# Majorization Properties for Certain Subclasses of Meromorphic Function of Complex Order

Authors: Neelam Khan; Muhammad Arif; Maslina Darus
Journal: Complexity
 (2022)

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## Abstract

By making use ofq−differential operators, many distinct subclasses of analytic and meromorphic functions have already been defined and investigated from numerous perspectives. In this article, we investigated several majorization results for the class of meromorphic univalent functions of complex order, defined by q−differential operator. Moreover, we pointed out some new or known consequences of our results, which is in the form of corollaries.


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## Body

## 1. Introduction and Definitions

Letℳ represent the class of meromorphic functions f in the form of(1)fz=1z+∑n=0∞anzn,which are analytic in the punctured disc U∘=z:0<z<1=U\0, where U=U∘∪0. For the two functions fz and gz belonging to the class ℳ, there exists a Schwartz function w, which is analytic in U with wz≤z and w0=0, such that fz=gwz, and the function f is subordinate to g, written as f≺g. The following relationship holds if g is univalent:(2)f≺g⟺f0=g0,andfU∘⊆gU∘.Because of its use in a variety of mathematical sciences, the study ofq−calculus (quantum calculus) has fascinated and motivated many scholars. One of the primary contributors among all the mathematicians who introduced the concept of q−calculus theory was Jackson [1, 2]. The formulation of this concept is widely used to investigate the nature of different structures of function theory, such as q−calculus was used in other branches of mathematics. Though the authors of the first article [3] discussed the geometric nature of q−starlike functions, Srivastava [4] laid a solid foundation for the use of q−calculus in the context of function theory. Also, in [5], Srivastava provided a brief overview of basic or q−calculus operators and fractional q−calculus operators, as well as their applications in the geometric function theory of complex analysis. Later, the authors [6–8] investigated a number of useful properties for the newly defined q−linear differential operator, and Mehmood and Sokół [9] discussed the Ruscheweyh q−differential operator, while Srivastava et al. [10] introduced a generalized operator for meromorphic harmonic functions by using the idea of convolution.Let0<q<1. For any nonnegative integer n, the q-integer number n is defined by(3)nq=1−qn1−q=1+q+q2+⋯+qn+1,0q=0.In general, we will denote(4)δq=1−qδ1−q,for a noninteger number δ. Also, the q-number shifted factorial is defined by(5)nq!=nqn−1q⋯2q1q,0q!=1.Clearly,(6)limq⟶1−nq=n,limq⟶1−nq=n!.Leta,q∈ℂq<1 and n∈ℕ0=ℕ∪0. Then, the q− shifted factorial a;qn is defined by(7)a;q0=1,a;qn=∏j=1n1−aqj−1,n∈ℕ.Letx∈ℂ−−n:n∈ℕ0. Then, q−gamma function is as follows:(8)Γqx=q;q∞qx;q∞1−q1−x,0<q<1.In a subset ofℂ, the q−derivative (or q−difference) operator Dqf of function f is defined by(9)Dqfz=fz−fqzz1−q,z≠0,f′0,z=0,provided that f′0 exists. We can easily observe from the definition of (9) that Dqfzlimq⟶1−=f′z.Suppose thatq∈0,1, then q− analog derivative of f as(10)Dqfz=fz−fqzz1−q,z∈U,or(11)Dqfz=−1qz2+∑n=1∞nqanzn.In 1967, Mac Gregor [11] introduced the notion of majorization as follows.Definition 1.
Let complex-valued functionsf and g be analytic in U. We say that f is majorized by g in U and write(12)fz≪gzz∈U,if there exists a function φzcomplex−valued function inU, satisfying(13)φz≤1andfz=φzgzz∈U.Majorization (12) is closely related to the concept of quasi-subordination between analytic functions in U. Several researchers have published articles on this topic; for example, Tang et al. [12] gave the concept of majorization for subclasses of starlike functions based on the sine and cosine functions, Arif et al. [13] discussed majorization for various new defined classes, Cho et al. [14] obtained coefficient estimates for majorization, and Tang and Deng [15] defined the majorization problem connected with Liu-Owa integral operator and exponential function. This concept is also defined for p−valent function by Altintas and Srivastava [16] and for complex order by Altintas et al. [17].The basic goal of this article is to examine and explain the idea of majorization in the context of the meromorphic function. Many researchers have shown their interest in this site. Goyal and Goswami [18, 19] studied this concept for majorization for meromorphic function with the integral operator, Tang et al. [12] discussed it for meromorphic sin and cosine functions, Bulut et al., Tang et al., and Janani [20–22] explained this concept for meromorphic multivalent functions, Rasheed et al. [23] investigated a majorization problem for the class of meromorphic spiral-like functions related with a convolution operator, and Panigrahi and El-Ashwah [24] defined majorization for subclasses of multivalent meromorphic functions through iterations and combinations of the Liu–Srivastava operator and Cho–Kwon–Srivastava operator and much more. In addition, there are several other articles on this topic [18].Here is the definition of our main function.Definition 2.
A functionfz∈ℳ is said to be in the class ℳSqγ of meromorphic functions of complex order γ≠0 in U∘, if(14)1−1γzqDqfzfz+1≺Ψz.
Now, we are going to choose some particular functions instead ofΨz. These choices are(15)Ψz=1+sinz,orΨz=cosz,orΨz=1+z,orΨz=1+z1−z,and by applying the above-mentioned concepts, we now consider the following cases:(16)ℳSsinqγ=fz∈ℳℋ:1−1γzDqfzfz+1≺1+sinz,ℳScosqγ=fz∈ℳℋ:1−1γzDqfzfz+1≺cosz,ℳSSLqγ=fz∈ℳℋ:1−1γzDqfzfz+1≺1+z,ℳSSqγ=fz∈ℳℋ:1−1γzDqfzfz+1≺1+z1−z.
In the present article, we discussed majorization problems for each of the above-defined classes ofℳSqγ.


## 2. Majorization Problem for the ClassesℳSqγ

We state the followingq−analogue of the result given by Nehari [25] and Salvakumaran et al. [26].Lemma 1 (see [27]).
If the functionφz is analytic and φz<1 in U∘, then(17)Dqφz≤1−φz21−z2.Theorem 1.
Let the functionfz∈ℳ and suppose g∈ℳSsinqγ if fz is majorized by gz in U∘, i.e.,(18)fz≪gz.Then, forz≦r1,(19)qzDqfz≤qzDqgz,where r1 is the smallest positive root of the following equation:(20)1−r21−γsinhr−2qr=0.Proof.
Sinceg∈ℳSsinqγ, by using (19), we can find if(21)1−1γzqDqgzgz+1≺Ψz,z∈U∘ and(22)Ψz=1+sinz.
By Lemma1, there exists a bounded analytic function w in U and(23)1−1γzqDqgzgz+1=1+sinwz,with w∞=∞. From (24), we obtain(24)zqDqgzgz=−1+γsinwz.
Letwz= Re iθ with R≦z=r<1 and −π≦t≦π. By simple calculation, we show that(25)sinwz2=sinReit2=cos2Rcostsinh2Rsint+sin2Rcostcosh2Rsint≕δt.
We easily see that the equation,(26)δ′t=sinh2Rsint−sin2Rcost=0,has five roots in −π,π, that is, 0,±π/2 and ±π. Because δ−t=δt, we just need to consider t∈0,π. Also, noticing that δ0=δπ=sin2R, δπ/2=sinh2R and(27)maxδ0,δπ,δπ2=δπ2=sinh2R.
Thus, we have(28)sinwz=sinReit≦sinhR≦sinhr.
From (24) and (28), we find that(29)gzDqgz≤qz1+γsinwz≦qz1−γsinwz≦rq1−γsinhr.
Sincefz is majorized by gz in U∘, from (13), we have(30)fz=φzgz.
By applyingq−derivative on the previous equation w.r.t z as in [27] and then multiplying by qz, we have(31)qzDqfz=qzDqφzgz+qzφzDqgz=qzDqgzφz+DqφzgzDqgz.
Noting thatφz is the Schwartz function, so ℜφz>0 in U∘, φz≠0 for all z∈U∘, satisfies the q−analogue result given by [25] proved in Lemma 1.(32)Dqφz≤1−φz21−z2.
Now, using (29) and (32) in (31), we have(33)qzDqfz≦qzDqgzφz+1−φz21−z2.rq1−γsinhr.By setting(34)z=r<1andφz=ζ,0≤ζ≤1,we get the inequality(35)qzDqfz≤Yr,ζqzDqgz.
We define(36)Υr,ζ=ζ+rq1−ζ21−r21−γsinhr0≦ζ≦1,0<r<1.
To determiner1, it is sufficient to choose(37)r1=maxr∈0,1:Yr,ζ≦1,∀ζ∈0,1,equivalently,(38)r1=maxr∈0,1:Y∗r,ζ≧0,∀ζ∈0,1,where(39)Υ∗r,ζ=1−r21−γsinhr−rq1+ζ.
Clearly, whenζ=1, the above function Y∗r,ζ assumes its minimum value, namely,(40)minY∗r,ζ:ζ∈0,1=Y∗r,1≔ψ∗r,where(41)ψ∗r=1−r21−γsinhr−2qr.
Next, we obtain the following inequalities:(42)ψ∗0=1>0andψ∗1=−2q<0.There exists r1 such that ψ∗r≧0 for all r∈0,r1, where r1 is the smallest positive root of (20). The proof of Theorem 1 is completed.Theorem 2.
Let the functionfz∈ℳ and suppose g∈ℳScosqγ if fz is majorized by gz in U∘, i.e.,(43)fz≪gz.Then, forz≦r2,(44)qzDqfz≤qzDqgz,where r2 is the smallest positive root of the following equation:(45)1−r2γ1+coshr+1−2qr=0.Proof.
Sinceg∈ℳScosqγ, from (11) and the subordination relationship, we see that(46a)zqDqgzgz=γ1−coswz−1,where wz is as same as in (24). Similar to (28), we can verify that(47)coswz=cosReiθ≦coshR≦coshr,where wz= Re iθ with R≦z=r<1 and −π≦t≦π.
Combining (46a) and (47), it is easy to see that(48)gzqzDqgz≦rqγ1−coshr+1.
By virtue of (32) as well as (48) in (31), we immediately obtain(49)qzDqfz≦qzDqgzφz+1−φz21−z2.rqγ1−coshr+1.
In succession, according to (34) and just as the proof of Theorem 1, we can deduce the required result (45). Hence, we have completed the proof of Theorem 2.Theorem 3.
Let the functionfz∈ℳ and suppose g∈ℳSSLqγ if fz is majorized by gz in U∘, i.e.,(50)fz≪gz.Then, forz≦r3,(51)qzDqfz≤qzDqgz,where r3 is the smallest positive root of the following equation:(52)1−r2γ1−1−r−1−2qr=0.Proof.
Letgz∈ℳSSLqγ. Then, from definition (16) in terms of the Schwartz function, we have(53)1−1γzqDqgzgz+1=1+wz,which implies(54)1−1γzqDqgzgz+12=1+wz,1−1γzqDqgzgz+12=1+wz≦1−wz.
Now, aswz= Re iθ with wz≦R≦z=r<1 and −π≦t≦π we have(55)1−1γzqDqgzgz+1≦1−r,which implies(56)zqDqgzgz≦γ1−1−r−1.
Now, as in Theorem2, we use (32), as well as (56) in (31), and we obtain(57)qzDqfz≦qzDqgzφz+1−φz21−z2.rqγ1−1−r−1.
Let us takez=r<1 and φz=ζ, 0≦ζ≦1; we obtain(58)qzDqfz≤Yr,ζqzDqgz.
We define(59)Υr,ζ=ζ+rq1−ζ21−r2γ1−1−r−10≦ζ≦1,0<r<1.
To determiner3, it is sufficient to choose(60)r3=maxr∈0,1:Yr,ζ≦1,∀ζ∈0,1,equivalently,(61)r3=maxr∈0,1:Y∗r,ζ≧0,∀ζ∈0,1,where(62)Y∗r,ζ=1−r2γ1−1−r−1−rq1+ζ.
This clearly shows the result that, whenζ=1, the above function Y∗r,ζ assumes its minimum value, namely,(63)minΥ∗r,ζ:ζ∈0,1=Υ∗r,1≔ψ∗r,where(64)ψ∗r=1−r2γ1−1−r−1−2qr.
Next, we obtain the following inequalities:(65)ψ∗0=−1<0,ψ∗1=−2q<0,there exists r3 such that ψ∗r≧0 for all r∈0,r3, where r3 is the smallest positive root of (52). The proof of Theorem 3 is completed.Theorem 4.
Let the functionfz∈ℳ and suppose g∈ℳSSqγ if fz is majorized by gz in U∘, i.e.,(66)fz≪gz.Then, forz≦r4,(67)qzDqfz≤qzDqgz,where r4 is the smallest positive root of the following equation:(68)1+r1−2γr−1−2qr=0.Proof.
Sincegz∈ℳSSqγ, we readily obtained from definition (17) that(69)1−1γzqDqgzgz+1=Ψz,(70)Ψz=1+wz1−wz,where wz is the well-known class of bounded analytic functions in U such that(71)wz≦zz∈U.
From (69) and (70) and making use of (71), we obtain(72)zqDqgzgz≦1−2γz−11−z.
Now, just like the above theorems, we use (32) as well as (72) in (31), and we obtain(73a)qzDqfz≦qzDqgzφz+1−φz21−z2.rq1−z1−2γz−1.
Let us takez=r<1 and φz=ζ, 0≦ζ≦1; we obtain(74)qzDqfz≤Υr,ζqzDqgz.
We define(75)Yr,ζ=ζ+rq1−ζ21+r1−2γr−1,0≦ζ≦1,0<r<1.
To determiner3, it is sufficient to choose(76)r3=maxr∈0,1:Yr,ζ≦1,∀ζ∈0,1,equivalently,(77)r3=maxr∈0,1:Y∗r,ζ≧0,∀ζ∈0,1,where(78)Υ∗r,ζ=1+r1−2γr−1−rq1+ζ.
Clearly, whenζ=1, the above function Y∗r,ζ assumes its minimum value, namely,(79)minY∗r,ζ:ζ∈0,1=Y∗r,1≔ψ∗r,where(80)ψ∗r=1+r1−2γr−1−2qr.
Next, we obtained the following inequalities:(81)ψ∗0=−1<0andψ∗1=−2q+2γ<0,there exists r4 such that ψ∗r≧0 for all r∈0,r4, where r4 is the smallest positive root of (68). The proof of Theorem 4 is completed.


## 3. Conclusion

In this article, we investigated majorization and other results for such subclasses of meromorphic functions, such as the meromorphic univalent function of complex order associated with theq−differential operator. We also highlighted some special cases and new consequences of our main results. In order to conclude our current study, we attract the attention of interested readers to the potential of examining the fundamental or quantum (or q−) extensions of the results obtained in this work. Applications of the q th majorization in the real world will be an interesting and encouraging future study for researchers.



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