# On Certain Class of Bazilevič Functions Associated with the Lemniscate of Bernoulli

Authors: Tamer M. Seoudy; Amnah E. Shammaky
Journal: Journal of Function Spaces
 (2020)

---

## Abstract

Making use of the principle of subordination, we introduce a certain class of multivalently Bazilevic˘ functions involving the Lemniscate of Bernoulli. Also, we obtain subordination properties, inclusion relationship, convolution result, coefficients estimate, and Fekete–Szegö problem for this class.


---

## Body

## 1. Introduction

LetHU be the class of analytic functions in the open unit disk
(1)U=ζ∈ℂ:ζ<1,and let Ap denote the subclass of HU consisting of functions of the form:
(2)fζ=ζp+∑k=p+1∞akζkp∈ℕ=1,2,3,⋯.We writeA1=A. For f1,f2∈HU, we say that f1ζ is subordinate to f2ζ, written symbolically, f1≺f2 in U or f1ζ≺f2ζζ∈U, if there exists a Schwarz function ωζ, which (by definition) is analytic in U with ω0=0 and ωζ<1ζ∈U such that f1ζ=f2ωζζ∈U. Further more, if the function f2ζ is univalent in U, then we have the following equivalence (see [1, 2]):
(3)f1ζ≺f2ζζ∈U⇔f10=f20andf1U⊂f2U.Letϕ:ℂ2×U⟶ℂ and hζ be univalent in U. If gζ is analytic in U and satisfies the first order differential subordination:
(4)ϕgζ,ζg′ζ;ζ≺hζ,then gζ is a solution of the differential subordination (4). The univalent function qζ is called a dominant of the solutions of the differential subordination (4) if gζ≺qζ for all gζ satisfying (4). A univalent dominant q~ that satisfies q~≺q for all dominants of (4) is called the best dominant.Sokól and Stankiewicz [3] introduced the class SL∗ consisting of analytic functions f∈A satisfying the following condition
(5)ζf′ζfζ2−1<1,which is equivalent to
(6)ζf′ζfζ≺qζ=1+ζ,where the function
(7)qζ=1+ζζ∈U,maps U onto the domain O=w∈ℂ:Rw>0,w2−1<1, and its boundary ∂O is the right-half of the lemniscate of Bernoulli x2+y22−2x2−y2=0. Several geometric properties of SL∗ were investigated done by many authors in ([4–7]).Now, we define a classBpλ,α of Bazilevic˘ functions associated with lemniscate of Bernoullia by using the principle of differential subordination as follows.Definition 1.
A functionf∈Ap is said to be the class Bpλ,α if it satisfies the following subordination condition:
(8)1−λfζζpα+λζf′ζpfζfζζpα≺1+ζ,all the powers are principal values and throughout the paper unless otherwise mentioned the parameters λ, α, and p are constrained as λ∈ℂ, α>0, p∈ℕ, and ζ∈U.We note that(1)
B1λ,α=Bλ,α=f∈A:1−λfζ/ζα+λζf′ζ/fζfζ/ζα≺1+ζ(2)
Bpλ,1=Bpλ=f∈Ap:1−λfζ/ζp+λf′ζ/pζp−1≺1+ζ and B1λ=Bλ(3)
Bp1,α=Bpα=f∈Ap:ζf′ζ/pfζfζ/ζpα≺1+ζ and B1α=Bα(4)
Bp1,0=SLp∗=f∈Ap:ζf′ζ/pfζ≺1+ζ and SL1∗=SL∗In order to establish our main results, we need the following lemmas.Lemma 2 [8].
Let the functionh be analytic and convex (univalent) in U with h0=1. Suppose also that the function gζ given by
(9)gζ=1+c1ζ+c2ζ2+⋯,is analytic in U. If
(10)gζ+ζg′ζγ≺hζRγ≥0;γ≠0;ζ∈U,then
(11)gζ≺qζ=γζ−γ∫0ζhttγ−1dt≺hζ,and qζ is the best dominant.Lemma 3.
[9]. For real or complex numbers a,b,cc≠0,−1,−2,⋯ and ζ∈U,
(12)∫01tb−11−tc−b−11−tζ−adt=ΓbΓc−bΓc2F1a,b;c;ζRc>Rb>0,2F1a,b;c;ζ=1−ζ−a2F1a,c−b;c;ζζ−1.Lemma 4.
[10]. Let F be analytic and convex in U. If f,g≺F, then
(13)λf+1−λg≺F0≤λ≤1.Lemma 5 [11].
Letfζ=∑k=1∞akζk be analytic in U and gζ=∑k=1∞bkζk be analytic and convex in U. If f≺g, then
(14)ak<b1k∈ℕ.Lemma 6 [12].
Letgζ=1+∑k=1∞ckζk∈P, i.e., let g be analytic in U and satisfy Rgζ>0 for ζ∈U, then the following sharp estimate holds
(15)c2−vc12≤2max1,2v−1forallv∈ℂ.The result is sharp for the functions given by(16)gζ=1+ζ21−ζ2orgζ=1+ζ1−ζ.Lemma 7.
[12]. If gζ=1+∑k=1∞ckζk∈P, then
(17)c2−νc12≤−4ν+2ifν≤0,2if0≤ν≤1,4ν−2ifν≥1,,when υ<0 or ν>1, the equality holds if and only if gζ=1+ζ/1−ζ or one of its rotations. If 0<ν<1, then the equality holds if and only if gζ=1+ζ2/1−ζ2 or one of its rotations. If ν=0, the equality holds if and only if
(18)gζ=1+λ21+ζ1−ζ+1−λ21−ζ1+ζ0≤λ≤1,or one of its rotations. If ν=1, the equality holds if and only if g is the reciprocal of one of the functions such that equality holds in the case of ν=0.Also, the above upper bound is sharp, and it can be improved as follows when0<ν<1:
(19)c2−νc12+νc12≤20≤ν≤12,c2−νc12+1−νc12≤212≤ν≤1.In the present paper, we obtain subordination properties, inclusion relationship, convolution result, coefficients estimate, and Fekete–Szegö inequalities for the classBpλ,α.


## 2. Main Results

We begin by presenting our first subordination property given by Theorem8.Theorem 8.
Iff∈Bpλ,α with Rλ>0, then
(20)fζζpα≺Qζ≺1+ζ,where the function Qζ given by
(21)Qζ=1+ζ1/22F1−12,1;pαλ+1;ζ1+ζ,is the best dominant.Proof.
Letf∈Bpλ,α and suppose that
(22)gζ=fζζpαζ∈U.
Then, the functiongζ is of the form (9), analytic in U, and g0=1. By taking the derivatives in the both sides of (22), we get
(23)1−λfζζpα+λζf′ζpfζfζζpα=gζ+λpαζg′ζ.
Sincef∈Bpλ,α, we have
(24)gζ+λpαζg′ζ≺1+ζ.
Now, by using Lemma2 for γ=pα/λ, we deduce that
(25)fζζpα≺Qζ=pαλζ−pα/λ∫0ζtpα/λ−11+t1/2dt=pαλ∫01upα/λ−11+ζu1/2du=1+ζ1/22F1−12,1;pαλ+1;ζ1+ζ,where we have made a change of variables followed by the use of identities in Lemma 3 with a=−1/2, b=pα/λn, and c=b+1. This completes the proof of Theorem 8.For a functionf∈Ap given by (2), the generalized Bernardi-Libera-Livingston integral operator Fp,μ:Ap⟶Ap, with μ>−p, is defined by (see [13–16])
(26)Fp,μfζ=μ+pζμ∫0ζtμ−1ftdtμ>−p.It is easy to verify that for allf∈Ap, we have
(27)ζFp,μfζ′=μ+pfζ−μFp,μfζ.Theorem 9.
If the functionf∈Ap satisfies the subordination condition
(28)1−λFp,μfζζpα+λfζFp,μfζFp,μfζζpα≺1+ζ,and Fp,μ is the integral operator defined by (26), then
(29)Fp,μfζζpα≺Kζ≺1+ζ,where the function K given by
(30)Kζ=1+ζ1/22F1−12,1;αp+μλ+1;ζ1+ζ,is the best dominant of (28).Proof.
Let(31)gζ=Fp,μfζζpαζ∈U,then g is analytic in U. Differentiating (31) with respect to ζ and using the identity (28) in the resulting relation, we get
(32)1−λFp,μfζζpα+λfζFp,μfζFp,μfζζpα=gζ+λζg′ζαp+μ≺1+ζ.
Employing the same technique that we used in the proof of Theorem8, the remaining part of the theorem can be proved similarly.Theorem 10.
Ifλ2≥λ1≥0, then
(33)Bpλ2,α⊂Bpλ1,α.Proof.
Suppose thatf∈Bpλ2,α. We know that
(34)1−λ2fζζpα+λ2ζf′ζpfζfζζpα≺1+ζ,
Thus, the assertion of Theorem10 holds for λ2=λ1≥0. If λ2>λ1≥0, by Theorem 8 and (34), we have
(35)fζζpα≺1+ζ.
At the same time, we have(36)1−λ1fζζpα+λ1ζf′ζpfζfζζpα=1−λ1λ2fζζpα+λ1λ21−λ2fζζpα+λ2ζf′ζpfζfζζpα.
Moreover, since0≤λ1/λ2<1, and the function 1+ζζ∈U is analytic and convex in U.
Combining (34)–(36) and Lemma 4, we find that
(37)1−λ1fζζpα+λ1ζf′ζpfζfζζpα≺1+ζ,that is f∈Bpλ1,α, which implies that the assertion (33) of Theorem 10 holds.Theorem 11.
Iff∈Ap, then f∈Bpλ,αif and only if
(38)fζζpα∗1−Lζ+Mζ21−ζ2≠0ζ∈U,where
(39)L=1+λαpe−iθ1+1+eiθ+2M=e−iθ1+1+eiθ+1.Proof.
For any functionf∈Ap,we can verify that
(40)fζζpα=fζζpα∗11−ζ,(41)ζf′ζpfζfζζpα=fζζpα∗1−1−1/pαζ1−ζ2.
First, in order to prove that (38) holds, we will write (8) by using the principle of subordination, that is,
(42)1−λfζζpα+λζf′ζpfζfζζpα=1+wζ,where wζ is a Schwarz function, hence
(43)1−λfζζpα+λζf′ζpfζfζζpα≠1+eiθ,for all ζ∈U and θ∈0,2π). From (40) and (41), the relation (43) may be written as
(44)fζζpα∗1−1+eiθ−1−λ/αp−21+eiθζ−1+eiθζ21−ζ2≠0,which is equivalent to
(45)fζζpα∗1−1+λ/pαe−iθ1+1+eiθ+2ζ+e−iθ1+1+eiθ+1ζ21−ζ2≠0,that is (38).
Reversely, suppose thatf∈Ap satisfy the condition (38). Like it was previously shown, the assumption (38) is equivalent to (41), that is,
(46)1−λfζζpα+λζf′ζpfζfζζpα≠1+eiθζ∈U.
Denoting(47)φζ=1−λfζζpα+λζf′ζpfζfζζpαandψζ=1+ζ,the relation (46) could be written as φU∩ψ∂U=∅. Therefore, the simply connected domain φU is included in a connected component of ℂ\ψ∂U. From this fact, using that φ0=ψ0=1 together with the univalence of the function ψ, it follows that φζ≺ψζ, that is f∈Bpλ,α.Theorem 12.
Iffζ given by (2) belongs to Bpλ,α, then
(48)ap+1≤p2pα+λ.Proof.
Combining (2) and (8), we obtain
(49)1−λfζζpα+λζf′ζpfζfζζpα=1+pα+λpap+1ζ+⋯.≺1+ζ=1+12ζ−18ζ2+⋯.
An application of Lemma5 to (49) yields
(50)pα+λpap+1<12.
Thus, from (50), we easily obtain (48) asserted by Theorem 12.


## 3. Fekete–Szegö Problem

Many authors have considered the Fekete–Szegö problem for many subclasses of analytic functions (see, for instance, [17–21]). In this section, we evaluate the Fekete–Szegö inequalities for the class Bpλ,α.Theorem 13.
Iff given by (2) belongs to the class Bpλ,α, then
(51)ap+2−μap+12≤p2αp+2λmax1;141+pαp+2λα−1+2μαp+λ2.The result is sharp.Proof.
Iff∈Bpλ,α, then there is a Schwarz function ω in U such that
(52)1−λfζζpα+λζf′ζpfζfζζpα=ϕωζ,where ϕζ=1+ζ. Define the function gζ by
(53)gζ=1+ωζ1−ωζ=1+c1ζ+c2ζ2+⋯.
Sinceωζ is a Schwarz function, we see that g∈P with g0=1. Therefore,
(54)ϕωζ=ϕgζ−1gζ+1=2gζgζ+1=1+14c1ζ+14c2−532c12ζ2+.⋯
Now by substituting (54) in (52), we have
(55)1−λfζζpα+λζf′ζpfζfζζpα=1+14c1ζ+14c2−532c12ζ2+.⋯
Equating the coefficients ofζ and ζ2, we obtain
(56)ap+1=p4αp+λc1,ap+2=p4αp+2λc2−p325αp+2λ+pα−1αp+λ2c12.
Therefore,(57)ap+2−μap+12=p4αp+2λc2−vc12,where
(58)ν=185+pαp+2λα−1+2μαp+λ2.
Our result now follows by an application of Lemma6. The result is sharp for the functions
(59)1−λfζζpα+λζf′ζpfζfζζpα=ϕζ2,1−λfζζpα+λζf′ζpfζfζζpα=ϕζ.
This completes the proof of Theorem13.Puttingλ=1 and α=0 in Theorem 13, we obtain the following corollary.Corollary 14.
Iff given by (2) belongs to the class Bpα, then
(60)ap+2−μap+12≤p2αp+2max1;141+pαp+2α−1+2μαp+12.The result is sharp.Puttingλ=1 and α=0 in Theorem 13, we obtain the following corollary.Corollary 15.
Iff given by (2) belongs to the class SLp∗, then
(61)ap+2−μap+12≤p4max1;1+2p2μ−14.The result is sharp.Puttingp=λ=1 and α=0 in Theorem 13, we obtain the following corollary.Corollary 16.
Iff given by (2) (with p=1) belongs to the class SL∗, then
(62)a3−μa22≤14max1;4μ−14.The result is sharp.Applying Lemma7 to (57) and (58), we obtain the following theorem.Theorem 17.
Let(63)σ1=pαp+2λ1−α−5αp+λ22pαp+2λ,σ2=pαp+2λ1−α+3αp+λ22pαp+2λ,σ3=pαp+2λ1−α−αp+λ22pαp+2λ.Iff given by (2) belongs to the class Bpλ,α, then
(64)ap+2−μap+12≤−p8αp+2λ1+pαp+2λα−1+2μαp+λ2μ≤σ1,p2αp+2λσ1≤μ≤σ2,p8αp+2λ1+pαp+2λα−1+2μαp+λ2μ≥σ2.Further, ifσ1≤μ≤σ3, then
(65)ap+2−μap+12+125αp+λ2pαp+2λ+α−1+2μap+12≤p2αp+2λ.Ifσ3≤μ≤σ2, then
(66)ap+2−μap+12+123αp+λ2pαp+2λ−α+1−2μap+12≤p2αp+2λ.Puttingλ=1 in Theorem 17, we obtain the following result.Corollary 18.
Let(67)δ1=pαp+21−α−5αp+122pαp+2,δ2=pαp+21−α+3αp+122pαp+2,δ3=pαp+21−α−αp+122pαp+2.Iff given by (2) belongs to the class Bpα, then
(68)ap+2−μap+12≤−p8αp+21+pαp+2α−1+2μαp+12μ≤δ1,p2αp+2δ1≤μ≤δ2,p8αp+21+pαp+2α−1+2μαp+12μ≥δ2.Further, ifδ1≤μ≤δ3, then
(69)ap+2−μap+12+125αp+12pαp+2+α−1+2μap+12≤p2αp+2.Ifδ3≤μ≤δ2, then
(70)ap+2−μap+12+123αp+12pαp+2−α+1−2μap+12≤p2αp+2.Puttingλ=1 and α=0 in Theorem 17, we obtain the following result for the subclass SLp∗.Corollary 19.
Iff given by (2) belongs to the class SLp∗, then
(71)ap+2−μap+12≤−p1+2p2μ−116μ≤2p−54p,p42p−54p≤μ≤2p+34p,p1+2p2μ−116μ≥2p+34p.Further, if2p−5/4p≤μ≤2p−1/4p, then
(72)ap+2−μap+12+145p−2+4μap+12≤p4.If2p−1/4p≤μ≤2p+3/4p, then
(73)ap+2−μap+12+143p+2−4μap+12≤p4.Puttingλ=p=1 and α=0 in Theorem 17, we obtain the following result obtained by ([18], Theorem 2.1).Corollary 20.
([18], Theorem 2.1). If f given by (2) (with p=1) belongs to the class SL∗, then
(74)a3−μa22≤−1164μ−1μ≤−34,14−34≤μ≤54,1164μ−1μ≥54.Further, if−3/4≤μ≤1/4, then
(75)a3−μa22+143+4μa22≤14.If1/4≤μ≤5/4, then
(76)a3−μa22+145−4μa22≤14.



---

*Source: 6622230-2020-12-12.xml*