# Applications of Mittag-Leffer Type Poisson Distribution to a Subclass of Analytic Functions Involving Conic-Type Regions

Authors: Muhammad Ghaffar Khan; Bakhtiar Ahmad; Nazar Khan; Wali Khan Mashwani; Sama Arjika; Bilal Khan; Ronnason Chinram
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this article, we introduce a new subclass of analytic functions utilizing the idea of Mittag-Leffler type Poisson distribution associated with the Janowski functions. Further, we discuss some important geometric properties like necessary and sufficient condition, convex combination, growth and distortion bounds, Fekete-Szegö inequality, and partial sums for this newly defined class.


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## Body

## 1. Introduction, Definitions, and Motivation

LetA represent the collections of holomorphic (analytic) functions f defined in the open unit disc:(1)D=z:z∈ℂandz<1,such that the Taylor series expansion of f is given by(2)fz=z+∑n=2∞anznz∈D.By convention,S stands for a subclass of class A comprising of univalent functions of the form (2) in the open unit disc D. Let P represent the class of all functions p that are holomorphic in D with the condition(3)Rpz>0,and has the series representation(4)pz=1+∑n=1∞cnznz∈D.Next, we recall the definition of subordination, for two functionsh1,h2∈A, we say h1 is subordinated to h2 and is symbolically written as(5)h1≺h2,if there exists an analytic function wz with the properties(6)wz≤z,w0=0,such that(7)h1z=h2wz.Further ifh2∈S, then the above condition becomes(8)h1≺h2⇔h10=h20,h1D≤h2D.Now, recall the definition of convolution, letf∈A given by (2) and hz given by(9)hz=z+∑n=2∞bnzn,then their convolution denoted by f∗hz is given by(10)f∗hz=z+∑n=2∞anbnznz∈D.The most important and well-known family of analytic functions is the class of starlike functions denoted byS∗ and is defined as(11)S∗=f∈A:Rzf′zfz>0∀z∈D.Next, for−1≤B<A≤1, Janowski [1] generalized the class S∗ as follows.Definition 1.
A functionh with property that h0=1 is placed in the class PA,B if and only if(12)hz≺1+Az1+Bz−1≤B<A≤1.Janowski also proved that for a functionp∈P, a function hz belongs to PA,B if the following relation holds(13)hz=A+1pz−A−1B+1pz−B−1.Also, functionf of form (2) belongs to the class S∗A,B if(14)zf′zfz=A+1pz−A−1B+1pz−B−1−1≤B<A≤1.Kanas et al. (see [2, 3]; see also [4, 5]) were the first to define the conic domain Ωkk≧0 as follows:(15)Ωk=u+iv:u>ku−12+v2.Moreover, for fixedk, Ωk represents the conic region bounded successively by the imaginary axis k=0. For k=1, it is a parabola, and for 0<k<1, it is the right-hand branch of the hyperbola, and for k>1, it represents an ellipse.For these conic regions, the following functions play the role of extremal functions:(16)pkz=χ1k,z,k=0,χ2k,z,k=1,χ3k,z,0≦k<1,χ4k,z,k>1,where(17)χ1k,z=1+z1−z=1+2z+2z2+⋯,χ2k,z=1+2π2log1+z1−z2,χ3k,z=1+21−k2sinh22πarccoskarctanhz,χ4k,z=1+1k2−1sinπ2Kκ∫0uz/κdt1−t21−κ2t2+1k2−1,uz=z−κ1−κz∀z∈D,and κ∈0,1 is chosen such that λ=coshπK′κ/4Kκ. Here Kκ is Legendre’s complete elliptic integral of first kind and K′κ=K1−κ2, that is, K′κ is the complementary integral of Kκ. Assume that(18)pkz=1+P1z+P2z2+⋯∀z∈D.Then, in [6], it has been shown that, for (16), one can have(19)P1=2N21−k2,0≦k<1,8π2,k=1,π24k2κ21+κκ,k>1,(20)P2=DkP1,where(21)Dk=N2+23,0≦k<1,23,k=1,4Kκ2κ2+6κ+1−π224Kκ21+κκ,k>1,with(22)N=2πarccosk.Definition 2.
A functionf of the form (2) is said to be in class k−ST, if and only if(23)lzf′zfz≺pkz,k≥0.Noor and Malik [7] combined the concepts of the Janowski functions and the conic regions and gave the following definition.Definition 3.
A functionh∈P is said to be in the class k−PA,B if and only if(24)hz≺A+1pkz−A−1B+1pkz−B−1,−1≤B<A≤1,k≥0.Geometrically,hz∈k−PA,B takes all values in domain ΔkA,B, which is defined as follows(25)ΔkA,B=w:RB−1w−A−1B+1w−A+1>kB−1w−A−1B+1w−A+1−1,the domain ΔkA,B represents conic-type regions, which was introduced and studied by Noor and Malik [7] and is further generalized by the many authors, see for example [8] and the references cited therein.Definition 4 [7].
A functionf∈A is said to be in the class k−S∗A,B if and only if(26)zf′zfz≺A+1pkz−A−1B+1pkz−B−1.The generalized exponential series:(27)Eαz=∑k=0∞zkΓ1+αk,α,z∈ℂandRα>0,is one special-type function with single parameter α, was introduced by Mittag-Leffer (see [9]), and is therefore known as the Mittag-Leffler function. Another function Eα,βz with two parameters α and β having similar properties to those of Mittag-Leffler function is given by(28)Eα,βz=∑k=0∞zkΓβ+αkα,β,z∈ℂ,and was introduced by Wiman [10, 11] Agrawal [12], and by the many other (see for example [13–16]). It can be seen that the series Eα;βz converges for all finite values of z if(29)Rα>0,Rβ>0.During the last years, the interest in Mittag-Leffler type functions has considerably increased due to their vast potential of applications in applied problems such as fluid flow, electric networks, probability, and statistical distribution theory. For a detailed account of properties, generalizations and applications of functions (27) and (28), one may refer to [17–19] and [20].Geometric properties including starlikeness, convexity, and close-to-convexity for the Mittag-Leffler functionEα,βz were recently investigated by Bansal and Prajapat in [21]. Differential subordination results associated with generalized Mittag-Leffler function were also obtained in [22].A variableN is said to be Poisson distributed if it takes the values 0,1,2,3,⋯ with probabilities e−ψ,ψe−ψ/1!,ψ2e−ψ/2!,ψ3e−ψ/3!,⋯ respectively, where ψ is called the parameter. Thus,(30)PnN=n=ψne−ψn!n=0,1,2,3,⋯.It is easy to see that (30) is a mass probability function because(31)Pψ,α,β;nz≥0,∑n=0∞Pψ,α,β;nz=1.The power seriesYψ,z given by(32)Yψ,z=z+∑n=2∞ψn−1e−ψn−1!zn∀z∈Dandψ>0,which coefficients are probabilities of Poisson distribution is introduced by Porwal [23]. We can see that by ratio test the radius of convergence of Yψ,z is infinity. Porwal [23] also defined and introduced the following series:(33)Gψ,z=2z−Yψ,z=z−∑n=2∞ψn−1e−ψn−1!zn∀z∈Dandψ>0.The works of Porwal [23] motivate researchers to introduced a new probability distribution if it assumes the positive values and its mass function is given by (30) (see for example [24–26]).It was Porwal and Dixit [24] who studied and connected the Poisson distribution and the well-known Mittag-Leffer function systematically. They called it the Mittag-Leffer type Poisson distribution and prevailed moments. The Mittag-Leffer type Poisson distribution is given by (see [24])(34)Yψ,α,βz=z+∑n=2∞ψn−1Γαn−1+βEα;βψzn,where Yψ,α,βz is a normalized function of class S, since(35)Yψ,α,β0=0=Y′ψ,α,β0−1.The probability mass function for the Mittag-Leffer type Poisson distribution series is given by(36)Pψ,α,β;nz=ψnEα;βψΓαn+βn=0,1,2,3,⋯,where Eα;βψ is given by (28). It is worthy to note that the Mittag-Leffer type Poisson distribution is a generalization of Poisson distribution. Furtheremore, Bajpai [27] also studied and obtain some necessary and sufficient conditions for this distribution series.Very recently, using the Mittag-Leffer type Poisson distribution series, Alessa et al. [28] defined the convolution operator as(37)Ωψ,α,βfz=Yψ,α,β∗fz=z+∑n=2∞φψnα,βanzn,where(38)φψnα,β=ψn−1Γαn−1+βEα;βψ.Using this convolution operator, they defined and studied a new subclass of analytic function systematically. They obtained certain coefficient estimates, neighborhood results, partial sums, and convexity and compactness properties for their defined functions class.In recent years, binomial distribution series, Pascal distribution series, Poisson distribution series, and Mittag-Leffer type Poisson distribution series play important role in the geometric function theory of complex analysis. The sufficient ways were innovated for certain subclasses of starlike and convex functions involving these special functions (see for example [26, 29–32]). Motivated by the abovementioned works and from the work of Alessa et al. [28], in this article, by mean of certain convolution operator for Mittag-Leffer type Poisson distribution, we shall define a new subclass of starlike functions involving both the conic-type regions and the Janowski functions. We then obtain some interesting properties for this newly defined function class including for example necessary and sufficient condition, convex combination, growth and distortion bounds, Fekete-Szegö inequality, and partial sums. We now define a subclass of Janowski-type starlike functions involving the conic domains by mean of certain convolution operator for Mittag-Leffer type Poisson distribution as follows.Definition 5.
For−1≤B<A≤1, a function f∈A is in class k−ΩS∗α,β,A,B if(39)RB−1ϑf;ψ,α,β−A−1B+1ϑf;ψ,α,β−A+1−1≥kB−1ϑf;ψ,α,β−A−1B+1ϑf;ψ,α,β−A+1−1,where(40)ϑf;ψ,α,β=zΩψ,α,βfz′Ωψ,α,βfz.For the proofs of our key findings, we need the following lemma.Lemma 6 [33].
Letp∈P have the series expansion of form (4), then(41)a3−ζa22≤2max1,2ζ−1,whereζ∈ℂ.


## 2. Main Results

Theorem 7.
Letf∈k−ΩS∗α,β,A,B and is of the form (2), then(42)∑n=2∞2k+11−n+n1+B−1+Aφψnα,βan<B−A.The result is sharp for the function given in (51).Proof.
Suppose that inequality (42) holds true, then it is enough to show that(43)kB−1ϑf;ψ,α,β−A−1B+1ϑf;ψ,α,β−A+1−1−RB−1ϑf;ψ,α,β−A−1B+1ϑf;ψ,α,β−A+1−1<1.
For this, consider(44)kB−1ϑf;ψ,α,β−A−1B+1ϑf;ψ,α,β−A+1−1−RB−1ϑf;ψ,α,β−A−1B+1ϑf;ψ,α,β−A+1−1.
As we have set(45)ϑf;ψ,α,β=zΩψ,α,βfz′Ωψ,α,βfz,therefore, after some straightforward simplifications, we have(46)kB−1ϑf;ψ,α,β−A−1B+1ϑf;ψ,α,β−A+1−1−RB−1ϑf;ψ,α,β−A−1B+1ϑf;ψ,α,β−A+1−1,≤k+1B−1zΩψ,α,βfz′−A−1Ωψ,α,βfzB+1zΩψ,α,βfz′−A+1Ωψ,α,βfz−1,=2k+1Ωψ,α,βfz−zΩψ,α,βfz′B+1zΩψ,α,βfz′−A+1Ωψ,α,βfz,=2k+1∑n=2∞1−nφψnα,βanznB−Az+∑n=2∞n1+B−1+Aφψnα,βanzn,≤2k+1∑n=2∞1−nφψnα,βanB−A−∑n=2∞n1+B−1+Aφψnα,βan.
By using (42), the above inequality is bounded above by 1, and hence, the proof is completed.☐Example 8.
For the function(47)fz=z+∑n=2∞B−A2k+11−n+n1+B−1+Aφψnα,βxnznz∈D,such that(48)∑n=2∞xn=1,we have(49)∑n=2∞2k+11−n+n1+B−1+Aφψnα,βan=∑n=2∞2k+11−n+n1+B−1+Aφψnα,β·B−A2k+11−n+n1+B−1+Aφψnα,βxn=B−A∑n=2∞xn=B−A.
Hence,f∈k−ΩS∗α,β,A,B and the result is sharp.Corollary 9.
Let the functionf of the form (2) be in the class k−ΩS∗α,β,A,B. Then,(50)an≤B−A2k+11−n+n1+B−1+Aφψnα,β.
The result is sharp for the functionftz given by(51)ftz=z+B−A2k+11−n+n1+B−1+Aφψnα,βzn.Theorem 10.
The classk−ΩS∗α,β,A,B is closed under convex combination.Proof.
Letfkz∈k−ΩS∗α,β,A,B such that(52)lfkz=z+∑n=2∞an,kzn,k∈1,2.
It is enough to show that(53)tf1z+1−tf2z∈k−ΩS∗α,β,A,Bt∈0,1.
As(54)ltf1z+1−tf2z=z+∑n=2∞tan,1+1−tan,2zn.
Now, by Theorem7, we have(55)∑n=2∞2k+11−n+n1+B−1+A·φψnα,βtan,1+1−tan,2≤∑n=2∞2k+11−n+n1+B−1+A·φψnα,βtan,1+1−tan,2≤t∑n=2∞2k+11−n+n1+B−1+A·φψnα,βan,1+1−t·∑n=2∞2k+11−n+n1+B−1+Aφψnα,βan,2<tB−A+1−tB−A=B−A.
Hence,(56)tf1z+1−tf2z∈k−ΩS∗α,β,A,B,which completes the proof.☐Theorem 11.
Letf∈k−ΩS∗α,β,A,B, then for z=r(57)r−B−A2k+1+2B−A+1φψ2α,βr2≤fz≤r+B−A2k+1+2B−A+1φψ2α,βr2.The result is sharp for the function given in (51) for n=2.Proof.
Letf∈k−ΩS∗α,β,A,B. Using Theorem 7, we can deduce the following inequity:(58)fz≤z+∑n=2∞anzn≤z+z2∑n=2∞an≤r+B−A2k+1+2B−A+1φψ2α,βr2.
Similarly,(59)fz≥z−∑n=2∞anzn≥z−z2∑n=2∞an≥r−B−A2k+1+2B−A+1φψ2α,βr2.☐Theorem 12.
Letf∈k−ΩS∗α,β,A,B, then for z=r(60)f′z≤1+2B−A2k+1+2B−A+1φψ2α,βr,f′z≥1−2B−A2k+1+2B−A+1φψ2α,βr.The result is sharp for the function given in (51) for n=2.Proof.
The proof is quite similar to Theorem11, so left for reader.☐Now, we evaluate a kind of Hankel determinant problem, which is also known as the Fekete-Szegö functional.Theorem 13.
Iff is of the form (2) and belongs to k−ΩS∗α,β,A,B, then(61)a3−ξa22≤P1A−B4φψ3α,βmax1,B−2P2+1φψα,β−2ξA−BP122P1φψα,β,
whereP1 and P2 are defined by (19) and (20), respectively.Proof.
To prove inequality (61), we let(62)ϑf;ψ,α,β=zΩψ,α,βfz′Ωψ,α,βfz,then from (26), we have(63)ϑf;ψ,α,β≺A+1pkz−A−1B+1pkz−B−1=Φzsay.
Thus, if(64)pkz=1+P1z+P2z2+⋯,then by simple computation, we get(65)Φz=1+12P1A−Bz+14A−B2P2−1+BP12z2+⋯.
Now, from (63), there exists an analytic function hz such that(66)hz=1+Φ−1ϑf;ψ,α,β1−Φ−1ϑf;ψ,α,β=1+c1z+c2z2+⋯,is analytic and(67)Rhz>0,
in open unit discD. Also, we have(68)ϑf;ψ,α,β=Φhz−1hz+1,where(69)zϑf;ψ,α,β′ϑf;ψ,α,β=1+φψ2α,βa2z+2φψ3α,βa3−φψ4α,βa22z2+⋯.(70)Φhz−1hz+1=1+14A−BP1c1z+14A−BP1c2+P22−1+B4−P12c12z2+⋯.
After comparing the (69) and (70), we get(71)a2=14φψ2α,βA−BP1c1,(72)a3=18φψ3α,βA−BP1c2+P22−1+B4−P12c12.
Now, by making use of (71) and (72), in conjunction with Lemma, we have(73)a3−ξa22≤P1A−B4φψ3α,βmax1,B−2P2+1φψα,β−2ξA−BP122P1φψα,β,which is the required result.☐


## 3. Partial Sums

In this section, we will examine the ratio of a function of form (2) to its sequence of partial sums(74)fjz=z+∑n=2janzn,when the coefficients of f are sufficiently small to satisfy condition (42). We will determine sharp lower bounds for(75)Rfzfjz,Rfjzfz,Rf′zfj′z,Rfj′zf′z.Theorem 14.
Iff of form (2) satisfies condition (42), then(76)Rfzfjz≥1−1ρj+1∀z∈D,(77)Rfjzfz≥ρj+11+ρj+1∀z∈D,where(78)ρj=2k+11−n+n1+B−1+Aφψnα,βA−B.The result is sharp for the function given in (51).Proof.
It is easy to verify that(79)ρn+1>ρn>1forn>2.
Thus, in order to prove the inequality (76), we set(80)ρj+1fzfjz−1−1ρj+1=1+∑n=2janzn−1+ρj+1∑n=j+1∞anzn−11+∑n=2janzn−1=1+h1z1+h2z.
We now set(81)1+h1z1+h2z=1+wz1−wz.
Then, we find after some suitable simplification that(82)wz=h1z−h2z2+h1z+h2z.
Thus, clearly, we find that(83)wz=ρj+1∑n=j+1∞anzn−12+2∑n=2janzn−1+ρj+1∑n=j+1∞anzn−1.
By applying the trigonometric inequalities withz<1, we arrived at the following inequality:(84)wz≤ρj+1∑n=j+1∞an2−2∑n=2jan−ρj+1∑n=j+1∞an.
We can now see that(85)wz≤1,if and only if(86)2ρj+1∑n=j+1∞an≤2−2∑n=2jan,which implies that(87)∑n=2jan+ρj+1∑n=j+1∞an≤1.
Finally, to prove the inequality in (76), it suffices to show that the left-hand side of (87) is bounded above by the following sum:(88)∑n=2∞ρnan,which is equivalent to(89)∑n=2jρn−1an+∑n=j+1∞ρn−ρj+1an≥0.
In virtue of (89), the proof of inequality in (76) is now completed.
Next, in order to prove the inequality (77), we set(90)1+ρj+1fjzfz−ρj+11+ρj+1=1+∑n=2janzn−1−ρj+1∑n=j+1∞anzn−11+∑n=2∞anzn−1,=1+wz1−wz,where(91)wz≤1+ρj+1∑n=j+1∞an2−2∑n=2jan−ρj+1−1∑n=j+1∞an≤1.
This last inequality in (91) is equivalent to(92)∑n=2jan+ρj+1∑n=j+1∞an≤1.
Finally, we can see that the left-hand side of the inequality in (92) is bounded above by the following sum:(93)∑n=2∞ρnan,so we have completed the proof of the assertion (77).☐We next turn to ratios involving derivatives.Theorem 15.
Iff of the form (2) satisfies condition (42), then(94)Rf′zfj′z≥1−j+1ρj+1∀z∈D,Rfj′zf′z≥ρj+1ρj+1+j+1∀z∈D,
whereρj is given by (78). The result is sharp for the function given in (51).Proof.
The proof of Theorem15 is similar to that of Theorem 14; we here choose to omit the analogous details.☐


## 4. Concluding Remarks and Observations

In our present work, by making use of the idea of Mittag-Leffler type Poisson distribution, we have defined and studied certain new subclasses of starlike functions involving the Janowski functions. Further, we have discussed some important geometric properties like necessary and sufficient condition, convex combination, growth and distortion bounds, Fekete-Szegö inequality, and partial sums for this newly defined functions class.



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