# A Subclass of Bi-Univalent Functions Defined by Generalized Sãlãgean Operator Related to Shell-Like Curves Connected with Fibonacci Numbers

Authors: Gurmeet Singh; Gurcharanjit Singh; Gagandeep Singh
Journal: International Journal of Mathematics and Mathematical Sciences
 (2019)

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## Abstract

The aim of this paper is to study certain subclasses of bi-univalent functions defined by generalized Sãlãgean differential operator related to shell-like curves connected with Fibonacci numbers. We find estimates of the initial coefficientsa2 and a3 and upper bounds for the Fekete-Szegö functional for the functions in this class. The results proved by various authors follow as particular cases.


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## Body

## 1. Introduction and Preliminaries

LetA be the class of functions of the form(1)fz=z+∑k=2∞akzkwhich are analytic in the unit disc U=z:z<1 with normalization f(0)=f′(0)-1=0. By S, we denote the class of functions f(z)∈A and univalent in U.Let us denote byB the class of bounded or Schwarz functions w(z) satisfying w(0)=0 and wz≤1 which are analytic in the open unit disc U and of the form(2)wz=∑n=1∞cnzn,z∈U.Consider two functions f and g analytic in U. We say that f is subordinate to g (symbolically f≺g) if there exists a bounded function u(z)∈B for which f(z)=g(u(z)). This result is known as principle of subordination.ByS∗, we denote the class of starlike functions f∈S which satisfies the following condition:(3)Re⁡zf′zfz>0orzf′zfz≺1+z1-z,z∈U.By K, we denote the class of convex functions f∈S which satisfies the following condition:(4)Re⁡zf′z′f′z>0orzf′z′f′z≺1+z1-z,z∈U.A function f∈S is said to be α-convex if it satisfies the inequality(5)Re⁡1-αzf′zfz+αzf′z′f′z>00≤α≤1,z∈U.The class of α-convex functions is denoted by M(α) and was introduced by Mocanu [1]. In particular M(0)≡S∗ and M(1)≡K.Forδ≥1 and f∈A, Al-Oboudi [2] introduced the following differential operator:(6)Dδ0fz=fz,Dδ1fz=1-δfz+δzf′z,and in general,(7)Dδnfz=DDδn-1fz=1-δDδn-1fz+δzDδn-1fz′,n∈Nor equivalent to(8)Dδnfz=z+∑k=2∞1+k-1δnakzk,n∈N0=N∪0with Dδnf(0)=0. It is obvious that, for δ=1, the operator Dδnf(z) is equivalent to the Sãlãgean operator introduced in [3]. So the operator Dδnf(z) is named as the Generalized Sãlãgean operator.The inverse functions of the functions in the classS may not be defined on the entire unit disc U although the functions in the class S are invertible. However using Koebe-one quarter theorem [4] it is obvious that the image of U under every function f∈S contains a disc of radius 1/4. Hence every univalent function f has an inverse f-1, defined by(9)f-1fz=zz∈Uand(10)ff-1w=ww<r0f:r0f≥14where(11)gw=f-1w=w-a2w2+2a22-a3w3-5a23-5a2a3+a4w4+⋯A function f∈A is said to be bi-univalent in U if both f and f-1 are univalent in U.ByΣ, we denote the class of bi-univalent functions in U defined by (1).Lewin [5] discussed the class Σ of bi-univalent functions and obtained the bound for the second coefficient. Brannan and Taha [6] investigated certain subclasses of bi-univalent functions, similar to the familiar subclasses of univalent functions consisting of strongly starlike, starlike and convex functions. They introduced bi-starlike functions and bi-convex functions and obtained estimates on the initial coefficients.Sokól [7] introduced the class SL of shell-like functions f∈A defined as below.Definition 1.
A functionf∈A given by (1) is said to be in the class SL of starlike shell-like functions if it satisfies the following condition:(12)zf′zfz≺p~z=1+τ2z21-τz-τ2z2where τ=(1-5)/2≈-0.618.It should be observed thatSL is a subclass of the class S∗ of starlike functions.Later Dziok et al. [8] introduced the class KSL of convex functions related to a shell-like curve as below.Definition 2.
A functionf∈A given by (1) is said to be in the class KSL of convex shell-like functions if it satisfies the condition that(13)1+zf′′zf′z≺p~z=1+τ2z21-τz-τ2z2where τ=(1-5)/2≈-0.618.Again Dziok et al. [9] defined the following class of α-convex shell-like functions.Definition 3.
A functionf∈A given by (1) is said to be in the class SLMα of α-convex shell-like functions if it satisfies the condition that(14)1-αzf′zfz+α1+zf′′zf′z≺p~z=1+τ2z21-τz-τ2z2where τ=(1-5)/2≈-0.618.ObviouslySLM0≡SL and SLM1≡KSL.The functionp~ is not univalent in U, but it is univalent in the disc z<(3-5)/2≈0.38. For example, p~(0)=p~-1/2τ=1 and p~(e∓iarccos⁡(1/4))=5/5, and it may also be noticed that(15)1τ=τ1-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 p~ is a curve described by the equation given by(16)10x-5y2=5-2x5x-12,which is translated and revolved trisectrix of Maclaurin. The curve p~(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:(17)τn=unτ+un-1.Also the subclasses of bi-univalent functions related to shell-like curves were studied by various authors [10–12].The earlier work on bi-univalent functions related to shell-like curves connected with Fibonacci numbers motivated us to define the following subclass.To avoid repetition, throughout the paper we assume that0≤α≤1, τ=(1-5)/2≈-0.618 and z∈U.Definition 4.
A functionf∈Σ given by (1) is said to be in the class SLMα,Σδ(n,p~(z)) if it satisfies the following conditions:(18)1-αDδn+1fzDδnfz+αDδn+1fz′Dδnfz′≺p~z=1+τ2z21-τz-τ2z2and(19)1-αDδn+1gwDδngw+αDδn+1gw′Dδngw′≺p~w=1+τ2w21-τw-τ2w2where g=f-1 and z,w∈U.The following observations are obvious:(i)SLMα,Σ1(n,p~(z))≡SLMα,Σ(n,p~(z)), the class of bi-univalent functions defined by Sãlãgean operator related to shell-like curves.(ii)SLMα,Σ1(0,p~(z))≡SLMα,Σ(p~(z)), the class of bi-univalent α-convex shell-like functions studied by Güney et al. [13].(iii)SLM0,Σ1(0,p~(z))≡SLΣ(p~(z)).(iv)SLM1,Σ1(0,p~(z))≡KSLΣ(p~(z)).In this paper, we study the classSLMα,Σδ(n,p~(z)) and obtain estimates of the initial coefficients a2 and a3 and upper bounds for the Fekete-Szegö functional for the functions in this class.


## 2. Coefficient Bounds for the Function ClassSLMα,Σδ(n,p~(z))

For deriving our results, we need the following lemma.Lemma 5 (see [14]).
Ifp∈P is family of all functions p analytic in U for which Re⁡[p(z)]>0 and has the form p(z)=1+p1z+p2z2+⋯ for z∈U, then |pn|≤2 for each n.Theorem 6.
Iff∈SLMα,Σδ(n,p~(z)), then(20)a2≤τδ21+δ2n1+α21-3τ+τδ21+2α1+2δn-1+3α1+δ2nand(21)a3≤τ1+δ2nδ21+α21-3τ-τδ1+3α21+2αδ1+2δnδ21+δ2n1+α21-3τ+τδ21+2α1+2δn-1+3α1+δ2nProof.
Asf∈SLMα,Σδ(n,p~(z)), so by Definition 4 and using the principle of subordination, there exist Schwarz functions r(z) and s(z) such that(22)1-αDδn+1fzDδnfz+αDδn+1fz′Dδnfz′=p~rzand(23)1-αDδn+1gwDδngw+αDδn+1gw′Dδngw′=p~swwhere r(z)=1+r1z+r2z2+⋯ and s(w)=1+s1w+s2w2+⋯.
On expanding, it yields(24)1-αDδn+1fzDδnfz+αDδn+1fz′Dδnfz′=1+1+αδ1+δna2z+δ21+2α1+2δna3-1+3α1+δ2na22z2+⋯and(25)1-αDδn+1gwDδngw+αDδn+1gw′Dδngw′=1-1+αδ1+δna2w+δ-21+2α1+2δna3+41+2α1+2δn-1+3α1+δ2na22w2+⋯Again(26)p~rz=1+p~1c1z2+12c2-c122+c124p~2z2+⋯and(27)p~sz=1+p~1d1w2+12d2-d122+d124p~2w2+⋯Using (24) and (26) in (22) and equating the coefficients of z and z2, we get(28)1+αδ1+δna2=c1τ2and(29)δ21+2α1+2δna3-1+3α1+δ2na22=12c2-c122τ+c1243τ2.Again using (25) and (27) in (23) and equating the coefficients of w and w2, we get(30)-1+αδ1+δna2=d1τ2and(31)δ-21+2α1+2δna3+41+2α1+2δn-1+3α1+δ2na22=12d2-d122τ+d1243τ2.From (28) and (30), it is clear that(32)c1=-d1and(33)2a22=c12+d12τ24δ21+δ2n1+α2.Adding (29) and (31), it yields(34)δ41+2α1+2δn-21+3α1+δ2na22=12c2+d2τ-14c12+d12τ+34c12+d12τ2.Putting (33) in (34), we get(35)δτ41+2α1+2δn-21+3α1+δ2n+21-3τδ21+δ2n1+α2a22=12c2+d2τ2.Using Lemma 5 and on applying triangle inequality in (35), (20) can be easily obtained.
Now subtracting (31) from (29), we obtain(36)4δ1+2α1+2δna3-4δ1+2α1+2δna22=12c2-d2τ.Applying triangle inequality and using Lemma 5 and (35) in (36), it yields(37)4δ1+2α1+2δna3≤2τ+4δ1+2α1+2δna22.From (37), result (21) is obvious.Forδ=1, Theorem 6 gives the following result.Corollary 7.
Iff∈SLMα,Σ(n,p~(z)), then(38)a2≤τ4n1+α2+21+2α3n-3α2+9α+44nτand(39)a3≤τ4n1+α2-3α2+9α+4τ21+2α3n4n1+α2+21+2α3n-3α2+9α+44nτFor δ=1, n=0, Theorem 6 gives the following result due to Güney et al. [13].Corollary 8.
Iff(z)∈SLMα,Σ(p~(z)), then(40)a2≤τ1+α2-1+α2+3ατand(41)a3≤τ1+α2-3α2+9α+4τ21+2α1+α1+α-2+3ατ.Forδ=1,n=0,α=0, Theorem 6 agrees with the following result proved by Güney et al. [13] (Corollary 1).Corollary 9.
Iff(z)∈SLΣ(p~(z)), then(42)a2≤τ1-2τand(43)a3≤τ1-4τ21-2τ.Forδ=1, n=0, α=1, Theorem 6 agrees with the following result proved by Güney et al. [13] (Corollary 2).Corollary 10.
Iff(z)∈KSLΣ(p~(z)), then(44)a2≤τ4-10τand(45)a3≤τ1-4τ32-5τ.


## 3. Fekete-Szegö Inequality for the Function ClassSLMα,Σδ(n,p~(z))

Theorem 11.
Letf(z)∈SLMα,Σδ(n,p~(z)), then(46)a3-μa22≤τ2δ1+2α1+2δn,μ-1≤τδ21+2α1+2δn-1+3α1+δ2n+δ21+α21+δ2n1-3τ2τδ1+2α1+2δn;1-μτ2τδ21+2α1+2δn-1+3α1+δ2n+δ21+α21+δ2n1-3τ,μ-1≥τδ21+2α1+2δn-1+3α1+δ2n+δ21+α21+δ2n1-3τ2τδ1+2α1+2δn.Proof.
From (35) and (36), it yields(47)a3-μa22=1-μτ2c2+d24τδ21+2α1+2δn-1+3α1+δ2n+4δ21+δ2n1+α21-3τ+τc2-d28δ1+2α1+2δn.Equation (47) can be expressed as(48)
a
3
-
μ
a
2
2
=
h
μ
+
τ
8
δ
1
+
2
α
1
+
2
δ
n
c
2
+
h
μ
-
τ
8
δ
1
+
2
α
1
+
2
δ
n
d
2where(49)hμ=1-μτ24τδ21+2α1+2δn-1+3α1+δ2n+4δ21+δ2n1+α21-3τ.Taking modulus, we obtain(50)a3-μa22≤τ2δ1+2α1+2δn,0≤hμ≤τ8δ1+2α1+2δn;4hμ,hμ≥τ8δ1+2α1+2δn.So (46) can be easily obtained from (50).Forδ=1, Theorem 11 gives the following result.Corollary 12.
Letf(z)∈SLMα,Σ(n,p~(z)), then(51)a3-μa22≤τ21+2α3n,μ-1≤21+2α3n-3α2+9α+44nτ+1+α24n2τ1+2α3n;1-μτ221+2α3n-3α2+9α+44nτ+1+α24n,μ-1≥21+2α3n-3α2+9α+44nτ+1+α24n2τ1+2α3n.Forδ=1, n=0, Theorem 11 gives the following result due to Güney et al. [13].Corollary 13.
Letf(z)∈SLMα,Σ(p~(z)), then(52)a3-μa22≤τ21+2α,μ-1≤1+α1+α-2+3ατ2τ1+2α;1-μτ21+α1+α-2+3ατ,μ-1≥1+α1+α-2+3ατ2τ1+2α.Forδ=1, n=0, α=0, Theorem 11 agrees with the following result proved by Güney et al. [13] (Corollary 4).Corollary 14.
Iff(z)∈SLΣ(p~(z)), then(53)a3-μa22≤τ2,μ-1≤1-2τ2τ;1-μτ21-2τ,μ-1≥1-2τ2τ.Forδ=1, n=0, α=1, Theorem 11 agrees with the following result proved by Güney et al. [13] (Corollary 5).Corollary 15.
Iff(z)∈KSLΣ(p~(z)), then(54)a3-μa22≤τ6,μ-1≤2-5τ3τ;1-μτ222-5τ,μ-1≥2-5τ3τ.



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