# Mittag-Leffler Operator Connected with Certain Subclasses of Bazilevic̆ Functions

Authors: Om Ahuja; Asena Çetinkaya; Naveen Kumar Jain
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, we introduce a new generalized class of analytic functions involving the Mittag-Leffler operator and Bazilevic̆ functions. We examine inclusion properties, radius problems, and an application of the generalized Bernardi–Libera–Livingston integral operator for this function class.


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## Body

## 1. Introduction

LetA be the family of all functions of the form(1)fz=z+∑n=2∞anzn,which are analytic in the open unit disc D≔z:|z|<1. Denote by S the subfamily of A consisting of functions that are univalent in D. Let K, S∗, and C be the well-known subclasses of S consisting of functions that are, respectively, convex, starlike (with respect to the origin), and close-to-convex in D. The class P consists of analytic functions that are analytic in D and satisfies the conditions p0=1 and Repz>0 in D and is also well-known in the theory of univalent functions. For definitions, properties, and history of these classes, one may refer to a survey article by the first author [1, 2]. Recently, Ali et al. [3] and Anand et al. [4] studied these classes to find various radius problems.The Mittag-Leffler functionEα, defined by(2)Eαz=∑n=0∞znΓαn+1,α∈ℂ;Reα>0;z∈ℂ,was introduced in 1903 by Mittag-Leffler [5, 6] in connection with his method of summation of some divergent series. A general form of this special function (2) given by(3)Eα,βz=∑n=0∞znΓαn+β,α,β∈ℂ;Reα>0;Reβ>0;z∈ℂ,which was studied by Wiman [7] in 1905. During these last twenty-five years, interest in Mittag-Leffler type functions (2) and (3) has significantly increased among engineers and scientists due to their applications in numerous applied problems, such as fluid flow, diffusive transport skin to diffusion, electric networks, probability, and statistical distribution theory. For detailed account of various properties and references related to applications, one may refer to [8, 9]. Motivated by Sivastava and Tomovski [10], Attiya [11] studied certain applications of the generalized Mittag-Leffler operator involving differential subordination.Corresponding to the functionEα,β, Elhaddad et al. [12] introduced the Mittag-Leffler linear operator ℰλ,α,βmf:A⟶A given by(4)ℰλ,α,βmfz=z+∑n=2∞Γβ1+n−1λmΓαn−1+βanzn,where m∈ℕ0=ℕ∪​0, λ≥0, α,β∈ℂ, Reα>0, Reβ>0 and when Reα=0 and β≠0. From (4), the following recurrence formula can be easily obtained:(5)ℰλ,α,βm+1fz=1−λℰλ,α,βmfz+λzℰλ,α,βmfz',where m∈ℕ0 and λ≥0. For suitable values of the parameters m,α,β, and λ, we may get several linear operators; for example,(1)
Forα=0 and β=1, we get Al-Oboudi operator [13].(2)
Forα=0, β=1, and λ=1, we get Sălăgean operator [14].(3)
Form=0 and λ=1, we get the operator(6)Eα,βz=zΓβEα,βz=z+∑n=2∞ΓβΓαn−1+βzn.Let us denote byℬϑ,τ,g,p or briefly denote by ℬ, a class of functions f∈A for which p∈P, g∈S∗, and real numbers ϑ,τ with ϑ>0 such that(7)fz=ϑ+iτ∫​0zptgtϑtiτ−1dt1/ϑ+iτ,where powers are taken as principal values. Bazilevich [15] proved that ℬ⊂S. In fact, it is known that K⊂S∗⊂C⊂ℬ⊂S. For ϑ>0 and ρ<1, Ponnusamy and Karunakaran [16] showed that(8)ℬ1ϑ,ρ=f∈A:Rezf'zf1−ϑzgϑz>ρ,z∈D.Forgz≡z, these authors observed that(9)ℬ2ϑ,ρ=f∈A:Rezf'zfzfzzϑ>ρ,z∈D,ℬ3ρ≔ℬ20,ρ=f∈A:Rezf'zfz>ρ,z∈D=S∗ρ,ℬ4ρ≔ℬ21,ρ=f∈A:Ref'z>ρ,z∈D=P'ρ.In view of (7), Singh [17] observed that ℬ1ϑ,0, ℬ2ϑ,0, ℬ30=S∗, and ℬ40=P' are subclasses of ℬ. For further details, one may refer to [15, 18].In 1976, Padmanabhan and Parvatham [19] introduced the class Pkρ of analytic functions p defined in D satisfying the properties p0=1 and(10)∫​02πRepz−ρ1−ρdθ≤kπ,where z=reiθ, k≥2, and 0≤ρ<1. . In fact, he proved the following important result.Lemma 1 (see [19]).
Ifp∈Pkρ, then(11)pz=12∫​02π1+1−2ρze−iθ1−ze−iθdμθ,where μθ is a function with bounded variation on 0,2π such that(12)∫​02πdμθ=2π,∫​02π|dμθ|≤kπ.From (10), it is observed that p∈Pkρ if and only if there exists p1,p2∈Pρ such that (see [18])(13)pz=k4+12p1z−k4−12p2z,z∈D.We note that, forρ=0, we obtain the class Pk0≔Pk defined by Pinchuk [20]. For k=2, we get P2ρ≔Pρ the class of analytic functions with positive real part greater than ρ, and for k=2 and ρ=0, we have the class P20≔P of functions with positive real part.Motivated by many researchers in [5–7, 10–12, 15, 17, 18, 21], we introduce the following generalized class of Mittag-Leffler–Bazilevic̆ operator involving the class Pkρ.Definition 1.
Letk≥2, 0≤ρ<1, ϑ>0, m∈ℕ0, λ≥0, and γ∈ℂ be such that Reγ>0. Then, a function f∈A given by (1) is in the class ℳλ,α,βm,γk,ϑ,ρ if it satisfies the condition(14)1−γℰλ,α,βmfzzϑ+γℰλ,α,βm+1fzzℰλ,α,βmfzzϑ−1∈Pkρ,where z∈D.
This new classℳλ,α,βm,γk,ϑ,ρ involves several subclasses of the family ℬ of Bazilevic̆ functions defined by (7); see [15]. For example, ℳ1,0,10,12,ϑ,0=ℬ2ϑ,0, ℳ1,0,10,12,0,0=ℬ30, and ℳ1,0,10,12,1,0=ℬ40 are subclasses of ℬ; see [15, 17]. In fact, for different values of m,ϑ,λ,α,β,γ,k and ρ, the class ℳλ,α,βm,γk,ϑ,ρ reduces to many important subclasses studied by various researchers. For instance,(i)
Settingm=0, λ=1, α=0, and β=1, we get (see [18])(15)ℳ1,0,10,γk,ϑ,ρ≕ℳγk,ϑ,ρ=f∈A:1−γfzzϑ+γzf'zfzfzzϑ∈Pkρ,0≤ρ<1.(ii)
Settingm=0, λ=1, α=0, γ=1, β=1, and k=2, we get (see [15])(16)ℳ1,0,10,12,ϑ,ρ≕ℬ2ϑ,ρ=f∈A:zf'zfzfzzϑ∈Pρ,0≤ρ<1.(iii)
Settingm=0, λ=1, α=0, β=1, γ=0, ϑ=1, and k=2, we get (see [21])(17)ℳ1,0,10,02,1,ρ≕ℳρ=f∈A:fzz∈Pρ,0≤ρ<1.
In view of the above examples, we conclude that the notion of generalized classℳλ,α,βm,γk,ϑ,ρ unifies several known subclasses of A.
In this paper, we study various properties of the classℳλ,α,βm,γk,ϑ,ρ. In particular, we investigate inclusion properties, radius problem, and an application of the generalized Bernardi–Libera–Livingston integral operator for this function class.
We list some preliminary lemmas required for proving our main results.Lemma 2 (see [22]).
Letu=u1+iu2 and v=v1+iv2, and suppose Ψu,v is a complex function satisfying the following conditions:(i)
Ψu,v is continuous in a domain D⊂ℂ2(ii)
1,0∈D and ReΨ1,0>0(iii)
ReΨiu2,v1≤0, whenever iu2,v1∈D and v1≤−1+u22/2Ifpz=1+c1z+c2z2+… is an analytic function in D such that pz,zp'z∈D and ReΨpz,zp'z>0 for z∈D, then Repz>0 in D.Lemma 3 (see [23]).
Ifp is in P, and if ζ is a complex number satisfying Reζ≥0,ζ≠0, then Repz+ζzp'z>ρ0≤ρ<1 implies that(18)Repz>ρ+1−ρ2ι1−1,where(19)ι1=∫​011+tReζ−1dt,ι1 is an increasing function of Reζ and 1/2≤ι1<1. This estimate cannot be improved in general.


## 2. Inclusion Properties

In this section, we examine some inclusion properties for the classℳλ,α,βm,γk,ϑ,ρ.Theorem 1.
Letγ>0 and f∈ℳλ,α,βm,γk,ϑ,ρ. Then,(20)ℰλ,α,βmfzzϑ∈Pkρ1,where ρ1 is given by(21)ρ1=2ϑρ+λγ2ϑ+λγ.Proof.
In view of (20), let(22)ℰλ,α,βmfzzϑ=1−ρ1pz+ρ1,where p0=1 and(23)pz=k4+12p1z−k4−12p2z.
Thus, by using (14) and (22), we obtain(24)1−γℰλ,α,βmfzzϑ+γℰλ,α,βm+1fzzℰλ,α,βmfzzϑ−1=1−γ1−ρ1pz+ρ1+γℰλ,α,βm+1fzℰλ,α,βmfz1−ρ1pz+ρ1.
Taking logarithmic differentiation of (22), we get(25)ϑzℰλ,α,βmfz'ℰλ,α,βmfz−1=1−ρ1zp'z1−ρ1pz+ρ1.
By using the identity (5) in the last expression, we obtain(26)ℰλ,α,βm+1fzℰλ,α,βmfz=λ1−ρ1zp'zϑ1−ρ1pz+ρ1+1.
Substituting (26) into (24) and using (13), we arrive at(27)1−ρ1pz+ρ1+λγ1−ρ1zp'zϑ=k4+121−ρ1p1z+ρ1+λγ1−ρ1zp1'zϑ−k4−121−ρ1p2z+ρ1+λγ1−ρ1zp2'zϑ.
Sincef∈ℳλ,α,βm,γk,ϑ,ρ, it follows that(28)1−ρ1piz+ρ1+λγ1−ρ1zpi'zϑ∈Pρ,0≤ρ<1,i=1,2.
That is,(29)11−ρ1−ρ1piz+ρ1+λγ1−ρ1zpi'zϑ−ρ∈P.
To prove the theorem, we will show thatpi∈Pi=1,2. We form the functional Ψu,v by taking u=u1+iu2 and v=v1+iv2 such that(30)Ψu,v=1−ρ1u+ρ1−ρ+λγ1−ρ1vϑ.
Using (29), it is easy to show that the first two conditions of Lemma 2 are satisfied. To verify condition (iii), we obtain(31)ReΨiu2,v1=ρ1−ρ+Reλγ1−ρ1v1ϑ≤ρ1−ρ−λγ1−ρ11+u222ϑ=2ϑρ1−2ϑρ−λγ1−ρ11+u222ϑ=2ϑρ1−ρ−λγ1−ρ1−λγ1−ρ1u222ϑ≔A+Bu222C,where v1≤−1+u22/2. Now, ReΨiu2,v1≤0 if(32)A=2ϑρ1−ρ−λγ1−ρ1≤0,B=−λγ1−ρ1≤0,C=ϑ>0.
It follows fromA≤0, ρ1 given by (21) and B≤0 that 0≤ρ1<1. In view of Lemma 2, for pz=piz, pi∈Pi=1,2, we get p∈Pkρ1. This proves the result.
Form=0, λ=1, α=0, and β=1, Theorem 1 reduces to the following new result.Corollary 1.
Letγ>0 and f∈ℳγk,ϑ,ρ. Then, fz/zϑ∈Pkρ1, where ρ1 is given by(33)ρ1=2ϑρ+γ2ϑ+γ.By using the inclusion relation given in Theorem1, we prove the following result.Theorem 2.
Letϑ>0 and 0≤γ1<γ2. Then, ℳλ,α,βm,γ2k,ϑ,ρ⊂ℳλ,α,βm,γ1k,ϑ,ρ.Proof.
Letf∈ℳλ,α,βm,γ2k,ϑ,ρ. Then, we have(34)H2z=1−γ2ℰλ,α,βmfzzϑ+γ2ℰλ,α,βm+1fzzℰλ,α,βmfzzϑ−1∈Pkρ.
In view of Theorem1, we conclude that(35)ℰλ,α,βmfzzϑ≔H1z∈Pkρ1⊂Pkρ.
Thus, forγ1≥0, we have(36)1−γ1ℰλ,α,βmfzzϑ+γ1ℰλ,α,βm+1fzzℰλ,α,βmfzzϑ−1=1−γ1γ2H1z+γ1γ2H2z.
Because the classPkρ is a convex set (see [18]), it follows that the right side of (36) belongs to Pkρ, and therefore f∈ℳλ,α,βm,γ1k,ϑ,ρ.
Form=0, λ=1, α=0, and β=1, Theorem 2 reduces to the following inclusion result.Remark 1 (see [18]).
Letϑ>0 and 0≤γ1<γ2. Then, ℳγ2k,ϑ,ρ⊂ℳγ1k,ϑ,ρ.


## 3. Radius Problem

In this section, we examine certain radius problems.Theorem 3.
If a functionf∈A satisfies(37)ℰλ,α,βmfzzϑ∈Pkρ,then f∈ℳλ,α,βm,γk,ϑ,ρ for |z|<r1, where(38)r1=λγ+ϑ−λ2γ2+2λγϑϑ.Proof.
In view of (37), we have(39)ℰλ,α,βmfzzϑ=1−ρpz+ρ,where p∈Pk. Hence, by using (5) and (39), we easily get(40)11−ρ1−γℰλ,α,βmfzzϑ+γℰλ,α,βm+1fzzℰλ,α,βmfzzϑ−1−ρ=pz+λγzp'zϑ=k4+12p1z+λγzp1'zϑ−k4−12p2z+λγzp2'zϑ,where p1,p2∈P and z∈D.
Now, by using well-known estimates (see [2]) for the class P given by(41)|zpi'z|≤2rRepiz1−r2,Repiz≥1−r1+r,|z|<r<1;i=1,2;z∈D,we have(42)Repiz+λγzpi'zϑ≥Repiz−λγ|zpi'z|ϑ≥Repiz1−2λγrϑ1−r2=Repizϑ1−r2−2λγrϑ1−r2.
The right hand side of the last inequality is positive if|z|=r<r1, where r1 is given by (38).
Lettingm=0, λ=1, α=0, and β=1, Theorem 3 reduces to the following new result.Corollary 2.
If a functionf∈A satisfies fz/zϑ∈Pkρ, then f∈ℳγk,ϑ,ρ for |z|<r2, where(43)r2=γ+ϑ−γ2+2γϑϑ.Remark 2.
Ifm=α=0, γ=β=λ=ϑ=1, and k=2 in Theorem 3, then f is in ℳ1,0,10,12,1,ρ≔ℬ211,1,1,ρ,1,0 for z<2−3≈0.2679. This result was proved in Theorem 3.4 in [24].


## 4. Application of an Integral Operator

In this section, we consider an application of the generalized Mittag-Leffler operator given by (4) involving the generalized Bernardi–Libera–Livingston integral operator ℒσ:A⟶A given by(44)ℒσfz=σ+1zσ∫​0ztσ−1ftdt,σ>−1.From this operator, we easily get(45)zℰλ,α,βmℒσfz'=σ+1ℰλ,α,βmfz−σℰλ,α,βmℒσfz.For several special cases of this operator and related operators, one may refer to a survey article by the first two authors [25] and related references therein.Theorem 4.
Letf∈A and ℒσf be given by (44). If(46)1−γℰλ,α,βmℒσfzz+γℰλ,α,βmfzz∈Pkρ,then(47)ℰλ,α,βmℒσfzz∈Pkι,z∈D,where ι is given by(48)ι=ρ+1−ρ2ι1−1,ι1=∫​011+tγσ+1−1dt.Proof.
Consider the function(49)ℰλ,α,βmℒσfzz=pz=k4+12p1z−k4−12p2z.
Differentiating both sides and using (45), we get(50)ℰλ,α,βmfzz=pz+zp'zσ+1.
If we use the identity given by (46), we obtain(51)1−γℰλ,α,βmℒσfzz+γℰλ,α,βmfzz=pz+γzp'zσ+1∈Pkρ.
This implies that(52)Repiz+γzpi'zσ+1>ρ,i=1,2.
By using Lemma3, we see that Repiz>ι, where ι is given by (48). Thus, we arrive at p∈Pkι. This completes the proof.
Settingm=0, λ=1, α=0, and β=1 in operator ℰλ,α,βmf, Theorem 4 gives the following result.Corollary 3.
Letf∈A and ℒσf be given by (44). If(53)1−γℒσfzz+γfzz∈Pkρ,then ℒσfz/z∈Pkι, where ι is given by(54)ι=ρ+1−ρ2ι1−1,ι1=∫​011+tγσ+1−1dt.


## 5. Conclusion

We conclude our investigation by remarking that the defined new generalized class of analytic functions involving the Mittag-Leffler operator and Bazilevic̆ functions gives various well-known subclasses of Bazilevic̆ functions as particular cases which in turn yields many known results as corollaries.



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