# Study on Certain Subclass of Analytic Functions Involving Mittag-Leffler Function

Authors: Nazek Alessa; B. Venkateswarlu; K. Loganathan; P. Thirupathi Reddy; A. Shashikala; Ngawang Namgyel
Journal: Journal of Function Spaces
 (2021)

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## Abstract

We propose and explore a new subclass of regular functions described by a new derivative operator in this paper. Some coefficient estimations, growth and distortion aspects, extreme points, star-like radii, convexity, Fekete-Szego inequality, and partial sums are derived.


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## Body

## 1. Introduction

LetA represent the regular function class u defined on the disk U=w:w<1 normalized by u (i.e., u0=0 and u′0=1). The origin of the form is about the Taylor series expansion of such an equation
(1)uw=w+∑η=2∞aηwη.S indicates a subclass of A consists entirely of mappings that are the same as U.Foru∈A presented by (2) and gw specified by
(2)gw=w+∑η=2∞bηwη,their convolution, represented by u∗g, is specified as
(3)u∗gw=w+∑n=2∞anbnwn=g∗uww∈U.TheA subclass consisting of the u-type function is specified by T as
(4)uw=w−∑η=2∞aηwη.Silverman [1] extensively examined this subclass.The study of operators plays an important role in geometric function theory in complex analysis and its related fields. Many derivative and integral operators can be written in terms of convolution of certain analytic functions. It is observed that this formalism brings an ease in further mathematical exploration and also helps to better understand the geometric properties of such operators. The Mittag-Leffler function [2, 3] is defined by the following power series, convergent in the whole complex plane:
(5)Eυw≔∑η=0∞wηΓυη+1,υ>0,w∈ℂ.We recognize that it is an entire function of order1/υ providing a simple generalization of the exponential function expw to which it reduces for υ=1. For detailed information on the Mittag-Leffler-type functions and their laplace transforms, the reader may consult, e.g., [4–6] and the recent treatise by Gorenflo et al. [7].We also note that for the convergence of the power series in (5), the parameter υ may be complex provided that Rυ>0. The most interesting properties of the Mittag-Leffler function are associated with its asymptotic expansions as w⟶∞ in various sectors of the complex plane. A more general function Eυ,τ generalizing Eυw was introduced by Wiman [8] and defined by
(6)Eυ,τw=∑η=0∞wηΓυη+τw,υ,τ∈ℂ,Rυ>0,Rτ>0.Observe that the functionEυ,τ contains many well-known functions as its special case, for example, E1,1w=ew,E1,2w=ew−1/w,E2,1w2=coshw,E2,1−w2=cosw,E2,2w2=sinhw/w,E2,2−w2=sinhw/w,E4=1/2cosw1/4+coshw1/4, and E3=1/2ew1/3+2e−1/3cos3/2w1/3.The Mittag-Leffler function arises naturally in the solution of fractional-order differential and integral equations and especially in the investigations of fractional generalization of kinetic equation, random walks, Levy flights, and super diffusive transport and in the study of complex systems. Several properties of Mittag-Leffler function and generalized Mittag-Leffler function can be found, e.g., in [9–16]. Observe that Mittag-Leffler function Eυ,τw does not belong to the family A. Thus, it is natural to consider the following normalization of Mittag-Leffler functions as below:
(7)Eυ,τ=wΓτEυ,τw=w+∑η=2∞ΓτΓυη−1+τwη.It holds for complex parametersυ,τ and w∈ℂ.The functionQυ,τw is specified by
(8)Qυ,τw=wΓτEυ,τw.Now, foru∈A, the derivative operator that follows is defined by Dℏmυ,τu:A⟶A by
(9)Dℏ0υ,τuw=uw∗Qυ,τw,Dℏ1υ,τuw=1−ℏuw∗Qυ,τw+ℏwuw∗Qυ,τw′,⋮Dℏmυ,τuw=Dℏ1Dℏm−1υ,τuw.Ifu is specified by (1), then from the operator’s definition Dℏmu, it is clear to see that
(10)Dℏmυ,τuw=w+∑η=2∞ϕηmℏ,υ,τaηwη,where
(11)ϕηmℏ,υ,τ=ΓτΓυη−1+τℏη−1+1m.Keep in mind that(1)
the Al-Oboudi operator [17] is achieved when υ=0 and τ=1(2)
we get the Salagean operator [18] when υ=0,τ=1, and ℏ=1(3)
whenm=0, we get Eυ,τw, according to Srivastava et al. [19]Ifu∈T is represented by (4), then we have got it.
(12)Dℏmυ,τuw=w−∑η=2∞ϕηmℏ,υ,τaηwη.Now, by utilizing the differential operator,Dℏmυ,τu, a new subclass of functions belonging to the class A is specified.Definition 1.
For0≤ν≤1,ℓ≥1,k≥0, and 0≤℘<1, a mapping in a class is referred to as u∈ASℏ,υ,τmν,ℓ,k,℘, if it satisfies the case
(13)Rℓwℵ′wℵw−ℓ−1>kℓwℵ′wℵw−ℓ+℘,where
(14)ℵw=1−νDℏmuw+νwDℏmuw′.We also defineTSℏ,υ,τmν,ℓ,k,℘=Sℏ,υ,τmν,ℓ,k,℘∩T.For special situations of characteristics,Sℏ,υ,τmν,ℓ,k,℘ and TSℏ,υ,τmν,ℓ,k,℘, it can be reduced to new or known categories of functions studied in recent research [20–25].The objective of this review is to look into a variety of properties for functions in the aforesaid class. For specific parameter instances.


## 2. Coefficient Estimates

To get our results, we will require the subsequent lemma.Lemma 2 (see [26]).
Let℘ be a real and z be a complex number. Then, Rz≥℘.
(15)⟺z+1−℘−z−1+℘≥0.
For beginnings, we have a coefficient that is relevant for functions in the classsSℏ,υ,τmν,ℓ,k,℘.Theorem 3.
Letu∈A indicated by (1). If
(16)∑η=2∞1−℘+ℓη−11+kχηℏ,ν,m,υ,τaη≤1−℘,where
(17)χηℏ,ν,m,υ,τ=1+νm−1ϕηmℏ,υ,τ,then u∈Sℏ,υ,τmν,ℓ,k,℘.Proof.
In the definition by consequence of1 and Lemma 2, it is enough to demonstrate that
(18)ℓwℵ′wℵw−ℓ−1−kℓwℵ′wℵw−ℓ−1+℘≤ℓwℵ′wℵw−ℓ−1−kℓwℵ′wℵw−ℓ+1+℘.
For the R.H.S and L.H.S of (18), we may, respectively, write
(19)R=ℓwℵ′wℵw−ℓ−1−kℓwℵ′wℵw−ℓ+1−℘=1ℵwℓwℵ′w−ℓ−1ℵw−keiθℓwℵ′w−ℓℵw+1−℘ℵw>wℵw2−℘−∑η=2∞2−℘+ℓη−1k+1χηℏ,ν,m,υ,τaη,and similarly,
(20)L=ℓwℵ′wℵw−ℓ−1−kℓwℵ′wℵw−ℓ−1+℘=1ℵwℓwℵ′w−ℓ−1ℵw−keiθℓwℵ′w−ℓℵw−1+℘ℵw<wℵw℘+∑η=2∞ℓη−11+k−℘χηℏ,ν,m,υ,τaη.
Then,(21)R−L>wℵw21−℘−2∑η=2∞1−℘+ℓη−11+kχηℏ,ν,m,υ,τaη≥0.
The condition (16) required is fulfilled.We have a necessary and adequate situation in the next theorem for a functionu∈T to be in the class TSℏ,υ,τmν,ℓ,k,℘.Theorem 4.
Letu∈T indicated by (3). Then, u∈TSℏ,υ,τmν,ℓ,k,℘.
(22)⟺∑η=2∞1−℘+ℓη−11+kχηℏ,ν,m,υ,τaη≤1−℘,where χηℏ,ν,m,υ,τ is defined by (17).Proof.
We can only prove the requirement in view3 of the theorem. If u∈TSℏ,υ,τmν,ℓ,k,℘ and w is real, then
(23)1−∑η=2∞1+ℓη−1χηℏ,ν,m,υ,τaηwη−11−∑η=2∞χηℏ,ν,m,υ,τaηwη−1−℘>k∑η=2∞ℓη−1χηℏ,ν,m,υ,τaηwη−11−∑η=2∞χηℏ,ν,m,υ,τaηwη−1.
We get the desired inequality from lettingw⟶1−.Corollary 5.
Ifu∈TSℏ,υ,τmν,ℓ,k,℘, then
(24)aη≤1−℘1−℘+ℓη−11+kχηℏ,ν,m,υ,τη≥2.


## 3. Growth and Distortion Theorem

Theorem 6.
Letu∈TSℏ,υ,τmν,ℓ,k,℘. Then, for w=r<1,
(25)r−1−℘B2ℏ,ν,m,υ,τ,ℓ,k,℘r2≤uw≤r+1−℘B2ℏ,ν,m,υ,τ,ℓ,k,℘r2,(26)1−21−℘B2ℏ,ν,m,υ,τ,ℓ,k,℘r2≤u′w≤1+21−℘B2ℏ,ν,m,υ,τ,ℓ,k,℘r,where
(27)Bηℏ,ν,m,υ,τ,ℓ,k,℘=1−℘+ℓη−11+kχηℏ,ν,m,υ,τη≥2.
Equations (25) and (26) are sharp for the u given function
(28)uw=w−1−℘B2ℏ,ν,m,υ,τ,ℓ,k,℘w2.Proof.
Sinceu∈TSℏ,υ,τmν,ℓ,k,℘ and it follows from 4 of the theorem,
(29)∑η=2∞Bηℏ,ν,m,υ,τ,ℓ,k,℘aη≤1−℘,where Bηℏ,ν,m,υ,τ,ℓ,k,℘ is given by (27), we have
(30)B2ℏ,ν,m,υ,τ,ℓ,k,℘∑η=2∞aη=∑η=2∞B2ℏ,ν,m,υ,τ,ℓ,k,℘aη≤∑η=2∞Bηℏ,ν,m,υ,τ,ℓ,k,℘aη≤1−℘,and therefore,
(31)∑η=2∞aη≤1−℘B2ℏ,ν,m,υ,τ,ℓ,k,℘.
Sinceu is given by (3), we get
(32)uw≤w+w2∑η=2∞aηwη−2≤r+r2∑η=2∞aη≤r+1−℘B2ℏ,ν,m,υ,τ,ℓ,k,℘r2,uw≥w−w2∑η=2∞aηwη−2≥r−r2∑η=2∞aη≥r−1−℘B2ℏ,ν,m,υ,τ,ℓ,k,℘r2.
In light of Theorem4, we have
(33)B2ℏ,ν,m,υ,τ,ℓ,k,℘2∑η=2∞ηaη=∑η=2∞B2ℏ,ν,m,υ,τ,ℓ,k,℘2ηaη≤∑η=2∞Bηℏ,ν,m,υ,τ,ℓ,k,℘aη≤1−℘,which yields
(34)∑η=2∞ηaη≤21−℘B2ℏ,ν,m,υ,τ,ℓ,k,℘.
Thus,(35)u′w≤1+∑η=2∞ηaηwη−1≤1+r∑η=2∞ηaη≤1+21−℘B2ℏ,ν,m,υ,τ,ℓ,k,℘r,u′w≥1−∑η=2∞ηaηwη−1≥1−r∑η=2∞ηaη≥1−21−℘B2ℏ,ν,m,υ,τ,ℓ,k,℘r.
Hence, the proof is complete.Consider thatuw=w−1−℘/B2ℏ,ν,m,υ,τ,ℓ,k,℘B2ℏ,ν,m,υ,τ,ℓ,k,℘w2 is sharp is to (25).Andu′w=1−21−℘21−℘/B2ℏ,ν,m,υ,τ,ℓ,k,℘B2ℏ,ν,m,υ,τ,ℓ,k,℘w is sharp is to (26).


## 4. Extreme Points

Now, for the function class, we look at the extreme pointsTSℏ,υ,τmν,ℓ,k,℘.Theorem 7.
Let the functionsu1w=w and
(36)uηw=w−1−℘Bηℏ,ν,m,υ,τ,ℓ,k,℘wηη≥2.
Then,u∈TSℏ,υ,τmν,ℓ,k,℘.
(37)⟺uw=∑η=2∞ℏηuηww∈U,where ℏη≥0η≥1 and ∑η=1∞ℏη=1.Proof.
Assume that it is possible to writeu as in (37). Then,
(38)uw=ℏ1w+∑η=2∞ℏηw−1−℘Bηℏ,ν,m,υ,τ,ℓ,k,℘wη=w−∑η=2∞ℏη1−℘Bηℏ,ν,m,ℓ,k,℘wη,since
(39)∑η=2∞Bηℏ,ν,m,υ,τ,ℓ,k,℘ℏη1−℘Bηℏ,ν,m,υ,τ,ℓ,k,℘=1−℘∑η=2∞ℏη=1−℘1−ℏ1≤1−℘.
By virtue4 of the theorem, it follows that u∈TSℏ,υ,τmν,ℓ,k,℘.
Conversely, supposeu∈TSℏ,υ,τmν,ℓ,k,℘ and consider
(40)ℏη=Bηℏ,ν,m,υ,τ,ℓ,k,℘1−℘aη,η≥2,ℏ1=1−∑η=2∞ℏη.
Then,uw=∑η=1∞ℏηunw, hence the theorem.


## 5. Radii of Starlikeness, Convexity, and Close-to-Convexity

Theorem 8.
Letu∈TSℏ,υ,τmν,ℓ,k,℘. Then, u is star-shaped of order ρ0≤ρ<1 in w<r1ℏ,ν,m,υ,τ,ℓ,k,℘, where
(41)r1ℏ,ν,m,υ,τ,ℓ,k,℘=infη≥21−ρBηℏ,ν,m,υ,τ,ℓ,k,℘η−ρ1−℘1/η−1.Proof.
To be able to prove the theorem, we have to demonstrate that(42)wu′wuw−1≤1−ρ,0≤ρ<1 for w∈U with w<r1ℏ,ν,m,υ,τ,ℓ,k,℘. We have
(43)wu′wuw−1=−∑η=2∞η−1aηwη−11−∑η=2∞aηwη−1≤∑η=2∞η−1aηwη−11−∑η=2∞aηwη−1.
Thus,(44)wu′wuw−1≤1−ρif∑η=2∞η−ρ1−ρaηwη−1≤1.
In virtue of (22), we have
(45)∑η=2∞Bηℏ,ν,m,υ,τ,ℓ,k,℘1−℘aη≤1.
The inequality of (43) would then be valid if
(46)η−ρ1−ρwη−1≤Bηℏ,ν,m,υ,τ,ℓ,k,℘1−℘η≥2,or if
(47)w≤1−ρBηℏ,ν,m,υ,τ,ℓ,k,℘η−ρ1−℘1/η−1η≥2.
Hence, the proof is complete.The evidence9 and 10 of the subsequent theorems is comparable to 8 of the theorem, so the evidence is excluded.Theorem 9.
Letu∈TSℏ,υ,τmν,ℓ,k,℘. Then, u is convex of order ρ0≤ρ<1 in w<r2ℏ,ν,m,υ,τ,ℓ,k,℘, where
(48)r2ℏ,ν,m,υ,τ,ℓ,k,℘=infη≥21−ρBηℏ,ν,m,υ,τ,ℓ,k,℘ηη−ρ1−℘1/η−1.Theorem 10.
Let the functionu given by (3) be in the class TSℏ,υ,τmν,ℓ,k,℘. Then, u in close-to-convex of order ρ0≤ρ<1 in w<r3ℏ,ν,m,υ,τ,ℓ,k,℘, where
(49)r3ℏ,ν,m,υ,τ,ℓ,k,℘=infη≥21−ρBηℏ,ν,m,υ,τ,ℓ,k,℘η1−℘1/η−1.


## 6. Fekete-Szego Inequality

In this section, for the mapping in the class, we get the Fekete-Szego inequalitySℏ,υ,τmν,ℓ,k,℘. To illustrate our fundamental result, we will identify the appropriate lemma.Lemma 11 (see [27]).
Ifpw=1+c1w+c2w+c3w2+⋯ is an analytic mapping with positive real part in U, then
(50)c2−ȷc12=−4ȷ+2,ȷ≤0,2,0≤ȷ≤1,4ȷ−2,ȷ≥1.
Whenȷ<0 or ȷ>1, the inequality holds iff pw=1+w/1−w or one of its rotations. If 0<ȷ<1, then the equality holds iff
(51)pw=1+w21−w2or one of its rotations. If ȷ=0, the equality holds iff
(52)pw=1+δ21+w1−w+1−δ21−w1+w0≤δ≤1or one of its rotations.
Ifȷ=1, the equality holds iff pw is the reciprocal of one of the mapping such that the equality holds when it comes to ȷ=0.Theorem 12.
Letℓ≥1,0≤k≤℘<1. If u∈Sℏ,υ,τmν,ℓ,k,℘ is given by (1), then
(53)a3−μa22=1−℘ℓ21−k2χ3ℏ,ν,m,υ,τℓ1−k+21−℘−4μ1−℘χ3ℏ,ν,m,υ,τχ22ℏ,ν,m,υ,τ,μ≤σ1,1−℘ℓ1−kχ3ℏ,ν,m,υ,τ,σ1≤μ≤σ2,−1−℘ℓ21−k2χ3ℏ,ν,m,υ,τℓ1−k+21−℘−4μ1−℘χ3ℏ,ν,m,υ,τχ22ℏ,ν,m,υ,τ,μ≥σ2,where
(54)σ1=χ22ℏ,ν,m,υ,τ2χ3ℏ,ν,m,υ,τ,σ2=χ22ℏ,ν,m,υ,τ1−℘+ℓ1−k2χ3ℏ,ν,m,υ,τ1−℘.The outcome is sharp.Proof.
Since, for complex numbers,Rz≤z, u∈Sℏ,υ,τmν,ℓ,k,℘ implies that
(55)Rℓwℵ′wℵw−ℓ−1>kRℓwℵ′wℵw−ℓ+℘or that
(56)Rwℵ′wℵw>℘−1+ℓ1−kℓ1−k.
Hence,(57)ℵ∈S∗℘−1+ℓ1−kℓ1−k.
Let(58)pw=wℵ′w/ℵw−℘−1+ℓ1−k/ℓ1−k1−℘/ℓ1−k=1+c1w+c2w2+⋯.
We then have, by way of (10) and (14),
(59)a2=1−℘ℓ1−kχ2ℏ,ν,m,υ,τc1,a3=1−℘2ℓ1−kχ2ℏ,ν,m,υ,τc2+1−℘ℓ1−kc12.
Therefore, we obtain(60)a3−μa22=1−℘2ℓ1−kχ3ℏ,ν,m,υ,τc2+1−℘ℓ1−kc12−μ1−℘2ℓ21−k2χ22ℏ,ν,m,υ,τc12=1−℘2ℓ1−kχ3ℏ,ν,m,υ,τc2−1−℘ℓ1−kc122μχ3ℏ,ν,m,υ,τA12ℏ,ν,m,υ,τ−1.
We write(61)a3−μa22=1−℘2ℓ1−kχ3ℏ,ν,m,υ,τc2−ρc12,where
(62)ρ=1−℘ℓ1−k2μχ3ℏ,ν,m,υ,τχ22ℏ,ν,m,υ,τ−1.
The implementation of the lemma above follows our conclusion. Denote(63)ξ=℘−1+ℓ1−kℓ1−k.
Ifμ<σ1 or μ>σ2, it is true that equality exists.
(64)⟺ℵw=w1−eiθw21−ξθ∈ℝ.
Whenσ1<μ<σ2, it is true that equality exists, iff
(65)ℵw=w1−eiθw21−ξθ∈ℝ.
Ifμ=σ1, then it is true that equality exists, iff
(66)ℵw=w1−eiθw21−ξ1+δ/2w1+eiθw21−ξ1−δ/2=w1−eiθw1+δ1+eiθw1−δ1−ξ,0≤δ≤1,θ∈ℝ.
Finally, if it is true that equality exists⟺pwμ=σ2, it is the inverse of one of the equality functions and holds true in the case of μ=σ2


## 7. Partial Sums

Consider the recent works on partial analytic function sums by Silverman [28] and Silvia [29]. Partial function in this class is considered in this section to be TSℏ,υ,τmν,ℓ,k,℘ giving sharp lower boundaries to the reap part ratios of uw to uqw and u′w to u′qw.Theorem 13.
Letu∈TSℏ,υ,τmν,ℓ,k,℘ and indicate u1w and uqw as partial sums
(67)u1w=w,uqw=w+∑η=2qaηwηq∈ℕ\1.
Suppose that(68)∑η=2∞dη∣aη∣≤1,where
(69)dη=1−℘+ℓη−11+kAηℏ,ν,m,υ,τ1−℘.
Then,u∈TSℏ,υ,τmν,ℓ,k,℘.
Furthermore,(70)Ruwuqw>1−1dq+1w∈U,q∈ℕ,(71)Ruqwuw>dq+11+dq+1.Proof.
It is not crucial to verify that thedη coefficients supplied by (69) are correct.
(72)dη+1>dη>1.
So we have(73)∑η=2qaη+dq+1∑η=q+1∞aη≤∑η=2∞dηaη≤1.
The hypothesis used (69), by setting
(74)g1w=dq+1uwuqw−1−1dq+1=1+dq+1∑η=q+1∞aηwη−11+∑η=2qaηwη−1.
If we use and apply (73), we find that
(75)g2w−1g2w+1≤dq+1∑η=q+1∞aη2−2∑η=2qaη−dq+1∑η=q+1∞aη≤1.
That immediately leads in a conclusion (70) of Theorem 13. To find out that
(76)uw=w+wq+1dq+1gives sharp result, we observe that forw=reiπ/q,
(77)uwuqw=1+wqdq+1⟶1−1dq+1asw⟶1−.
Similarly, if we take(78)g2w=1+dq+1uqwuw−dq+11+dq+1=1−1+dη+1∑η=q+1∞aηwη−11+∑η=2∞aηwη−1,we can deduce, and make use of (73), that
(79)g2w−1g2w+1≤1+dq+1∑η=q+1∞aη2−2∑η=2q∣aη∣−1−dq+1∑η=q+1∞aη.
This leads directly to the statement (71) of Theorem 13.
For eachq∈ℕ with the external mapping uw, the bound in (71) is sharp indicated by (76).
Thus, the evidence of the Theorem13 is complete.Theorem 14.
Letu∈TSℏ,υ,τmν,ℓ,k,℘ and fulfill (16). Then,
(80)Ru′wuq′w≥1−q+1dq+1.Proof.
By setting(81)gw=dq+1u′wuq′w−1−q+1dq+1=1+dq+1/q+1q+1∑η=q+1∞naηwη−1+∑η=2∞ηaηwη−11+∑η=2∞ηaηwη−1=1+dq+1/q+1q+1∑η=q+1∞ηaηwη−11+∑η=2∞ηaηwη−1.
Now,(82)gw−1gw+1≤dq+1/q+1q+1∑η=q+1∞ηaη2−2∑η=2qηaη−dq+1/q+1q+1∑η=q+1∞ηaη.
Now,(83)gw−1gw+1≤1if∑η=2qηaη+dq+1q+1∑η=q+1∞ηaη≤1,since the L.H.S. of (83) is bounded above by ∑η=2qdηaη if
(84)∑η=2qdη−ηaη+∑η=q+1∞dη−dq+1q+1ηaη≥0,and the proof is complete.The consequence of the extreme function is sharpuw=w+wq+1/dq+1.Theorem 15.
Letu∈TSℏ,υ,τmν,ℓ,k,℘ and fulfill (16). Then,
(85)Ru′qwu′w≥dq+1q+1+dq+1.Proof.
By setting(86)gw=q+1+dq+1uq′wu′w−dq+1q+1+dq+1=1−1+dq+1/q+1q+1∑η=q+1∞ηaηwη−11+∑η=2qηaηwη−1.
Using (84) and making use of it, we deduce that
(87)gw−1gw+1≤1+dq+1/q+1q+1∑η=q+1∞ηaη2−2∑η=2qηaη−1+dq+1/q+1q+1∑η=q+1∞ηaη≤1that immediately leads us to the statement 15 of the theorem.


## 8. Conclusions

This research has introduced study a new differential operator related to analytic function and studied some basic properties of geometric function theory. Accordingly, some results to coefficient estimates, grouth and distortion theorem, Fekete-Szego inequalityy, and partial sums have also been considered, inviting future research for this field of study.



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