# Bi-Univalent Function Classes Defined by Using a Second Einstein Function

Authors: Alaa H. El-Qadeem; Saleh A. Saleh; Mohamed A. Mamon
Journal: Journal of Function Spaces
 (2022)

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## Abstract

Motivated by q-calculus, subordination principle, and the second Einstein function, we define two families of bi-univalent analytic functions on the open unit disc of the complex plane. We deduce estimates for the first two Maclaurin’s coefficients and the Fekete-Sezgö functional inequalities for the functions that belong to these families of functions.


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## Body

## 1. Introduction and Basic Concepts

LetA denotes the collection of all functions f with the following series representation:
(1)fz=z+∑n=2∞anzn,which are analytic and univalent in the open unit disc U=z:z<1 and satisfy the usual normalization condition f0=f′0−1=0. Also, an important class of functions will be called P, P defines the family of functions ϕ with the restrictions that the image domain of ϕ (ϕ is a convex function with Reϕ>0 in U) is symmetric along the real axis and starlike about ϕ0=1 with ϕ′0>0.In 1980, Gradshteyn and Ryzhik [1] give an expression of the Bernoulli polynomials which have important applications in number theory and classical analysis. They appear in the integral representation of differentiable periodic functions since they are employed for approximating such functions in terms of polynomials. They are also used for representing the remainder term of the composite Euler-Maclaurin quadrature rule.The Bernoulli polynomialsBnx are usually defined (see, e.g., [2]) by means of the generating function:
(2)Gx,t≔textet−1=∑n=0∞Bnxn!tn,t<2π,where Bnx are polynomials in x, for each nonnegative integer n.The Bernoulli polynomials are easily computed by recursion since(3)∑j=0n−1njBjx=nxn−1,n=2,3,⋯.The first few Bernoulli polynomials are(4)B0x=1,B1x=x−12,B2x=x2−x+16,B3x=x3−32x2+12x,⋯.Furthermore, Bernoulli numbersBn≔Bn0 are produced directly by putting x=0 in Bernoulli polynomials. The first few Bernoulli numbers are
(5)B0=1,B1=−12,B2=16,B4=−130,⋯,B2n+1=0,∀n=1,2,⋯.Moreover, Bernoulli numbersBn can be generated by means of the so-called Einstein function Ez:
(6)Ez≔zez−1=∑n=0∞Bnn!zn.In mathematics, the Einstein function is a name occasionally used for one of the functions (see [3–6]):
(7)E1z≔zez−1,E2z≔z2ezez−12,E3z≔log1−e−z,E4z≔zez−1−log1−e−z.It is easily noticed that bothE1 and E2 have these nice properties, but E3 and E4 are not (see Figure 1); the range of E1 and E2 (E1 and E2 are convex functions) is symmetric along the real axis and starlike about E10=E20=1 and RE1z>0,RE2z>0∀z∈U.Figure 1
The images of unit discU by the Einstein functions E1, E2, E3, and E4.
(a)
E1U(b)
E2U(c)
E3U(d)
E4UThe series representation is given by (one can type “Maclaurin series forz/−1+ez” in [7])
(8)E1z=1+∑n=1∞Bnn!zn,E2z=1+∑n=1∞1−nBnn!zn,where Bn is the nth Bernoulli number.El-Qadeem et al. [8] have introduced some results related to the first Einstein function E1. Here, we will deal with the second Einstein function E2. Note that E2′0≯0 (indeed E2′0=0), i.e., E2∉P. Thus, we shall define the function:
(9)Eμz≔E2z+μz,where μ∈I≔0.28,0.92. It is obvious that EμU is a convex domain, symmetric along the real axis and starlike about Eμ0=1 and REμz>0∀z∈U; moreover, Eμ′0=μ>0. This proves that Eμ∈P.Example 1.
(i)
Ifμ<0.28, then E2z+μz is not a convex function, see Figure 2(a)(ii)
Ifμ>0.92, then ∃z∈U s.t. RE2z+μz≯0, see Figure 2(b)(iii)
If0.28≤μ≤0.92, then RE2z+μz>0∀z∈U; also, E2z+μz is a convex function, see Figure 2(c)Figure 2
EμU for three different values of μ.
(a)
E2z+0.27z(b)
E2z+0.93z(c)
E2z+0.5zNow, letS be the subfamily of A consisting of all functions of the form (1) which are univalent in U. It is well known, by using the Koebe one-quarter theorem [9], that every univalent function f∈S containing a disc of radius 1/4 has an inverse function f−1, which is defined by
(10)f−1fz=z,z∈U,ff−1ω=ωω∈Δ=ω∈ℂ:ω<14.A functionf∈S is said to be bi-univalent in U if both f and f−1 are univalent in U. Let Σ denotes the subfamily of S, consisting of all biunivalent functions defined on the unit disc U. Since f∈Σ has the Maclaurin series expansion given by (1), a simple calculation shows that its inverse g=f−1 has the series expansion:
(11)gω=f−1ω=w−a2w2+2a22−a3w3−⋯.Examples of functions in the classΣ are
(12)z1−z,−log1−z,12log1+z1−z,and so on. However, the familiar Koebe function is not a member of Σ. Other common examples of functions in S such as
(13)z−z22,z1−z2are also not members of Σ.Now, we introduce some notes about the q-difference operator which uses in investigating our main families. In view of Annaby and Mansour [10], the q-difference operator is defined by
(14)∂qfz=fqz−fzzq−1,z≠0;f′0,z=0;∂q0fz=fz,∂q1fz=∂qfz,∂qmfz=∂q∂qm−1fzm∈ℕ.Thus, for the functionf∈Σ denoted by (1), we have
(15)∂qfz=1+∑n=2∞nqanzn−1z≠0,where
(16)nq=qn−1q−1=∑j=0n−1qj,n∈ℕ.Definition 1 (see [11, 12]).
An analytic functionf is said to be subordinate to another analytic function g, written as fz≺gzz∈U, if there exists a Schwarz function ω, which is analytic in U with ω0=0 and ωz<1z∈U, such that fz=gwz. In particular, if the function g is univalent in U, then we have the following equivalence:
(17)fz≺gz⇔f0=g0,fU⊂gU.
Our aim in this article is to introduce two families of analytic bi-univalent function related to the modified Einstein functionEμz. Furthermore, we get estimations to a2, a3, and also the Fekete Sezgö inequalities for the functions that belong to these two families.Definition 2.
Consider0≤δ≤1, 0≤λ≤1, q∈0,1, and μ∈0.28,0.92. The function f∈Σ is said to be in MΣqδ,λ;Eμ if it is satisfying
(18)1−δzfz1−λ∂qfz+δ∂qz∂qfz∂qfz≺Eμz,(19)1−δωgω1−λ∂qgω+δ∂qω∂qgω∂qgω≺Eμω,where g=f−1 is given by (11) and z,ω∈U.Definition 3.
Consider0≤α≤1, 0≤β≤1, q∈0,1, and μ∈0.28,0.92. The function f∈Σ is said to be in SΣqα,β;Eμ if it is satisfying
(20)1−αfzz+α∂qfz+βz∂q2fz≺Eμz,1−αgωω+α∂qgω+βω∂q2gω≺Eμω,where g=f−1 is given by (11) and z,ω∈U.Remark 4.
Specializing the parametersδ, λ, α, β, and q in Definitions 2 and 3, then we have the following subfamilies:
(i)
Ifδ=0 and q⟶1−, then MΣλ;Eμ is the subfamily of Σ, consisting of functions f which satisfy the following conditions:(21)zfz1−λf′z≺Eμz,ωgω1−λg′ω≺Eμω,where g=f−1 is given by (11), Eμ is defined by (9), and z,ω∈U(ii)
Ifδ=1, λ=0, and q⟶1−, then let KMΣEμ be the family of bi-univalent convex functions with respect to Einstein function, consisting of the functions f which satisfy the following conditions:(22)1+zf′′zf′z≺Eμz,1+ωg′′ωg′ω≺Eμω,where g=f−1 is given by (11), Eμ is defined by (9), and z,ω∈U(iii)
Ifδ=λ=0 and q⟶1−, then let S∗MΣEμ be the family of bi-univalent starlike functions with respect to Einstein function, consisting of functions f which satisfy the following conditions:(23)zf′zfz≺Eμz,ωg′ωgω≺Eμω,where g=f−1 is given by (11), Eμ is defined by (9), and z,ω∈U(iv)
Ifα=β=1 and q⟶1−, then KΣEμ is the subfamily of Σ, consisting of functions f which satisfy the following conditions:(24)f′z+zf′′z≺Eμz,g′ω+ωg′′ω≺Eμω,where g=f−1 is given by (11), Eμ is defined by (9), and z,ω∈U(v)
Ifα=1, β=0, and q⟶1−, then SΣEμ is the subfamily of Σ, consisting of functions f which satisfy the following conditions:(25)f′z≺Eμz,g′ω≺Eμω,where g=f−1 is given by (11), Eμ is defined by (9), and z,ω∈ULemma 5 (see [13, 14]).
Letl1,l2∈ℝ and p1,p2∈ℂ. If p1,p2<ζ, then
(26)l1+l2p1+l1−l2p2≤2l1ζ,l1≥l2,2l2ζ,l1≤l2.Lemma 6 (see [9]).
Suppose thatχz is analytic in the unit open disc U with χ0=0, χz<1, and that
(27)χz=ρ1z+∑n=2∞ρnznforallz∈U,then
(28)ρ1≤1,ρn≤1−ρ12n∈ℕ\1.


## 2. Main Results

Unless otherwise mentioned, we assume in the remainder of this article that0≤δ≤1, 0≤λ≤1, 0≤α≤1, 0≤β≤1, q∈0,1, μ∈0.28,0.92, and z,ω∈U.Theorem 7.
Letf∈MΣqδ,λ;Eμ, then
(29)a2≤2μK2+K4+K12/6μ2+2/μK12,a3≤μK2+K4K3K2+K4,where
(30)K1=1−δ2q+λ−1+δ2q2q−1K2=1−δλ−12q+λ2−1−δ2q2q−1K3=1−δ3q+λ−1+δ3q3q−1K4=1−δλ−12q+λ2+1−δ2q22q−1+21−δ3q+2δ3q3q−1.Proof.
Letf and g be in MΣqδ,λ;Eμ, then, they satisfy (18) and (19), respectively. But according to Definition 1 and Lemma 6, there exist two functions uz and vω of the form
(31)uz=∑n=1∞cnzn,vω=∑n=1∞dnωn,such that
(32)1−δzfz1−λ∂qfz+δ∂qz∂qfz∂qfz=μuz+E2uz,(33)1−δωgω1−λ∂qgω+δ∂qω∂qgω∂qgω=μvω+E2vω.
After some simple calculations, we deduce(34)μuz+E2uz=1+μuz−uz212+uz4240+⋯=1+μc1z+μc2−c1212z2+⋯,(35)μvω+E2vω=1+μvω−vω212+vω4240+⋯=1+μd1ω+μd2−d1212ω2+⋯.
Also,(36)1−δzfz1−λ∂qfz+δ∂qz∂qfz∂qfz=1+K1a2z+K3a3+K2a22z2+⋯.(37)1−δωgω1−λ∂qgω+δ∂qω∂qgω∂qgω=1−K1a2ω+K4a22−K3a3ω2+⋯,where Kj:j=1,2,3,4 are stated in (30).
By substituting from (34), (35), (36), and (37) into (32) and (33) with comparing the coefficient in both sides, we obtain
(38)K1a2=μc1,(39)K3a3+K2a22=μc2−c1212,(40)−K1a2=μd1,(41)−K3a3+K4a22=μd2−d1212.
As a direct result of equations (38) and (40), we get
(42)c1=−d1,and also,
(43)c12+d12=2μ2K12a22.
By adding (39) to (41), then using (43), we obtain
(44)K2+K4+K126α2a22=μc2+d2.
Equations (42) and (44) together with using Lemma 6 implies that
(45)K2+K4+K126α2a22≤2μ1−c12.
But from equation (38), we can deduce
(46)c12=K12μ2a22.
By using (46) into (45), we obtain
(47)a2≤2μK2+K4+K12/6μ2+2K12/μ.
Further, from (39) and (41) with using (42), we get
(48)K3K2+K4a3=μc2K4−K2d2−c1212K4−K2.
Thus, by virtue of Lemma6, we find
(49)K3K2+K4a3≤K2+K4μ+c12112−μ.
Since1/12−μ<0 for all μ∈0.28,0.92. Then, we conclude
(50)a3≤μK2+K4K3K2+K4.
Thus, the proof is completed.Theorem 8.
Letf∈SΣqα,β;Eμ, then
(51)a2≤2μYα,β,μ;q+2/μ1+α2q−1+β2q2,a3≤μ1+α3q−1+β2q3q,1+α2q−1+β2q2μ1+α3q−1+β2q3q≥1,Θα,β,μ,q+μ1+α3q−1+β2q3q,1+α2q−1+β2q2μ1+α3q−1+β2q3q<1,where
(52)Yα,β,μ;q=21+α3q−1+β2q3q+112μ21+α2q−1+β2q2,(53)Θα,β,μ,q=2μ1+α3q−1+β2q3q−21+α2q−1+β2q21+α3q−1+β2q3qYα,β,μ;q+2/μ1+α2q−1+β2q2.Proof.
Supposef and g be in SΣqα,β;Eμ, then it satisfies (18) and (19). According to Definition 1 and Lemma 6, there exist two functions uzandvω of the form
(54)uz=∑n=1∞cnzn,vω=∑n=1∞dnωn,such that
(55)1−αfzz+α∂qfz+βz∂q2fz=μuz+E2uz,(56)1−αgωω+α∂qgω+βω∂q2gω=μvω+E2vω.
With some simple calculations, we get(57)1−αfzz+α∂qfz+βz∂q2fz=1+∑n=2∞1+αnq−1+βn−1qnqanzn,(58)1−αgωω+α∂qgω+βω∂q2gω=1−1+α2q−1+β2qa2ω+1+α3q−1+β2q3qa2ω2+⋯.
By substituting from (34), (35), (57), and (58) into (55) and (56) with comparing the coefficient in both sides, we conclude
(59)1+α2q−1+β2qa2=μc1,(60)1+α3q−1+β2q3qa3=μc2−c1212,(61)−1+α2q−1+β2qa2=μd1,(62)1+α3q−1+β2q3q2a22−a3=μd2+d1212.
From (59) and (61), we obtain
(63)c1=−d1,and also,
(64)c12+d12=1+α2q−1+β2q2μ2a22.
By adding (60) to (62) with using (64), we get
(65)Yα,β,μ;qa22=μc2+d2.
In view of Lemma6, equation (65) together with (63) implies that
(66)Yα,β,μ;qa22≤2μ1−c12.
On the other hand, from equation (59), we can write
(67)c12=1+α2q−1+β2q2μ2a22.
By using (67) into (66), we get
(68)a2≤2μYα,β,μ;q+2/μ1+α2q−1+β2q2,where Yα,β,μ;q is defined in (52).
Further, by subtracting (62) from (60) and using (63), we have
(69)a3=a22+μc2−d221+α3q−1+β2q3q.
In view of Lemma6, equation (69) together with (67) implies that
(70)a3≤1−1+α2q−1+β2q2μ1+α3q−1+β2q3qa22+μ1+α3q−1+β2q3q.
By the virtue of (68), we can get the desired result. Thus, we complete the proof.Theorem 9.
Supposef∈MΣqδ,λ;Eμ and η∈ℂ, then
(71)a3−ηa22≤μΦδ,λ,μ,η;qK3,Φδ,λ,μ,η;q≥1,μK3,Φδ,λ,μ,η;q≤1,where
(72)Φδ,λ,μ,η;q=K4−K2−2ηK3K2+K4+K12/6μ2,and K1,K2,K3, and K4 are given by (30).Proof.
To investigate the desired result, we first subtract (41) from (39) with using (42), we get
(73)a3=K4−K22K3a22+μc2−d22K3.
Thus,(74)a3−ηa22=K4−K2−2ηK32K3a22+μc2−d22K3.
As a result of subsequent computations performed by using (44), we obtain
(75)a3−ηa22≤μ2K3Φδ,λ,μ,η;q+1c2+Φδ,λ,μ,η;q−1d2,where Φδ,λ,μ,η;q given by (72).
But in view of Kanas et al. [15] and (28), we can obtain
(76)c2≤1−c12≤1,d2≤1−d12≤1.
Now, applying Lemma5 to (75), we can obtain the desired result directly. Thus, we complete the proof.Theorem 10.
Let us considerf∈SΣqα,β;Eμ and η∈ℂ, then
(77)a3−ηa22≤2μ1−ηYα,β,μ;q,1−ηYα,β,μ;q≥121+α3q−1+β2q3q,μ1+α3q−1+β2q3q,1−ηYα,β,μ;q≤121+α3q−1+β2q3q,where Yα,β,μ;q is defined by (52).Proof.
In order to investigate the desired result (77), we first subtract (62) from (60) with taking in consideration (63), we conclude
(78)a3−ηa22=1−ηa22+μc2−d221+α3q−1+β2q3q.
By virtue of (65), we can get that
(79)a3−ηa22=μc21−ηYα,β,μ;q+121+α3q−1+β2q3q+μd21−ηYα,β,μ;q−121+α3q−1+β2q3q.
By applying Lemma5 to (79) with using (76), we obtain the required result which completes the proof.


## 3. Set of Corollaries

In this part, we introduce some corollaries by specializing the values of the given parametersλ,δ,α, and β, and taking the limit when q⟶1−, in our main results.Putδ=0 and q⟶1−in Theorems 7, 9, then we have the following corollary:Corollary 11.
Letfz belong to MΣλ;Eμ, then
(80)a2≤12μ3λ+1λ6μ2+12μ+1+12μ2+12μ+1,a3≤2μλ2+3λ+2,a3−ηa22≤2μ1−ηλ+1λ+2+λ+1/6μ2,1−η≥λ+1λ+2+λ+1/6μ22λ+4,μλ+2,1−η≤λ+1λ+2+λ+1/6μ22λ+4.Putδ=λ=0 and q⟶1−in Theorems 7, 9, then we get the following corollary:Corollary 12.
Letfz belong to S∗MΣEμ, then
(81)a2≤12μ312μ2+12μ+1,a3≤μ,a3−ηa22≤12μ31−η12μ2+1,1−η≥12μ2+124μ2,μ2,1−η≤12μ2+124μ2.Putδ=1,λ=0, and q⟶1−in Theorems 7, 9, then we obtain the following:Corollary 13.
Letfz belong to KMΣEμ, then
(82)a2≤3μ39μ2+12μ+1,a3≤518μ,a3−ηa22≤μ35−6η29μ2+1,5−6η≥9μ2+13μ2,μ6,5−6η≤9μ2+13μ2.Putα=β=1 and q⟶1−in Theorems 8, 10, then we have the following corollary:Corollary 14.
Letfz belong to KΣEμ, then
(83)a2≤3μ327μ2+48μ+2,a3≤μ9,a3−ηa22≤μ6μ21−η54μ2+4,1−η≥54μ2+427μ2,19,1−η≤54μ2+427μ2.Putα=1,β=0, and q⟶1− in Theorems 8, 10, then we have the following corollary:Corollary 15.
Letfz belong to SΣEμ, then
(84)a2≤6μ318μ2+24μ+1,a3≤μ3,a3−ηa22≤μ6μ21−η18μ2+1,1−η≥18μ2+118μ2,13,1−η≤18μ2+118μ2.



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