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

Department of Complex Analysis, Faculty of Mathematics and Computer Science, University of Warmia and Mazury in Olsztyn, ul. SÅ‚onecz na 54, 10-710 Olsztyn, Poland Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology (Deemed to be University), Vellore 632014, India Department of Mathematics, University College of Engineering Tindivanam, Anna University, Tindivanam 604001, India


Introduction and Preliminary Results
Let H be the class comprising of all holomorphic functions in the unit disc D ≔ fς ∈ ℂ : jςj < 1g. Also, let A signify the subclass of H entailing of functions h ∈ A be of the form with the normalization hð0Þ = h′ð0Þ − 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 and are correspondingly known as starlike and convex functions of order αð0 ≤ α < 1Þ. It is well known that S * ðαÞ ⊂ S and CðαÞ ⊂ S: In interpretation of Alexander's relation, h ∈ CðαÞ ⇔ ςh ′ ðςÞ ∈ S * ðαÞ for ς ∈ D: For α = 0, the class S * ≔ S * ð0Þ condenses to the well-known class of normalized starlike univalent functions, and C ≔ Cð0Þ reduces to the normalized convex univalent functions. A function f ∈ H is subordinate to g ∈ H written as f ≺ g if there exists ω ∈ H with ωð0Þ = 0 and ωðDÞ ⊂ D such that f ðςÞ = gðωðςÞÞ for every ς ∈ D: In precise, if g is univalent, then f ≺ g if and only if f ð0Þ = gð0Þ and f ðDÞ ⊂ gðDÞ: Let P symbolize the class of functions p ∈ H with the normalization pð0Þ = 1, i.e., of the form and such that RpðςÞ > 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 * ð1Þ comprising of all Φ that is univalent in D with ΦðDÞ is symmetric with respect to the real axis (2) Starlike with respect to 1 He also represented the class Φ ∈ P * ð1Þ by The class P * ð1Þ plays a vital part in defining generalized form of holomorphic functions. Ma and Minda [2] considered the function Φ ∈ P * ð1Þ and defined S * ðΦÞ as the class of all h ∈ A such that ςh′ðςÞ/hðςÞ ≺ ΦðςÞ for ς ∈ D: The above functions defined are called as functions of Ma and Minda kind. Observe that S * ðαÞ = S * ðΦÞ with ΦðςÞ = ð1 + ð1 − 2αÞςÞ/ð1 − ςÞ, ς ∈ D: There are recent articles ( [3][4][5][6]) where subclasses of A were defined by using subordination satisfying the relation ςh ′ ðςÞ/hðςÞ ≺ ΦðςÞ for ς ∈ D (see also [7,8]). In particular, the exponential function Φ e ðςÞ = e ς ≔ exp ðςÞ, an entire function in ℂ has positive real part in D, Φ e ð0Þ = 1, Φ e ′ð0Þ = 1, and Φ e ðDÞ = fw ∈ ℂ : jlog wj < 1g, is symmetric with respect to the real axis and starlike with respect to 1. Further, Φ e ∈ P * ð1Þ and therefore, it is now to make a remark that the class is 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][12][13]). We recall the class of close-to-convex functions denoted by K 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 ∈ ð−π/2, π/2Þ such that 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 functions g ∈ 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 with 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 such that proving that G ⊂ 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 be the Pick function. By using the Pick function Pðς ; MÞ, the author in [23] considered another closely related class to G, the family GðMÞ, M > 1, comprising of all g ∈ H of the form (10) such that In [24], Todorov established a structural formula and coefficient estimates by associating G with a functional f ðς Þ/1 − ς for ς ∈ D: For g ∈ H in (10), Obradovic and Owa [25] and Silverman and Silvia [26] separately introduced the classes 2 Journal of Function Spaces where α ∈ ½0, 1Þ: The authors in [26] confirmed a remarkable fact that for each α ∈ ½0, 1Þ, the class G α is a subclass of G * : Clearly, G 1/2 = G and appealing coefficient inequalities of G were established in [27].
For g ∈ H assumed as in (10) and −1 < E ≤ 1 ; −E < F ≤ 1 , Jakubowski and Włodarczyk [28] defined the class GðE, FÞ as where By desirable quality of the initiative proposed in [2], Mohd and Darus in [29] presented a new class S * b ðΦÞ, where Φ ∈ P * ð1Þ, of all g ∈ H of the form (10) such that 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. For g ∈ H and as assumed in (10), we let a new class G e as Remark 2. Note that the condition (18) is well defined, for is holomorphic in D: Based on the description of the class G e and on the analytical characterization of the class G * of starlike functions with respect to a boundary point, we can prepare the next result.

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 class G e .

Theorem 3. A function g ∈ G e if and only if there exists
Proof. Let us suppose that g ∈ G e , then, a function p defined by (19) is holomorphic and satisfies p ≺ Φ e : Also, (19) can be rewritten in the type This upon integration give This in essence gives which imply (20).☐ Let us presume p ≺ Φ e . By defining a function g as in (20), and by observing that pð0Þ = 1, it is noticeable that g is holomorphic in D: A working out shows that g satisfies (21); so, (19). Thus, g ∈ G e , which ends the confirmation of the theorem.
Let Ψ e be a holomorphic function which is the solution of the differential equation (see also [[10], p. 367]) i.e., Next, we present few examples for the class G e : Example 4.
(1) For a specified A ∈ ℝ and ς ∈ D, let us name 3 Journal of Function Spaces Note down that g A ∈ H with g A ð0Þ = 1. Observe that We finish that g A ∈ G e for |A | ≤1 − 1/e.
Proof. Let g ∈ G e .
(i) Describe the function Obviously, h is a holomorphic function in D, and an uncomplicated working out yields It is straightforward to witness from the above that g ∈ G e if and only if By the result of Corollary 1 ′ of [2], we obtain i.e., by using (34), which gives (32).
(ii) By (36), a function h defined by (34) belongs to S * ðΦ e Þ. Due to Corollary 3 ′ of [2], the inequality is valid. Using now (34) in turn yields (33).☐ Next, we ascertain some coefficient results for the class g ∈ G e . Let B ≔ fω ∈ H : jωðςÞj ≤ 1, ς ∈ Dg and B 0 be the subclass of B consisting of functions ω such that ωð0Þ = 0: We comment at this time that the elements of B 0 are termed as Schwarz functions.
We will pertain two lemmas below to prove our main results. Lemma 6. (see [2]). If p ∈ P is of the form (3), then for μ ∈ ℂ, In particular, if μ is a real number, then When μ < 0 or μ > 1, the equality holds true if and only if pðςÞ = ð1 + ςÞ/ð1 − ςÞ ≕ LðςÞ, ς ∈ D, or one of its rotations. If 0 < μ < 1, then the equality holds true if and only if pðς = Lðς 2 Þ,ς ∈ D, or one of its rotations. If μ = 0, the equality 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]). If p ∈ P is of the form (3) and βð2β − 1Þ ≤ δ ≤ β, then 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 for f ∈ G e . Theorem 8. If g ∈ G e is of the form (10), then and for δ ∈ ℝ, Inequalities (44), (45), (46), (47), and (48) are sharp.