# Majorization Results for Certain Subfamilies of Analytic Functions

Authors: Muhammad Arif; Miraj Ul-Haq; Omar Barukab; Sher Afzal Khan; Saleem Abullah
Journal: Journal of Function Spaces
 (2021)

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## Abstract

Leth1z and h2z be two nonvanishing holomorphic functions in the open unit disc with h10=h20=1. For some holomorphic function qz, we consider the class consisting of normalized holomorphic functions f whose ratios fz/zqz and qz are subordinate to h1z and h2z, respectively. The majorization results are obtained for this class when h1z is chosen either h1z=cosz or h1z=1+sinz or h1z=1+z and h2z=1+sinz.


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## Body

## 1. Introduction

In order to better explain the terminology included in our key observations, some of the essential relevant literature on geometric function theory needs to be provided and discussed here. We start with symbolA which represents the class of holomorphic functions in the region of open unit disc Ud=z∈ℂ:z<1, and if fz is in A, then it satisfies the relationship f0=f′0−1=0. Also, the family S⊂A contains all univalent functions. Though function theory was started in 1851, in 1916, due to coefficient conjecture provided by Bieberbach [1], this field emerged as a good area of new research. This conjecture was proved by De-Branges [2] in 1985. Many good scholars of the period attempted to prove or disprove this conjecture between the years 1916 and 1985. As a result, they identified several subfamilies of a class S of univalent functions linked to various image domains. The families of star-like S∗ and convex K functions are the most basic, mostly studied, and beautiful geometric representations of these subfamilies, which are described as
(1)S∗≕f∈S:Rzf′zfz>0,z∈Ud,K≕f∈S:Rzf′z′f′z>0,z∈Ud.In 1970, Roberston [3] established the idea of quasisubordination among holomorphic functions. Two functions F1z,F2z∈A are related to the relationship of quasisubordination, denoted mathematically by F1z≺qF2z, if there exist functions φz,uz∈A such that zf′z/φz is holomorphic in Ud with φz≦1,u0=0, and uz≦z satisfying the relationship
(2)F1z=φzF2uz,z∈Ud.Also, by choosinguz=z and φz≡1, we obtain the most useful concepts of geometric function theory known as subordination between analytic functions. In fact, if F2z∈S, then, for F1z,F2z∈A, the subordination relationship has
(3)F1z≺F2z⇔F1Ud⊂F2UdwithF10=F20.By takinguz=z, the above definition of quasisubordination becomes the majorization between holomorphic functions and is written mathematically by F1z≪F2z, for F1z,F2z∈A. That is; F1z≪F2z, if a function φz∈A exists with φz≦1 in such a way that
(4)F1z=φzF2z,z∈Ud.This idea was introduced by MacGregor [4] in 1967. Numerous articles have been published in which this idea was used. The work of Altintas and Srivastava [5], Cho et al. [6], Goswami and Aouf [7], Goyal and Goswami [8, 9], Li et al. [10], Panigraht and El-Ashwah [11], Prajapat and Aouf [12], and the authors [13, 14] are worth mentioning on this topic.The general form of the classS∗ was studied in 1992 by Ma and Minda [15] and was given by
(5)S∗Λ=f∈S:zf′zfz≺Λzz∈Ud,where Λz is a regular function with positive real part and Λ′0>0. Also, the function Λz maps Ud onto a star-shaped region with respect to Λ0=1 and is symmetric about the real axis. They addressed some specific results such as distortion, growth, and covering theorems. In recent years, several subfamilies of the set A were studied as a special case of the class S∗Λ. For example,
(i)
if we takeΛz=1+Mz/1+Nz with −1≤N<M≤1, then the deduced family S∗M,N≡S∗1+Mz/1+Nz is described by the functions of the Janowski star-like family established in [16] and later studied in different directions in [17, 18](ii)
the familySL∗≡S∗Λz with Λz=1+z was developed in [19] by Sokól and Stankiewicz. The image of the function Λz=1+z demonstrates that the image domain is bounded by the Bernoullis lemniscate right-half plan specified by w2z−1<1(iii)
by selectingΛz=1+sinz, the class S∗Λz leads to the family Ssin∗ which was explored in [20] while Se∗≡S∗ez has been produced in the article [21] and later studied in [22](iv)
the familySR∗≡S∗Λz with Λz=1+z/JJ+z/J−z,J=2+1 is studied in [23] while Scos∗≔S∗cosz and Scosh∗≔S∗coshz were recently examined by Raza and Bano [24], and Abdullah et.al [25], respectivelyNow, let us take the nonvanishing analytic functionsh1z and h2z in Ud with h10=h20=1. Then, the families defined in this article consist of functions fz∈A whose ratios fz/zqz and qz are subordinated to h1z and h2z, respectively, for some analytic function qz with q0=1 as
(6)fzzqz≺h1z,qz≺h2z.We are now going to choose some particular functions instead ofh1z and h2z. These choices are
(7)h1z=1+sinzorh1z=coszorh1z=1+z,h2z=1+sinz, and by applying the above-mentioned concepts, we now consider the following classes:
(8)Fcos=fz∈A:fzzqz≺cosz&qz≺h2z,z∈Ud,(9)FSL=fz∈A:fzzqz≺1+z&qz≺h2z,z∈Ud,(10)Fsin=fz∈A:fzzqz≺1+sinz&qz≺h2z,z∈Ud.In the present article, we discuss majorization problems for each of the above-defined classesFcos,FSL, and Fsin.


## 2. Main Results

To prove majorization results for the classesFcos,FSL, and Fsin, we need the following lemma.Lemma 1.
Letqz≺1+sinz and for z≤r. Then, qz satisfies the following inequalities:
(11)1−sinrcoshr≤qz≤1+sinrcoshr,(12)zq′zqz≤rcoshr1−r21−sinhr.Proof.
Ifqz≺1+sinz, then
(13)qz=1+sinwz,for some Schwartz function wz. Now, after some easy calculations, we have
(14)zq′zqz=zw′zcoswz1+sinwz.
Letwz=Reiθ with z=r≤R,−π≤θ≤π.A calculation shows that
(15)RcosReiθ=cosRxcoshRy,where
(16)x=cosθ,y=sinθ,forx,y∈−1,1.
Now, we can write(17)cosRx≥cosR≥cosr,1≤coshRy≤coshR≤coshr.
Thus, we have(18)Rcoswz≥cosr.
Now, consider(19)sinReiθ2=cosRcosθ2sinhRsinθ2+sinRcosθ2coshRsinθ2=Ψθ.
A calculation shows that the numbers,0,±π,±π/2, are the roots of equation (19) in −π,π. Since Ψθ is an even function, it is sufficient to consider θ∈0,π. We observe that Ψπ/2=sinh2R and Ψ0=sin2R. Now, we can write
(20)maxΨ0,Ψπ2,Ψπ=sinh2R.
Hence,(21)sinReiθ≤sinhR≤sinhr.
Similarly, one can easily show that(22)cosr≤coswz≤coshr.
Now, from well-known inequality for Schwartz functionwz, we obtain
(23)w′z≤1−wz21−z2=1−R21−z2≤11−r2.Now, by applying (21), (22), and (23) in (14), we get (12).Theorem 2.
Let the functionsfz∈A,gz∈Fcos and also suppose that fz≪gz in Ud. Then, for z≤r1,
(24)f′z≦g′z,where r1 is the smallest positive root of the equation
(25)1−r2−r1+ρcosr−rsinhr1−sinhr−rcosrcoshr=0.Proof.
Ifgz∈Fcos, then by the subordination relationship, we have
(26)gzzqz=coswz.
Now, after simple calculations, we have(27)zg′zgz=1+zq′zqz−zw′zsinwzcoswz.
Now, by using (21), (22), and (23) along with Lemma 1, we obtain
(28)gzg′z=z1+zq′z/qz−zw′zsinwz/coswz≦z1−zq′z/qz−zw′zsinwz/coswz≦r1−r21−sinhrcosr1−r21−sinhrcosr−rcosrcoshr−rsinhr1−sinhr.
From (4), we can write
(29)fz=φzgz.
Differentiating the above equality on both sides, we get(30)f′z=φ′zgz+φzg′z=g′zφz+φ′zzg′zgz.
Also, the Schwartz functionφz fulfils the below inequality:
(31)φ′z≦1−φz21−z2=1−φz21−r2z∈Ud.
Now, applying (28) and (31) in (30), we have
(32)f′z≦φz+r1−φz21−sinhrcosr1−r21−sinhrcosr−rcosrcoshr−rsinhr1−sinhrg′z,which by putting
(33)φ′z=ρ0≦ρ≦1becomes the inequality
(34)f′z≦Φ1r,ρg′z,where
(35)Φ1r,ρ=φz+r1−φz21−sinhrcosr1−r2cosr−rsinhr1−sinhr−rcosrcoshr.
To determiner1, it is sufficient to choose
(36)r1=maxr∈0,1:Φ1r,ρ≦1,∀ρ∈0,1,or, equivalently,
(37)r1=maxr∈0,1:Ψ1r,ρ≥0,∀ρ∈0,1,where
(38)Ψ1r,ρ=1−r2−r1+ρcosr−rsinhr1−sinhr−cosrcoshr.
Clearly, whenρ=1, the function Ψ1r,ρ assumes its minimum value, namely,
(39)minΨ1r,ρ,ρ∈0,1=Ψ1r,1=Ψ1r,where
(40)Ψ1r=1−r2−2rcosr−rsinhr1−sinhr−rcosrcoshr.
Next, we have the following inequalities:(41)Ψ10=1>0,Ψ11=−0.43851<0.There exists r1 such that Ψ1r≥0 for all r∈0,r1, where r1 is the smallest positive root of equation (25). Thus, the proof is completed.Theorem 3.
Letfz∈A,gz∈Fsin and also suppose that fz≪gz in Ud. Then, for z≤r2,
(42)f′z≦g′z,where r2 is the root smallestpositive of the equation:
(43)1−r21−sinhr−2rcoshr+1−sinhr=0.Proof.
Ifgz∈Fsin, then by using (10) along with the subordination relationship, a holomorphic function wz in Ud occurs with w0=0 and wz≤z in such a way that
(44)gzzqz=1+sinwzhold. Now, after simple calculations, we have
(45)zg′zgz=1+zq′zqz+zw′zcoswz1+sinwz.
Using (21), (22), and (23) along with Lemma 1, we obtain
(46)gzg′z=z1+zp′z/pz+zw′zcoswz/1+sinwz≦r1−r21−sinhr1−r21−sinhr−2rcoshr.
Also, with the use of (31) and (46) in (30), we easily get
(47)f′z≦φz+r1−φz21−sinhr1−r21−sinhr−2rcoshrg′z.Now, by the similar lines of Theorem2 along with the virtue of (33), we easily obtain the required result.Theorem 4.
Letfz∈A,gz∈FSL and also suppose that fz is majorized by gz in Ud. Then, for z≤r4,
(48)f′z≦g′z,where r4 is the positive smallest root of the equation
(49)1−2r2−5r1−sinhr−2rcoshr=0.Proof.
Letgz∈FSL. Then, a holomorphic function wz in Ud occurs with w0=0 and wz≤z so that
(50)gzzqz=1+wz.
Now, after simple calculations, we have(51)zg′zgz=1+zq′zqz+zw′z21+wz.
Using (23), we obtain
(52)zw′z21−wz≤z1+wz21−z2≤z1+z21−z2=z21−z≤r21−r.
By virtue of (23) and Lemma 1, we obtain
(53)gzg′z≦z1−zp′z/pz−zw′z/21+wz≦2r1−r21−sinhr21−r21−sinhr−2rcoshr−r1+r1−sinhr.
Now, using (31) and (53) in (30), we get
(54)f′z≦φz+2r1−φz21−sinhr21−r21−sinhr−2rcoshr−r1+r1−sinhrg′z.The required result follows directly using similar calculations as Theorem2 along with the use of (33).


## 3. Conclusion

For some particular subfamilies of holomorphic functions which are connected with different shapes of image domains, we studied the problems of majorization. These problems can be examined for some other families such as for the families of meromorphic functions as well as for the families of harmonic functions.



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*Source: 5548785-2021-04-08.xml*