f ∈ Σ of theform f ( z ) = z + ∞ X n =2 a n z n , is said to be in KSL Σ (˜ p ) , if the following conditions1 + z f ′′ ( z ) f ′ ( z ) ≺ g p ( z ) = 1 + τ z − τ z − τ z , z ∈ D and for g ( w ) = f − ( w )1 + w g ′′ ( w ) g ′ ( w ) ≺ ] p ( w ) = 1 + τ w − τ w − τ w , w ∈ D , hold, where τ = 1 − √ ≈ − . . Remark . For γ = 0 , PSL Σ (0 , ˜ p ) ≡ SL Σ (˜ p ) and γ = 1 , PSL Σ (1 , ˜ p ) ≡ KSL Σ (˜ p )the classes were introduced and studied G¨uney et al. [10]. HELL-LIKE CURVES CONNECTED WITH FIBONACCI NUMBERS 7
In order to prove our results for the function in the classes
WSL Σ ( γ, λ, α, ˜ p ) , RSL Σ ( γ, λ, ˜ p ), SLB Σ ( λ ; ˜ p ) and PSL Σ ( λ ; ˜ p ), we need the following lemma. Lemma 1.7. [14] If p ∈ P , then | p i | ≦ for each i, where P is the family of all functions p, analytic in D , for which ℜ{ p ( z ) } > z ∈ D ) , where p ( z ) = 1 + p z + p z + · · · ( z ∈ D ) . In this investigation, we find the estimates for the coefficients | a | and | a | for functionsin the subclass WSL Σ ( γ, λ, α, ˜ p ), RSL Σ ( γ, λ, ˜ p ), SLB Σ ( λ ; ˜ p ) and PSL Σ ( λ ; ˜ p ) Also, weobtain the upper bounds using the results of | a | and | a | . Initial Coefficient Estimates and Fekete-Szeg¨o Inequalities
In the following theorem, we obtain coefficient estimates for functions in the class f ∈ WSL Σ ( γ, λ, α, ˜ p ) . Theorem 2.1.
Let f ( z ) = z + ∞ P n =2 a n z n be in the class WSL Σ ( γ, λ, α, ˜ p ) . Then | a | ≤ | γ | | τ | p γτ (1 + 2 α + 2 λ ) + (1 − τ )(1 + α ) , | a | ≤ | γ | | τ | { (1 − τ )(1 + α ) } (1 + 2 α + 2 λ ) [ γτ (1 + 2 α + 2 λ ) + (1 − τ )(1 + α ) ] and (cid:12)(cid:12) a − µa (cid:12)(cid:12) ≤ γ | τ | (1 + 2 α + 2 λ ) ; 0 ≤ | h ( µ ) | ≤ γ | τ | α + 2 λ )4 | h ( µ ) | ; | h ( µ ) | ≥ γ | τ | α + 2 λ ) , where h ( µ ) = (1 − µ ) γ τ (cid:2) γτ (1 + 2 α + 2 λ ) + (1 + α ) (1 − τ ) (cid:3) . Proof.
Since f ∈ WSL Σ ( γ, λ, α, ˜ p ), from the Definition 1.1 we have1 + 1 γ (cid:18) (1 − α + 2 λ ) f ( z ) z + ( α − λ ) f ′ ( z ) + λzf ′′ ( z ) − (cid:19) = ^ p ( u ( z )) (2.1)and 1 + 1 γ (cid:18) (1 − α + 2 λ ) g ( w ) w + ( α − λ ) g ′ ( w ) + λwg ′′ ( w ) − (cid:19) = ^ p ( v ( w )) , (2.2)where z, w ∈ D and g = f − . Using the fact the function p ( z ) of the form p ( z ) = 1 + p z + p z + . . . and p ≺ ˜ p. Then there exists an analytic function u such that | u ( z ) | < D and p ( z ) = ˜ p ( u ( z )) . Therefore, define the function h ( z ) = 1 + u ( z )1 − u ( z ) = 1 + c z + c z + . . . HELL-LIKE CURVES CONNECTED WITH FIBONACCI NUMBERS 8 is in the class P . It follows that u ( z ) = h ( z ) − h ( z ) + 1 = c z + (cid:18) c − c (cid:19) z (cid:18) c − c c + c (cid:19) z . . . and ˜ p ( u ( z )) = 1 + ˜ p (cid:18) c z + (cid:18) c − c (cid:19) z (cid:18) c − c c + c (cid:19) z . . . (cid:19) +˜ p (cid:18) c z + (cid:18) c − c (cid:19) z (cid:18) c − c c + c (cid:19) z . . . (cid:19) +˜ p (cid:18) c z + (cid:18) c − c (cid:19) z (cid:18) c − c c + c (cid:19) z . . . (cid:19) + . . . = 1 + ˜ p c z + (cid:18) (cid:18) c − c (cid:19) ˜ p + c p (cid:19) z + (cid:18) (cid:18) c − c c + c (cid:19) ˜ p + 12 c (cid:18) c − c (cid:19) ˜ p + c p (cid:19) z + . . . . (2.3)Similarly, there exists an analytic function v such that | v ( w ) | < D and p ( w ) = ˜ p ( v ( w )) . Therefore, the function k ( w ) = 1 + v ( w )1 − v ( w ) = 1 + d w + d w + . . . is in the class P . It follows that v ( w ) = k ( w ) − k ( w ) + 1 = d w + (cid:18) d − d (cid:19) w (cid:18) d − d d + d (cid:19) w . . . and˜ p ( v ( w )) = 1 + ˜ p (cid:18) d w + (cid:18) d − d (cid:19) w (cid:18) d − d d + d (cid:19) w . . . (cid:19) +˜ p (cid:18) d w + (cid:18) d − d (cid:19) w (cid:18) d − d d + d (cid:19) w . . . (cid:19) +˜ p (cid:18) d w + (cid:18) d − d (cid:19) w (cid:18) d − d d + d (cid:19) w . . . (cid:19) + . . . = 1 + ˜ p d w + (cid:18) (cid:18) d − d (cid:19) ˜ p + d p (cid:19) w + (cid:18) (cid:18) d − d d + d (cid:19) ˜ p + 12 d (cid:18) d − d (cid:19) ˜ p + d p (cid:19) w (2.4)+ . . . . By virtue of (2.1), (2.2), (2.3) and (2.4), we have1 γ (1 + α ) a = c τ , (2.5) a γ (1 + 2 α + 2 λ ) = 12 (cid:18) c − c (cid:19) τ + 3 c τ , (2.6) HELL-LIKE CURVES CONNECTED WITH FIBONACCI NUMBERS 9 − γ (1 + α ) a = d τ , (2.7)and (1 + 2 α + 2 λ ) γ (2 a − a ) = 12 (cid:18) d − d (cid:19) τ + 3 d τ . (2.8)From (2.5) and (2.7), we obtain c = − d , and 2 γ (1 + α ) a = ( c + d ) τ a = γ ( c + d ) τ α ) . (2.9)By adding (2.6) and (2.8), we have2 γ (1 + 2 α + 2 λ ) a = 12 ( c + d ) τ − (cid:0) c + d (cid:1) τ + 34 (cid:0) c + d (cid:1) τ . (2.10)By substituting (2.9) in (2.10), we reduce that a = γ ( c + d ) τ γτ (1 + 2 α + 2 λ ) + (1 − τ )(1 + α ) ] . (2.11)Now, applying Lemma 1.7, we obtain | a | ≤ | γ | | τ | p γτ (1 + 2 α + 2 λ ) + (1 − τ )(1 + α ) . (2.12)By subtracting (2.8) from (2.6), we obtain a = γ ( c − d ) τ α + 2 λ ) + a . (2.13)Hence by Lemma 1.7, we have | a | ≤ | γ | ( | c | + | d | ) | τ | α + 2 λ ) + | a | ≤ | γ | | τ | (1 + 2 α + 2 λ ) + | a | . (2.14)Then in view of (2.12), we obtain | a | ≤ | γ | | τ | { (1 − τ )(1 + α ) } (1 + 2 α + 2 λ ) [ γτ (1 + 2 α + 2 λ ) + (1 − τ )(1 + α ) ]From (2.13), we have a − µa = γ ( c − d ) τ α + 2 λ ) + (1 − µ ) a . (2.15)By substituting (2.11) in (2.15), we have a − µa = γ ( c − d ) τ α + 2 λ ) + (1 − µ ) (cid:18) γ ( c + d ) τ γτ (1 + 2 α + 2 λ ) + (1 − τ )(1 + α ) ] (cid:19) = (cid:18) h ( µ ) + γ | τ | α + 2 λ ) (cid:19) c + (cid:18) h ( µ ) − γ | τ | α + 2 λ ) (cid:19) d , (2.16) HELL-LIKE CURVES CONNECTED WITH FIBONACCI NUMBERS 10 where h ( µ ) = (1 − µ ) γ τ (cid:2) γτ (1 + 2 α + 2 λ ) + (1 + α ) (1 − τ ) (cid:3) . Thus by taking modulus of (2.16), we conclude that (cid:12)(cid:12) a − µa (cid:12)(cid:12) ≤ γ | τ | (1 + 2 α + 2 λ ) ; 0 ≤ | h ( µ ) | ≤ γ | τ | α + 2 λ )4 | h ( µ ) | ; | h ( µ ) | ≥ γ | τ | α + 2 λ ) . (cid:3) Theorem 2.2.
Let f ( z ) = z + ∞ P n =2 a n z n be in the class RSL Σ ( γ, λ, ˜ p ) . Then | a | ≤ √ | γ | | τ | p γτ (2 + λ ) (1 + λ ) + 2(1 − τ )(1 + λ ) , | a | ≤ | γ | | τ | { γτ (2 + λ ) (1 + λ ) + 2(1 − τ )(1 + λ ) − λ ) γτ } (2 + λ ) [ γτ (2 + λ ) (1 + λ ) + 2(1 − τ )(1 + λ ) ] and (cid:12)(cid:12) a − µa (cid:12)(cid:12) ≤ | γ | | τ | λ ; 0 ≤ | µ − | ≤ M | γ | | τ | (2 + λ )2 | − µ | γ τ M ; | µ − | ≥ M | γ | | τ | (2 + λ ) , where M = γτ (2 + λ ) (1 + λ ) + 2 (1 + λ ) (1 − τ ) . Proof.
Since f ∈ RSL Σ ( γ, λ, ˜ p ), from the Definition 1.2 we have1 + 1 γ (cid:18) z − λ f ′ ( z )( f ( z )) − λ − (cid:19) = ^ p ( u ( z )) (2.17)and 1 + 1 γ (cid:18) w − λ g ′ ( w )( g ( w )) − λ − (cid:19) = ^ p ( v ( w )) . (2.18)By virtue of (2.17), (2.18), (2.3) and (2.4), we get1 γ (1 + λ ) a = c τ , (2.19)1 γ (2 + λ ) (cid:20) a + ( λ − a (cid:21) = 12 (cid:18) c − c (cid:19) τ + 3 c τ , (2.20) − γ (1 + λ ) a = d τ γ (2 + λ ) (cid:20) (3 + λ ) a − a (cid:21) = 12 (cid:18) d − d (cid:19) τ + 3 d τ . (2.22)From (2.19) and (2.21), we obtain c = − d , HELL-LIKE CURVES CONNECTED WITH FIBONACCI NUMBERS 11 and 2 γ (1 + λ ) a = ( c + d ) τ a = γ ( c + d ) τ λ ) . (2.23)By adding (2.20) and (2.22), we have1 γ (2 + λ ) (1 + λ ) a = 12 ( c + d ) τ − (cid:0) c + d (cid:1) τ + 34 (cid:0) c + d (cid:1) τ . (2.24)By substituting (2.23) in (2.24), we reduce that a = γ ( c + d ) τ γτ (2 + λ ) (1 + λ ) + 2(1 − τ )(1 + λ ) ] . (2.25)Now, applying Lemma 1.7, we obtain | a | ≤ √ | γ | | τ | p γτ (2 + λ ) (1 + λ ) + 2(1 − τ )(1 + λ ) . (2.26)By subtracting (2.22) from (2.20), we obtain a = γ ( c − d ) τ λ ) + a . (2.27)Hence by Lemma 1.7, we have | a | ≤ | γ | ( | c | + | d | ) | τ | λ ) + | a | ≤ | γ | | τ | (2 + λ ) + | a | . (2.28)Then in view of (2.26), we obtain | a | ≤ | γ | | τ | { γτ (2 + λ ) (1 + λ ) + 2(1 − τ )(1 + λ ) − λ ) γτ } (2 + λ ) [ γτ (2 + λ ) (1 + λ ) + 2(1 − τ )(1 + λ ) ] . From (2.27), we have a − µa = γ ( c − d ) τ λ ) + (1 − µ ) a . (2.29)By substituting (2.25) in (2.29), we have a − µa = γ ( c − d ) τ λ ) + (1 − µ ) γ ( c + d ) τ (cid:2) γτ (2 + λ ) (1 + λ ) + 2 (1 + λ ) (1 − τ ) (cid:3) ! = (cid:18) h ( µ ) + | γ | | τ | λ ) (cid:19) c + (cid:18) h ( µ ) − | γ | | τ | λ ) (cid:19) d , (2.30)where h ( µ ) = (1 − µ ) γ τ (cid:2) γτ (2 + λ ) (1 + λ ) + 2 (1 + λ ) (1 − τ ) (cid:3) . HELL-LIKE CURVES CONNECTED WITH FIBONACCI NUMBERS 12
Thus by taking modulus of (2.30), we conclude that (cid:12)(cid:12) a − µa (cid:12)(cid:12) ≤ γ | τ | (2 + λ ) ; 0 ≤ | h ( µ ) | ≤ γ | τ | λ )4 | h ( µ ) | ; | h ( µ ) | ≥ γ | τ | λ ) . This gives desired inequality. (cid:3)
Theorem 2.3.
Let f ( z ) = z + ∞ P n =2 a n z n be in the class SLB Σ ( λ ; ˜ p ) . Then | a | ≤ | τ | p (2 λ −
1) [ τ (3 − λ ) + 2 λ − , | a | ≤ | τ | (cid:2) (2 λ − − (cid:0) λ − λ + 1 (cid:1) τ (cid:9) (2 λ − λ − − λ ) τ + 2 λ − and (cid:12)(cid:12) a − µa (cid:12)(cid:12) ≤ | γ | | τ | λ − ≤ | µ − | ≤ M | τ | (3 λ − | − µ | τ M ; | µ − | ≥ M | τ | (3 λ − , where M = (2 λ −
1) [ τ (3 − λ ) + 2 λ − . Proof.
Since f ∈ SLB Σ ( λ ; ˜ p ), from the Definition 1.4 we have z [ f ′ ( z )] λ f ( z ) = ^ p ( u ( z )) (2.31)and w [ g ′ ( w )] λ g ( w ) = ^ p ( v ( w )) . (2.32)By virtue of (2.31), (2.32), (2.3) and (2.4), we get(2 λ − a = c τ , (2.33)(3 λ − a + (2 λ − λ + 1) a = 12 (cid:18) c − c (cid:19) τ + 3 c τ , (2.34) − (2 λ − a = d τ λ + 2 λ − a − (3 λ − a = 12 (cid:18) d − d (cid:19) τ + 3 d τ . (2.36)From (2.33) and (2.35), we obtain c = − d , HELL-LIKE CURVES CONNECTED WITH FIBONACCI NUMBERS 13 and 2(2 λ − a = ( c + d ) τ a = ( c + d ) τ λ − . (2.37)By adding (2.34) and (2.36), we have2 λ (2 λ − a = 12 ( c + d ) τ − (cid:0) c + d (cid:1) τ + 34 (cid:0) c + d (cid:1) τ . (2.38)By substituting (2.37) in (2.38), we reduce that a = ( c + d ) τ λ −
1) [ τ (3 − λ ) + 2 λ − . (2.39)Now, applying Lemma 1.7, we obtain | a | ≤ | τ | p (2 λ −
1) [ τ (3 − λ ) + 2 λ − . (2.40)By subtracting (2.36) from (2.34), we obtain a = ( c − d ) τ λ −
1) + a . (2.41)Hence by Lemma 1.7, we have | a | ≤ ( | c | + | d | ) | τ | λ −
1) + | a | ≤ | τ | λ − | a | . (2.42)Then in view of (2.40), we obtain | a | ≤ | τ | (cid:2) (2 λ − − (cid:0) λ − λ + 1 (cid:1) τ (cid:9) (2 λ − λ − − λ ) τ + 2 λ − a − µa = ( c − d ) τ λ −
1) + (1 − µ ) a . (2.43)By substituting (2.39) in (2.43), we have a − µa = ( c − d ) τ λ −
1) + (1 − µ ) (cid:18) ( c + d ) τ λ −
1) [ τ (3 − λ ) + 2 λ − (cid:19) = (cid:18) h ( µ ) + | τ | λ − (cid:19) c + (cid:18) h ( µ ) − | τ | λ − (cid:19) d , (2.44)where h ( µ ) = (1 − µ ) τ λ −
1) [ τ (3 − λ ) + 2 λ − . Thus by taking modulus of (2.44), we conclude that (cid:12)(cid:12) a − µa (cid:12)(cid:12) ≤ | τ | (3 λ −
1) ; 0 ≤ | h ( µ ) | ≤ | τ | λ − | h ( µ ) | ; | h ( µ ) | ≥ | τ | λ − . HELL-LIKE CURVES CONNECTED WITH FIBONACCI NUMBERS 14
This gives desired inequality. (cid:3)
Theorem 2.4.
Let f ( z ) = z + ∞ P n =2 a n z n be in the class PSL Σ ( λ ; ˜ p ) . Then | a | ≤ | τ | q (1 + λ ) − τ (cid:0) λ + 2 λ + 1 (cid:1) , | a | ≤ | τ | (1 − τ ) (1 + λ ) λ ) (cid:2) (1 + λ ) − τ (cid:0) λ + 2 λ + 1 (cid:1)(cid:3) and (cid:12)(cid:12) a − µa (cid:12)(cid:12) ≤ | τ | λ ; 0 ≤ | µ − | ≤ M | τ | (1 + 2 λ ) | − µ | τ M ; | µ − | ≥ M | τ | (1 + 2 λ ) , where M = (1 + λ ) − τ (cid:0) λ + 2 λ + 1 (cid:1) . Proof.
Since f ∈ PSL Σ ( λ ; ˜ p ), from the Definition 1.5 we have zf ′ ( z ) + λz f ′′ ( z )(1 − λ ) f ( z ) + λzf ′ ( z ) = ^ p ( u ( z )) (2.45)and wf ′ ( w ) + λw g ′′ ( w )(1 − λ ) g ( w ) + λwg ′ ( w ) = ^ p ( v ( w )) . (2.46)By virtue of (2.45), (2.46), (2.3) and (2.4), we get(1 + λ ) a = c τ , (2.47)2(1 + 2 λ ) a − (1 + λ ) a = 12 (cid:18) c − c (cid:19) τ + 3 c τ , (2.48) − (1 + λ ) a = d τ − λ ) a − ( λ − λ − a = 12 (cid:18) d − d (cid:19) τ + 3 d τ . (2.50)From (2.47) and (2.49), we obtain c = − d , and 2(1 + λ ) a = ( c + d ) τ a = ( c + d ) τ λ ) . (2.51) HELL-LIKE CURVES CONNECTED WITH FIBONACCI NUMBERS 15
By adding (2.48) and (2.50), we have2 (cid:0) λ − λ (cid:1) a = 12 ( c + d ) τ − (cid:0) c + d (cid:1) τ + 34 (cid:0) c + d (cid:1) τ . (2.52)By substituting (2.51) in (2.52), we reduce that a = ( c + d ) τ (cid:2) (1 + λ ) − τ (cid:0) λ + 2 λ + 1 (cid:1)(cid:3) . (2.53)Now, applying Lemma 1.7, we obtain | a | ≤ | τ | q (1 + λ ) − τ (cid:0) λ + 2 λ + 1 (cid:1) . (2.54)By subtracting (2.50) from (2.48), we obtain a = ( c − d ) τ λ ) + a . (2.55)Hence by Lemma 1.7, we have | a | ≤ ( | c | + | d | ) | τ | λ ) + | a | ≤ | τ | λ + | a | . Then in view of (2.54), we obtain | a | ≤ | τ | (1 − τ ) (1 + λ ) λ ) (cid:2) (1 + λ ) − τ (cid:0) λ + 2 λ + 1 (cid:1)(cid:3) From (2.55), we have a − µa = ( c − d ) τ λ ) + (1 − µ ) a . (2.56)By substituting (2.53) in (2.56), we have a − µa = ( c − d ) τ λ ) + (1 − µ ) ( c + d ) τ (cid:2) (1 + λ ) − τ (cid:0) λ + 2 λ + 1 (cid:1)(cid:3) ! = (cid:18) h ( µ ) + τ λ ) (cid:19) c + (cid:18) h ( µ ) − τ λ ) (cid:19) d , (2.57)where h ( µ ) = (1 − µ ) τ (cid:2) (1 + λ ) − τ (cid:0) λ + 2 λ + 1 (cid:1)(cid:3) . Thus by taking modulus of (2.57), we conclude that (cid:12)(cid:12) a − µa (cid:12)(cid:12) ≤ | τ | λ ) ; 0 ≤ | h ( µ ) | ≤ | τ | λ )4 | h ( µ ) | ; | h ( µ ) | ≥ | τ | λ ) . This gives desired inequality. (cid:3)
HELL-LIKE CURVES CONNECTED WITH FIBONACCI NUMBERS 16 Corollaries and Consequences
Corollary 3.1.
Let f ( z ) = z + ∞ P n =2 a n z n be in the class F SL Σ ( γ, λ, ˜ p ) . Then | a | ≤ | γ | | τ | p γτ (1 + 2 λ ) + 4(1 − τ )(1 + λ ) , | a | ≤ | γ | | τ | (1 − τ )(1 + α ) λ ) [3 γτ (1 + 2 λ ) + 4(1 − τ )(1 + λ ) ] and (cid:12)(cid:12) a − µa (cid:12)(cid:12) ≤ | γ | | τ | λ ; 0 ≤ | h ( µ ) | ≤ | γ | | τ |
12 + 24 λ | h ( µ ) | ; | h ( µ ) | ≥ | γ | | τ |
12 + 24 λ , where h ( µ ) = (1 − µ ) γ τ γτ (1 + 2 λ ) + 4(1 − τ )(1 + λ ) ] . Corollary 3.2.
Let f ( z ) = z + ∞ P n =2 a n z n be in the class BSL Σ ( γ, α, ˜ p ) . Then | a | ≤ | γ | | τ | p γτ (1 + 2 α ) + (1 − τ )(1 + α ) , | a | ≤ | γ | | τ | { (1 − τ )(1 + α ) } (1 + 2 α ) [ γτ (1 + 2 α ) + (1 − τ )(1 + α ) ] and (cid:12)(cid:12) a − µa (cid:12)(cid:12) ≤ | γ | | τ | α ; 0 ≤ | h ( µ ) | ≤ | γ | | τ | α | h ( µ ) | ; | h ( µ ) | ≥ | γ | | τ | α , where h ( µ ) = (1 − µ ) γ τ (cid:2) γτ (1 + 2 α ) + (1 + α ) (1 − τ ) (cid:3) . Corollary 3.3.
Let f ( z ) = z + ∞ P n =2 a n z n be in the class HSL Σ ( γ, ˜ p ) . Then | a | ≤ | γ | | τ | p γτ + 4(1 − τ ) , | a | ≤ | γ | | τ | (1 − τ )3 [3 γτ + 4(1 − τ )] , and (cid:12)(cid:12) a − µa (cid:12)(cid:12) ≤ | γ | | τ | ≤ | h ( µ ) | ≤ | γ | | τ | | h ( µ ) | ; | h ( µ ) | ≥ | γ | | τ | , where h ( µ ) = (1 − µ ) γ τ γτ + 4 (1 − τ )] . HELL-LIKE CURVES CONNECTED WITH FIBONACCI NUMBERS 17
Corollary 3.4. [10]
Let f ( z ) = z + ∞ P n =2 a n z n be in the class SL Σ (˜ p ) . Then | a | ≤ | τ |√ − τ , | a | ≤ | τ | (1 − τ )2 − τ and (cid:12)(cid:12) a − µa (cid:12)(cid:12) ≤ | τ | ≤ | µ − | ≤ − τ | τ || − µ | τ − τ ; | µ − | ≥ − τ | τ | . Corollary 3.5. [10]
Let f ( z ) = z + ∞ P n =2 a n z n be in the class KSL Σ (˜ p ) . Then | a | ≤ | τ |√ − τ , | a | ≤ | τ | (1 − τ )6 − τ . and (cid:12)(cid:12) a − µa (cid:12)(cid:12) ≤ | τ | ≤ | µ − | ≤ − τ | τ || − µ | τ − τ ; | µ − | ≥ − τ | τ | . Remark . Results discussed in Corollaries 3.4 and 3.5 are coincide with bounds obtainedin [10, Corollary 1, Corollary 2, Corollary 4 and Corollary 5].
References [1] R. M. Ali, S. K. Lee, V. Ravichandran, S. Supramanian, Coefficient estimates for bi-univalent Ma-Minda starlike and convex functions, Appl. Math. Lett. (2012), no. 3, 344–351.[2] S¸. Altınkaya and S. Yal¸cin, On the (p, q)-Lucas polynomial coefficient bounds of the bi-univalentfunction class, Boletin de la Sociedad Matematica Mexicana, (2018), 1–9.[3] S. Bulut, Coefficient estimates for a class of analytic and bi-univalent functions, Novi Sad J. Math. (2013), no. 2, 59–65.[4] M. C¸ a˘glar, H. Orhan and N. Ya˘gmur, Coefficient bounds for new subclasses of bi-univalent functions,Filomat, (2013), no. 7, 1165–1171.[5] E. Deniz, Certain subclasses of bi-univalent functions satisfying subordinate conditions, J. ClassicalAnal. (2013), no. 1, 49–60.[6] J. Dziok, R. K. Raina, J. Sok´o l, Certain results for a class of convex functions related to a shell-likecurve connected with Fibonacci numbers, Comp. Math. Appl., (2011), 2605–2613.[7] J. Dziok, R. K. Raina, J. Sok´o l, On α − convex functions related to a shell-like curve connected withFibonacci numbers, Appl. Math. Comp., (2011), 996–1002.[8] P. L. Duren, Univalent Functions , Grundlehren der Mathematischen Wissenschaften Series, 259,Springer Verlag, New York, 1983.[9] V. B. Girgaonkar and S. B. Joshi, Coefficient estimates for certain subclass of bi-univalent functionsassociated with Chebyshev polynomial, Ganita, (2018), 79 – 85.[10] H. ¨O. G¨uney, G. Murugusundaramoorthy and J. Sok´o l, Subclasses of bi-univalent functions relatedto shell-like curves connected with Fibonacci numbers, Acta Univ. Sapientiae, Math., (2018),no. 1, 70–84.[11] J. M. Jahangiri, S. G. Hamidi and S. Abd. Halim, Coefficients of bi-univalent functions with positivereal part derivatives, Bull. Malays. Math. Sci. Soc. (2) (2014), no. 3, 633–640.[12] H. Orhan, N. Magesh and V. K. Balaji, Fekete-Szeg¨o problem for certain classes of Ma-Minda bi-univalent functions, Afr. Mat. (2016), no. 5-6, 889–897. HELL-LIKE CURVES CONNECTED WITH FIBONACCI NUMBERS 18 [13] H. Orhan, N. Magesh and V. K. Balaji, Certain classes of bi-univalent functions with boundedboundary variation, Tbilisi Math. J. (2017), no. 4, 17–27.[14] C. Pommerenke, Univalent Functions , Vandenhoeck & Ruprecht, G¨ottingen, 1975.[15] R. K. Raina, J. Sok´o l, Fekete-Szeg¨o problem for some starlike functions related to shell-like curves,Math. Slovaca, (2016), 135–140.[16] J. Sok´o l, On starlike functions connected with Fibonacci numbers, Folia Scient. Univ. Tech. Reso-viensis, (1999), 111–116.[17] H. M. Srivastava and D. Bansal, Coefficient estimates for a subclass of analytic and bi-univalentfunctions, J. Egyptian Math. Soc. (2015), no. 2, 242–246.[18] H. M. Srivastava, S. Bulut, M. C¸ a˘glar, N. Ya˘gmur, Coefficient estimates for a general subclass ofanalytic and bi-univalent functions, Filomat, (2013), no. 5, 831–842.[19] H. M. Srivastava, S. S. Eker and R. M. Ali, Coefficient bounds for a certain class of analytic andbi-univalent functions, Filomat (2015), no. 8, 1839–1845.[20] H. M. Srivastava, S. Gaboury and F. Ghanim, Coefficient estimates for some subclasses of M -foldsymmetric bi-univalent functions, Acta Univ. Apulensis Math. Inform. (2015), 153–164.[21] H. M. Srivastava, S. Gaboury and F. Ghanim, Initial coefficient estimates for some subclasses of m -fold symmetric bi-univalent functions, Acta Math. Sci. Ser. B Engl. Ed. (2016), no. 3, 863–871.[22] H. M.Srivastava, N. Magesh and J. Yamini, Initial coefficient estimates for bi- λ − convex and bi- µ − starlike functions connected with arithmetic and geometric means, Electronic J. Math. Anal. Appl. (2014), no. 2, 152 – 162.[23] H. M. Srivastava, A. K. Mishra and P. Gochhayat, Certain subclasses of analytic and bi-univalentfunctions, Appl. Math. Lett. (2010), no. 10, 1188–1192.[24] H. M. Srivastava, G. Murugusundaramoorthy and N. Magesh, Certain subclasses of bi-univalentfunctions associated with the Hohlov operator, Global J. Math. Anal. (2013), no. 2, 67–73.[25] H. Tang, G-T Deng and S-H Li, Coefficient estimates for new subclasses of Ma-Minda bi-univalentfunctions, J. Ineq. Appl. (2013), 1–10.[26] P. Zaprawa, On the Fekete-Szeg¨o problem for classes of bi-univalent functions, Bull. Belg. Math.Soc. Simon Stevin (2014), no. 1, 169–178. Post-Graduate and Research Department of Mathematics,, Government Arts Collegefor Men,, Krishnagiri 635001, Tamilnadu, India. e-mail: nmagi yahoo.co.in
Department of Mathematics, L.N. Govt College,, Ponneri, Chennai, Tamilnadu, India. e-mail: balajilsp @ yahoo.co.in Faculty of Engineering and Technology, SRM University, Kattankulathur-603203,Tamilnadu, India. e-mail: shreelekha07@