# Sufficient Condition for Strongly Starlikeness of Normalized Mittag-Leffler Function

Authors: Deepak Bansal; Sudhananda Maharana
Journal: Journal of Complex Analysis
 (2017)

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## Abstract

In the present investigation a sufficient condition is obtained for normalized Mittag-Leffler function to be starlike and strongly starlike in the open unit disk. Results obtained are new and their usefulness is depicted by deducing several interesting corollaries and examples.


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## Body

## 1. Introduction

LetH denote the class of analytic functions in the open unit disk: (1)D=z∈C:z<1and A denote the subclass of H, which are normalized by the condition f(0)=0=f′(0)-1 and have representation of the following form: (2)fz=z+∑n≥2anzn,z∈D.A function f is said to be univalent in a domain D if it is one to one in D. A function f∈A is called starlike, denoted by f∈S∗ if f is univalent in D and f(D) is a starlike domain with respect to the origin. For a given 0≤α<1, a function f∈A is called starlike function of order α, class of such function denoted by S∗(α), if and only if Rzf′(z)/f(z)>α, z∈D. Further, a function f∈A is called a convex function, if 1+Rzf′′(z)/f′(z)>0, z∈D. It is well known that S∗(0)=S∗. Further, let S~∗(α), 0<α≤1, be the class of strongly starlike functions of order α defined by (3)S~∗α=f∈A:arg⁡zf′zfz<απ2,z∈D.Note that S~∗(1)≡S∗. For more details about these classes, one can refer to [1].Iff,g∈H, then the function f is said to be subordinate to g, written as f(z)≺g(z)(z∈D), if there exists a Schwarz function w∈H with w(0)=0 and w(z)<1 (z∈D) such that f(z)=g(w(z)). In particular, if g is univalent in D, then we have the following equivalence: (4)fz≺gz⟺f0=g0,fD⊂gD.The function (5)Eλz=∑n=0∞znΓλn+1λ∈C,Rλ>0,z∈C,was introduced by Mittag-Leffler (see [2]) in connection with the summing of divergent series. An important generalization, (6)Eλ,μz=∑n=0∞znΓλn+μλ,μ∈C,Rλ>0,Rμ>0,z∈C,was introduced by Wiman [3, 4]. The Mittag-Leffler function has a close connection to differential equations of fractional order and integral equations of Abel type, such equations are becoming more and more popular in modelling natural and technical process [5–10]. 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, Lévy flights, and superdiffusive transport and in the study of complex systems. These functions interpolate between a purely exponential law and power-law like behavior of phenomena governed by ordinary kinetic equations and their fractional counterparts [6–8, 11]. The most essential properties of these entire functions, investigated by many mathematicians, can be found in [12, 13].Observe that Mittag-Leffler functionEλ,μ does not belong to the family A. Thus, it is natural to consider the following normalization of Mittag-Leffler function in D: (7)Eλ,μz=ΓμzEλ,μz=∑n=0∞ΓμΓλn+μzn+1λ,μ∈C;Rλ>0,μ≠0,-1,…;z∈D.Whilst formula (7) holds for complex-valued λ, μ, in this paper we shall restrict our attention to the case of real-valued λ, μ. It is easy to see that Eλ,μ satisfy the following relation: (8)λzEλ,μ′z=μ-1Eλ,μ-1z+λ-μ+1Eλ,μz.Several researchers studied families of analytic functions involving special functions to find conditions such that the images of functions underD have certain geometric properties like univalency, starlikeness, or convexity. In this context, many results are available in the literature regarding the hypergeometric functions [14], Bessel functions [15–17], Wright function [18], and Mittag-Leffler functions [19]. In this paper, we study some more properties of normalized Mittag-Leffler function Eλ,μ.


## 2. Strongly Starlikeness

To prove results in this section, we shall need the following lemma.Lemma 1 (Hallenbeck and Ruscheweyh [20]).
LetG(z) be convex and univalent in D with G(0)=1. Let F(z) be analytic in D, F(0)=1 and F(z)≺G(z) in D. Then, for all n∈N∪{0}, we have (9)n+1z-n-1∫0ztnFtdt≺n+1z-n-1∫0ztnGtdt.Further, it is easy to see that, under the hypothesisλ≥1, the inequality, Γ(n+μ)≤Γ(λn+μ), holds for all n∈N, which is equivalent to (10)ΓμΓλn+μ≤1μμ+1⋯μ+n-1,∀n∈N.Theorem 2.
Ifλ≥1 and μ≥3+17/2, then Eλ,μ∈S~∗(α), where (11)α=2πarcsin⁡η1-η24+η21-η2,for η=(3μ+2)/μ2.Proof.
Using inequality (10), we get (12)Eλ,μ′z-1≤∑n=1∞n+1ΓμΓλn+μzn≤∑n=1∞n+1μμ+1⋯μ+n-1=∑n=1∞nμμ+1⋯μ+n-1+∑n=1∞1μμ+1⋯μ+n-1<1μ+1μ1+1μ+1+1μ+12+⋯+1μ1+1μ+1+1μ+12+⋯(13)=3μ+2μ2=η.Note that, under the hypothesis, 0<η≤1. From (12), we conclude that (14)Eλ,μ′z≺1+ηzz∈D,which gives (15)arg⁡Eλ,μ′z<arcsin⁡η,z∈D.Using Lemma 1, for F(z)=Eλ,μ′(z) and G(z)=1+ηz with n=0, we get (16)Eλ,μzz≺1+η2z,z∈D.Consequently, (17)arg⁡Eλ,μzz<arcsin⁡η2,z∈D.
Now from (15) and (17), we conclude that (18)arg⁡zEλ,μ′zEλ,μz=arg⁡zEλ,μz+arg⁡Eλ,μ′z≤arg⁡zEλ,μz+arg⁡Eλ,μ′z<arcsin⁡η2+arcsin⁡η=arcsin⁡η1-η24+η21-η2,that is, Eλ,μ(z)∈S~∗(α), for α given in (11).Corollary 3.
Letλ≥1 and μ≥3+17/2. If 0<α≤1 and (19)η=3μ+2μ2=2ν5-41-ν216ν2+9,where ν=sinαπ/2, then Eλ,μ∈S~∗(α).Proof.
If we putη from (19) to (11), we obtain α.Puttingα=1 in Corollary 3, we get (20)ν=1⟹η=3μ+2μ2=25.Corollary 4.
Ifλ≥1 and μ=μ∗, where μ∗ is positive root of 2μ2-35μ-25=0, then Eλ,μ(z)∈S∗.Example 5.
Ifλ=1 and μ=4 then η=(3μ+2)/μ2=7/8. From (11), we get (21)α=2πarcsin⁡7128207+15and thus, from Theorem 2, we have (22)E1,4z=6ez-1-z-3z2z2∈S~∗α.


## 3. A Nonlinear Differential Equation

In order to prove results in this section, we required the following known results.Lemma 6 (see [9]).
LetΩ⊂C. Suppose that the function ψ(z):C2×D→C satisfies the condition ψ(Meiθ,Keiθ;z)∉Ω for all K≥M(M-|a|)/(M+|a|), θ∈R, and z∈D. Also, let p(z) be an analytic function of the following form: (23)pz=a+a1z+a2z2+⋯z∈D,such that ψ(p(z),zp′(z);z)∈Ω for all z∈D. Then |p(z)|<M, where 0≤|a|<M.Lemma 7 (see [21]).
Letf∈A be of form (2) which satisfies the inequality |f′′(z)|≤1,z∈D. Then f is starlike in D.Theorem 8.
For allλ>0,μ>0, let the analytic function Eλ,μ(z) in D satisfy the following inequality: (24)zEλ,μz<MM-aM+1M+a0≤a<M≤1;z∈D,and also let ϕ:=ϕ(z) be the (unique) solution of the initial value problem for higher-order linear differential equation given by (25)ϕn+1±Eλ,μzϕn=Eλ,μzz∈D;n∈N0≔N∪0ϕ0=0ϕ′0=1ϕk0=0k=2,3,…,n-1ϕn0=a,where ϕn(z)=dnϕ/dzn (n∈N0). Then the inequality |ϕ(n)(z)|<M(0≤|a|<M≤1) holds.Proof.
Definep(z) by (26)pz=ϕnzz∈D.Clearly, function p(z) belongs to (23), and then by the related implicit function it immediately follows that (27)zp′z1±pz=zϕn+1z1±ϕnz=zEλ,μzz∈D;ϕnz≠±1,and also denotes ψ(r,s;z) and Ω by (28)ψr,s;z≔s1±rr≠±1,Ω≔w∈C:w<MM-aM+1M+a0≤a<M≤1,respectively. Then, clearly, (29)ψpz,zp′z;z=zp′z1±pz=zϕn+1z1±ϕnz∈Ω∀z∈D.Further, for any (30)θ∈R,K≥MM-aM+a,z∈D,we have (31)ψMeiθ,Keiθ;z=Keiθ1±Meiθ≥MM-aM+1M+a,which gives (32)ψMeiθ,Keiθ;z∉Ω.Therefore, in view of Lemma 6, it follows that (33)pz=ϕnz<Mz∈D;0≤a<M≤1.This completes the proof.By takingn=2 in the above theorem we get the following.Corollary 9.
Let the analytic functionEλ,μ(z) in D satisfy inequality (24), and also let ϕ≔ϕ(z) be the (unique) solution of the initial value problem for third-order linear differential equation given by (34)ϕ′′′±Eλ,μzϕ′′=Eλ,μzz∈Dϕ0=0ϕ′0=1ϕ′′0=a.Then the inequality |ϕ′′(z)|<M(0≤|a|<M≤1) holds.Corollary 10.
If the analytic functionEλ,μ(z) in D satisfies the following inequality (35)zEλ,μz<1-a21+a0≤a<1and the function ϕ(z) is the (unique) solution of the initial value problem for third-order linear differential equation given by (34), then ϕ(z)∈S∗.Proof.
By takingM=1, in Corollary 9 (or in Theorem 8 with n=2), and then using Lemma 7, the proof can easily be obtained.



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*Source: 6564705-2017-11-05.xml*