# On a Subclass of Analytic Functions That Are Starlike with Respect to a Boundary Point Involving Exponential Function

Authors: Adam Lecko; Gangadharan Murugusundaramoorthy; Srikandan Sivasubramanian
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In the present exploration, the authors define and inspect a new class of functions that are regular in the unit discD≔ς∈ℂ:ς<1, by using an adapted version of the interesting analytic formula offered by Robertson (unexploited) for starlike functions with respect to a boundary point by subordinating to an exponential function. Examples of some new subclasses are presented. Initial coefficient estimates are specified, and the familiar Fekete-Szegö inequality is obtained. Differential subordinations concerning these newly demarcated subclasses are also established.


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## Body

## 1. Introduction and Preliminary Results

LetH be the class comprising of all holomorphic functions in the unit disc D≔ς∈ℂ:ς<1. Also, let A signify the subclass of H entailing of functions h∈A be of the form
(1)hς=ς+∑n=2∞anςn,ς∈D,with the normalization h0=h′0−1=0. Denote by S, the subclass of A comprising univalent functions. Two conversant subclasses of A are familiarized by Robertson [1], are defined with their analytical description as
(2)S∗α≔h∈A:Rςh′ςhς>α,ς∈D,Cα≔h∈A:R1+ςh′′ςh′ς>α,ς∈D,and are correspondingly known as starlike and convex functions of order α0≤α<1. It is well known that S∗α⊂S and Cα⊂S. In interpretation of Alexander’s relation, h∈Cα⇔ςh′ς∈S∗α for ς∈D. For α=0, the class S∗≔S∗0 condenses to the well-known class of normalized starlike univalent functions, and C≔C0 reduces to the normalized convex univalent functions.A functionf∈H is subordinate to g∈H written as f≺g if there exists ω∈H with ω0=0 and ωD⊂D such that fς=gως for every ς∈D. In precise, if g is univalent, then f≺g if and only if f0=g0 and fD⊂gD.LetP symbolize the class of functions p∈H with the normalization p0=1, i.e., of the form
(3)pς=1+∑n=1∞pnςn,ς∈D,and such that Rpς>0 for ς∈D. Functions in P are called familiarly as the Carathéodory class of functions. Ma and Minda [2] proposed a appropriate subclass of P denoted by P∗1 comprising of all Φ that is univalent in D with
(4)Φ0=1;Φ′0>0,ΦD is symmetric with respect to the real axis(2)
Starlike with respect to1He also represented the classΦ∈P∗1 by
(5)Φς=1+∑n=1∞Bnςn,B1>0;ς∈D.The classP∗1 plays a vital part in defining generalized form of holomorphic functions. Ma and Minda [2] considered the function Φ∈P∗1 and defined S∗Φ as the class of all h∈A such that ςh′ς/hς≺Φς for ς∈D. The above functions defined are called as functions of Ma and Minda kind. Observe that S∗α=S∗Φ with Φς=1+1−2ας/1−ς,ς∈D.There are recent articles ([3–6]) where subclasses of A were defined by using subordination satisfying the relation ςh′ς/hς≺Φς for ς∈D (see also [7, 8]). In particular, the exponential function Φeς=eς≔expς, an entire function in ℂ has positive real part in D, Φe0=1,Φe′0=1, and ΦeD=w∈ℂ:logw<1, is symmetric with respect to the real axis and starlike with respect to 1. Further, Φe∈P∗1 and therefore, it is now to make a remark that the class
(6)Se=f∈A:ςf′ςfς≺Φeς=eς,ς∈Dis well defined. For an attractive study on starlike functions connected with the exponential function, an individual can refer to Mendiratta et al. [9, 10] (see also the works of [11–13]).We recall the class of close-to-convex functions denoted byK introduced and studied by Kaplan [14]. A function h∈H is called to be close-to-convex if and only if there exist a function ψ∈C and β∈−π/2,π/2 such that
(7)Reiβh′ςψ′ς>0,ς∈D.Remarking at this time that even though starlikeness of a fixed order has been discussed and well thought-out in detail in countless articles in excess of a elongated stage of period, class of univalent functionsg∈H that maps D onto Ω, starlike domain with reverence to a boundary point is still a conception that is not exclusively explored. Robertson [15] recognized this examination and introduced a new subclass
(8)G∗=g∈H:Reiδgς>0;δ∈ℝ;∀ς∈D,with
(9)g0=1,g1≔limr⟶1−gr=0,and maps (univalently) D onto a domain starlike with respect to the origin. Presume in addition that the constant function g≡1∈G∗, in addition, Robertson through a conjecture that G∗ coincides with the class G of all g∈H of the structure
(10)gς=1+∑n=1∞ϑnςn,ς∈D,such that
(11)R2ςg′ςgς+1+ς1−ς>0,ς∈D,proving thatG⊂G∗. Definitely, in the same article Robertson shown that if g∈G and g≢1, then g∈K and so univalent in D. It is importance of citing that (11) was identified by much erstwhile by Styer [16]. This surmise of Robertson that G∗ coincide with the class G was soon after proved by Lyzzaik [17], where he established that G∗⊂G.A different analytical categorization of starlike functions with respect to a boundary point was proposed by Lecko [18] proving the necessity. The sufficiency part of the categorization was afterwards proved by Lecko and Lyzzaik [19] (see [[20], Chapter VII] as well). Encouraged by the article of Robertson [15], Aharanov et al. [21] (see also [22]) investigated about the class of functions that are sprirallike with respect to a boundary point. Let
(12)Pς;M≔4ς1−ς2+4ς/M+1−ς2,1≔1,ς∈D,be the Pick function. By using the Pick function Pς;M, the author in [23] considered another closely related class to G, the family GM, M>1, comprising of all g∈H of the form (10) such that
(13)R2ςg′ςgς+ςP′ς;MPς;M>0,ς∈D.In [24], Todorov established a structural formula and coefficient estimates by associating G with a functional fς/1−ς for ς∈D. For g∈H in (10), Obradovic˘ and Owa [25] and Silverman and Silvia [26] separately introduced the classes
(14)Gα=Rςg′ςgς+1−α1+ς1−ς>0,ς∈D,whereα∈0,1. The authors in [26] confirmed a remarkable fact that for each α∈0,1, the class Gα is a subclass of G∗. Clearly, G1/2=G and appealing coefficient inequalities of G were established in [27].Forg∈H assumed as in (10) and −1<E≤1;−E<F≤1, Jakubowski and Włodarczyk [28] defined the class GE,F as
(15)RJς>0,ς∈D,where
(16)Jς=2ςg′ςgς+1+Eς1−Fς.By desirable quality of the initiative proposed in [2], Mohd and Darus in [29] presented a new class Sb∗Φ, where Φ∈P∗1, of all g∈H of the form (10) such that
(17)2ςg′ςgς+1+ς1−ς≺Φς,ς∈D.An additional appealing class on the above direction was in recent times analyzed by Lecko et al. [30].The most important intend of the present article is to illustrate and do a organized inquiry of the function class defined as below.Definition 1.
Forg∈H and as assumed in (10), we let a new class Ge as
(18)Ge=g∈H:2ςg′ςgς+1+ς1−ς≺eς,ς∈D.Remark 2.
Note that the condition (18) is well defined, for
(19)pς≔2ςg′ςgς+1+ς1−ς,ς∈Dis holomorphic in D.Based on the description of the classGe and on the analytical characterization of the class G∗ of starlike functions with respect to a boundary point, we can prepare the next result.


## 2. Representation Theorem and Coefficient Results

Let us start the section with the following representation theorem which in fact offers a handy procedure to build functions in our new classGe.Theorem 3.
A functiong∈Ge if and only if there exists p∈H such that p≺Φe and
(20)gς=1−ςexp12∫0ςpζ−1ζdζ,ς∈D.Proof.
Let us suppose thatg∈Ge, then, a function p defined by (19) is holomorphic and satisfies p≺Φe. Also, (19) can be rewritten in the type
(21)2g′ςgς+21−ς=pς−1ς,ς∈D.
This upon integration give(22)loggς21−ς2=∫0ςpζ−1ζdζ,ς∈D,log1≔0.
This in essence gives(23)gς2=1−ς2exp∫0ςpζ−1ζdζ,ς∈D,which imply (20).☐Let us presumep≺Φe. By defining a function g as in (20), and by observing that p0=1, it is noticeable that g is holomorphic in D. A working out shows that g satisfies (21); so, (19). Thus, g∈Ge, which ends the confirmation of the theorem.LetΨe be a holomorphic function which is the solution of the differential equation (see also [[10], p. 367])
(24)ςΨe′ςΨeς=eς,ς∈D,Ψe0=0,Ψe′0=1,i.e.,(25)Ψeς=ςexp∫0ςeζ−1ζdζ=ς+ς2+34ς3+1736ς4+⋯,ς∈D.Next, we present few examples for the classGe.Example 4.
(1)
For a specifiedA∈ℝ and ς∈D, let us name(26)pAς≔1+Aς,gAς≔1−ςexpAς2,ς∈D.
Note down thatgA∈H with gA0=1. Observe that
(27)2ςgA′ςgAς+1+ς1−ς=pAς,ς∈D.We finish that gA∈Ge for ∣A∣≤1−1/e.
(2)
Given−1<A≤1 and −A<B<1, define(28)w=pA,Bς≔1+Aς1−Bς,ς∈D.
Then, we identify thatpA,BD is an open disk symmetrical with respect to the real axis centered at 1+AB/1−B2 of radius A+B/1−B2. In particular, for B=A, this disk is given by
(29)w−1+A21−A2<2A1−A2,with diametric end points xL≔1−A/1+A and xR≔1+A/1−A. Since xL≥1/e and xR≤e iff A≤e−1/e+1, we perceive that then pA,A≺Φe. As a result, a function g∈H with g0=1 defined by
(30)2ςg′ςgς+1+ς1−ς=pA,Aς,ς∈D,
i.e., the function(31)gς=1−ς1−Aς,ς∈D,belongs to the class Ge for ∣A∣≤e−1/e+1.Theorem 5.
Let0<r<1. If g∈Ge, then(i)(32)−Ψe−rr1−r≤∣gς∣≤Ψe−rr1+r,∣ς∣=r.(ii)(33)arggς01−ς02≤12max∣ς∣=rargΨeςς,∣ς0∣=r,arg1≔0.Proof.
Letg∈Ge.
(i)
Describe the function(34)hς≔ςgς21−ς2,ς∈D.
Obviously,h is a holomorphic function in D, and an uncomplicated working out yields
(35)ςh′ςhς=2ςg′ςgς+1+ς1−ς,ς∈D.
It is straightforward to witness from the above thatg∈Ge if and only if
(36)ςh′ςhς≺eς,ς∈D.
By the result of Corollary1′ of [2], we obtain
(37)−Ψe−r≤∣hς∣≤Ψer,∣ς∣=r,
i.e., by using (34),
(38)−Ψe−r≤ςgς21−ς2≤Ψer,∣ς∣=r,which gives (32).
(ii)
By (36), a function h defined by (34) belongs to S∗Φe. Due to Corollary 3′ of [2], the inequality(39)arghς0ς0≤max∣ς∣=rargΨeςς,∣ς0∣=ris valid. Using now (34) in turn yields (33).☐Next, we ascertain some coefficient results for the classg∈Ge. Let B≔ω∈H:ως≤1,ς∈D and B0 be the subclass of B consisting of functions ω such that ω0=0. We comment at this time that the elements of B0 are termed as Schwarz functions.We will pertain two lemmas below to prove our main results.Lemma 6. (see [2]).
Ifp∈P is of the form (3), then for μ∈ℂ,(40)p2−μp12≤2max1,2μ−1.In particular, ifμ is a real number, then
(41)p2−μp12≤−4μ+2,μ≤0,2,0≤μ≤1,4μ−2,μ≥1.Whenμ<0 or μ>1, the equality holds true if and only if pς=1+ς/1−ς≕Lς,ς∈D, or one of its rotations. If 0<μ<1, then the equality holds true if and only if pς=Lς2,ς∈D, or one of its rotations. If μ=0, the equality holds true if and only if
(42)pς=121+λLς+121−λL−ς,ς∈D,where 0≤λ≤1, or one of its rotations. If μ=1, then the equality holds true if p is a reciprocal of one of the functions such that the equality holds true in the case when μ=0.Lemma 7. (see [31]).
Ifp∈P is of the form (3) and β2β−1≤δ≤β, then
(43)p3−2βp1p2+δp13≤2.At the moment, we are in a position to state the theorem which give a few better bounds for early coefficients and the Fekete-Szegö inequalities forf∈Ge.Theorem 8.
Ifg∈Ge is of the form (10), then
(44)ϑ1+1≤12,(45)ϑ1≤32,(46)2ϑ2−ϑ12+1≤12,(47)∣ϑ2∣≤34,(48)3ϑ3−3ϑ1ϑ2+ϑ13+1≤12,and for δ∈ℝ,(49)ϑ2−δϑ12≤14max1,δ−1+22δ−1+4δ.Inequalities (44), (45), (46), (47), and (48) are sharp.Proof.
In view of (18), there exists ω∈B0 such that
(50)2ςg′ςgς+1+ς1−ς=Φeως=expως,ς∈D.
By an application of (10), one can easily obtain with simple computation that
(51)2ςg′ςgς+1+ς1−ς=1+2ϑ1+1ς+22ϑ2−ϑ12+1ς2+23ϑ3−3ϑ1ϑ2+ϑ13+1ς3+⋯,ς∈D.
Define the functionp by
(52)pς=1+ως1−ως=1+p1ς+p2ς2+⋯,ς∈D.
Clearly,p∈P. Moreover,
(53)ως=pς−1pς+1=p12ς+p22−p124ς2+p32−p1p22+p138ς3+⋯,ς∈D.
Hence,(54)expως=1+ως+ως22+ως36+⋯=1+p1ς2+p22−p128ς2+p32−p1p24+p1348ς3+⋯,ς∈D.☐Substituting (51) and (54) into (50), by comparing the corresponding coefficients, we obtain
(55)2ϑ1+1=p12,(56)22ϑ2−ϑ12+1=p22−p128,(57)23ϑ3−3ϑ1ϑ2+ϑ13+1=p32−p1p24+p1348.Since (e.g., ([[32]], Vol. I, p. 80)),
(58)pn≤2,n∈ℕ.From (55), we obtain (44). Rewriting (55) as ϑ1=p1/4−1, (45) easily follows. Further, (56) together with (40) yields
(59)22ϑ2−ϑ12+1=p22−p128≤1,which proves (46).Upon applying (55) for ϑ1 in (56), we get
(60)4ϑ2=p22−p1.Hence, by applying (41), we obtain (47).An application of (43) in (57) gives
(61)6ϑ3−6ϑ1ϑ2+2ϑ13+2=p32−p1p24+p1348≤1,i.e., the inequality (48).Using (60) and making use of the expression for ϑ1 and in turn by applying (41) and (58), we get
(62)ϑ2−δϑ12≤18p2−δ2p12+22δ−1p1+8δ,δ∈ℝ,which leads to the inequality (49).Equalities in (44) and (45) hold for the function p=L; in (46) for the function pς=Lς2,ς∈D, in (47) for the function pς=L−ς,ς∈D and in (48) for the function pς=Lς3,ς∈D.


## 3. Differential Subordination Results InvolvingGe

In this segment, we derive certain differential subordination result concerning the classGe.To demonstrate differential subordination results, we recollect the next lemma (see ([[33]], Theorem 8.4 h, p. 132)).
Q is starlike univalent in D, orh is convex univalent in DLemma 9.
Supposeq is univalent in D,θ and φ be holomorphic in a domain D containing qD with φw≠0 when w∈qD. Let Qς≔ςq′ςφqς and hς≔θqς+Qς for ς∈D. Suppose that eitherAssume also that(iii)(63)Rςh′ςQς>0,ς∈D.Ifp∈H with p0=q0,pD⊂D, and
(64)θpς+ςp′ςφpς≺θqς+ςq′ςφqς,ς∈D,then p≺q and q are the best dominant.Theorem 10.
Letg∈H and g0=1. If g satisfies the subordination condition,
(65)2ςg′ςgς+1+ς1−ς≺1+ς,ς∈D.Then,
(66)pς≔gς21−ς2≺eς,ς∈D.Proof.
LetD≔ℂ\0. Let θw≔1,w∈ℂ and φw≔1/w,w∈D. Note that ΦeD⊂D and θ and φ are holomorphic in D. Thus,
(67)Qς≔ςΦe′ςφΦeς=ςΦe′ςΦeς=ς,ς∈Dis well defined and holomorphic. Clearly, Q is a univalent starlike function and so for a function hς≔θΦeς+Qς=1+Qς,ς∈D, we achieve
(68)Rςh′ςQς=RςQ′ςQς=1>0,ς∈D.
Hence, for any functionp belonging to H with p0=Φe0=1 such that pD⊂D, i.e., for p nonvanishing in D, by applying Lemma 9, we infer that from the subordination
(69)1+ςp′ςpς≺1+ςΦe′ςΦeς=1+ς,ς∈D,
it follows the subordinationp≺Φe.☐Next, we at this time takeg∈H with g0=1 and gς be nonzero for ς∈D satisfying (65). Let a function p be taken as in (66). Then, one can notice that p0=Φe0=1, pς≠0, for ς∈D, and p is holomorphic. Since
(70)1+ςp′ςpς=2ςg′ςgς+1+ς1−ς,ς∈D,from (69), the conclusion (66) follows, which complete the proof.Theorem 11.
Letg∈H with g0=1. If g satisfies
(71)2ςg′ςgς+1+ς1−ς≺eς+ς,ς∈D,then
(72)pς≔ςgς1−ς2∫0ςgζ1−ζ2dζ−1≺eς,ς∈D.Proof.
LetD≔ℂ\0. Let ϕw≔w,w∈ℂ, and ψw≔1/w,w∈D. Note that ΦeD⊂D and ϕ and ψ are holomorphic in D. Thus, the function Q defined by (67), i.e., the identity function, is univalent starlike. Hence, for a function hς≔θΦeς+Qς=Φeς+Qς,ς∈D, we obtain
(73)Rςh′ςQς=RςΦe′ςQς+RςQ′ςQς=RΦeς+RςQ′ςQς>0,ς∈D.
Thus, for any functionp∈H with p0=Φe0=1 such that pD⊂D, i.e., pς≠0 for ς∈D, by applying Lemma 9, we deduce that from the subordination
(74)pς+ςp′ςpς≺Φeς+ςΦe′ςΦeς=eς+ς,ς∈D,
it follows the subordinationp≺Φe.☐Let now takeg∈H with g0=1 and gς≠0 for ς∈D satisfying (65). Define a function p as in (72). We see that
(75)p0=limς⟶0ςgς1−ς2∫0ςgζ1−ζ2dζ−1=g02limς⟶0ς∫0ςgζ1−ζ2dζ−1=1=Φe0,pς=0 for ς∈D and p is holomorphic. Since
(76)pς+ςp′ςpς=2ςg′ςgς+1+ς1−ς,ς∈D,from (74), (71) follows which completes the proof.



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