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

Abstract

In the present exploration, the authors define and inspect a new class of functions that are regular in the unit disc 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.

1. Introduction and Preliminary Results

Let be the class comprising of all holomorphic functions in the unit disc . Also, let signify the subclass of entailing of functions be of the form with the normalization . Denote by , the subclass of comprising univalent functions. Two conversant subclasses of are familiarized by Robertson [1], are defined with their analytical description as and are correspondingly known as starlike and convex functions of order . It is well known that and In interpretation of Alexander’s relation, for For , the class condenses to the well-known class of normalized starlike univalent functions, and reduces to the normalized convex univalent functions.

A function is subordinate to written as if there exists with and such that for every In precise, if is univalent, then if and only if and

Let symbolize the class of functions with the normalization i.e., of the form and such that for Functions in are called familiarly as the Carathéodory class of functions. Ma and Minda [2] proposed a appropriate subclass of denoted by comprising of all that is univalent in with   is symmetric with respect to the real axis(2)Starlike with respect to

He also represented the class by

The class plays a vital part in defining generalized form of holomorphic functions. Ma and Minda [2] considered the function and defined as the class of all such that for The above functions defined are called as functions of Ma and Minda kind. Observe that with

There are recent articles ([3–6]) where subclasses of were defined by using subordination satisfying the relation for (see also [7, 8]). In particular, the exponential function , an entire function in has positive real part in , , and , is symmetric with respect to the real axis and starlike with respect to 1. Further, 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 [11–13]).

We recall the class of close-to-convex functions denoted by introduced and studied by Kaplan [14]. A function is called to be close-to-convex if and only if there exist a function and 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 that maps onto 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) onto a domain starlike with respect to the origin. Presume in addition that the constant function , in addition, Robertson through a conjecture that coincides with the class of all of the structure such that

proving that Definitely, in the same article Robertson shown that if and then and so univalent in . It is importance of citing that (11) was identified by much erstwhile by Styer [16]. This surmise of Robertson that coincide with the class was soon after proved by Lyzzaik [17], where he established that

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 , the author in [23] considered another closely related class to , the family , , comprising of all of the form (10) such that

In [24], Todorov established a structural formula and coefficient estimates by associating with a functional for For in (10), Obradovi and Owa [25] and Silverman and Silvia [26] separately introduced the classes

where The authors in [26] confirmed a remarkable fact that for each , the class is a subclass of Clearly, and appealing coefficient inequalities of were established in [27].

For assumed as in (10) and , Jakubowski and Włodarczyk [28] defined the class as where

By desirable quality of the initiative proposed in [2], Mohd and Darus in [29] presented a new class where of all 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 and as assumed in (10), we let a new class as

Remark 2. Note that the condition (18) is well defined, for is holomorphic in

Based on the description of the class and on the analytical characterization of the class 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 class .

Theorem 3. A function if and only if there exists such that and

Proof. Let us suppose that , then, a function defined by (19) is holomorphic and satisfies Also, (19) can be rewritten in the type This upon integration give This in essence gives which imply (20).☐

Let us presume . By defining a function as in (20), and by observing that it is noticeable that is holomorphic in A working out shows that satisfies (21); so, (19). Thus, , which ends the confirmation of the theorem.

Let 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

Example 4. (1)For a specified and , let us nameNote down that with . Observe that We finish that for . (2)Given and defineThen, we identify that is an open disk symmetrical with respect to the real axis centered at of radius . In particular, for , this disk is given by with diametric end points and . Since and iff we perceive that then As a result, a function with defined by i.e., the function belongs to the class for .

Theorem 5. Let If then

(i)

(ii)

Proof. Let . (i)Describe the functionObviously, is a holomorphic function in , and an uncomplicated working out yields It is straightforward to witness from the above that if and only if By the result of Corollary of [2], we obtain i.e., by using (34), which gives (32). (ii)By (36), a function defined by (34) belongs to . Due to Corollary of [2], the inequalityis valid. Using now (34) in turn yields (33).☐

Next, we ascertain some coefficient results for the class . Let and be the subclass of consisting of functions such that We comment at this time that the elements of are termed as Schwarz functions.

We will pertain two lemmas below to prove our main results.

Lemma 6. (see [2]). If is of the form (3), then for

In particular, if is a real number, then

When or , the equality holds true if and only if or one of its rotations. If , then the equality holds true if and only if or one of its rotations. If , the equality holds true if and only if where or one of its rotations. If , then the equality holds true if is a reciprocal of one of the functions such that the equality holds true in the case when .

Lemma 7. (see [31]). If is of the form (3) and , 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 .

Theorem 8. If is of the form (10), then and for

Inequalities (44), (45), (46), (47), and (48) are sharp.

Proof. In view of (18), there exists such that By an application of (10), one can easily obtain with simple computation that Define the function by Clearly, Moreover, Hence, ☐

Substituting (51) and (54) into (50), by comparing the corresponding coefficients, we obtain

Since (e.g., ([[32]], Vol. I, p. 80)), From (55), we obtain (44). Rewriting (55) as , (45) easily follows. Further, (56) together with (40) yields which proves (46).

Upon applying (55) for in (56), we get

Hence, by applying (41), we obtain (47).

An application of (43) in (57) gives

i.e., the inequality (48).

Using (60) and making use of the expression for and in turn by applying (41) and (58), we get which leads to the inequality (49).

Equalities in (44) and (45) hold for the function in (46) for the function , in (47) for the function and in (48) for the function

3. Differential Subordination Results Involving

In this segment, we derive certain differential subordination result concerning the class .

To demonstrate differential subordination results, we recollect the next lemma (see ([[33]], Theorem 8.4 h, p. 132)).   is starlike univalent in or  is convex univalent in

Lemma 9. Suppose is univalent in and be holomorphic in a domain containing with when . Let and for Suppose that either

Assume also that

(iii)

If with and then and are the best dominant.

Theorem 10. Let and . If satisfies the subordination condition, Then,

Proof. Let Let and Note that and and are holomorphic in Thus, is well defined and holomorphic. Clearly, is a univalent starlike function and so for a function we achieve Hence, for any function belonging to with such that i.e., for nonvanishing in by applying Lemma 9, we infer that from the subordination it follows the subordination ☐

Next, we at this time take with and be nonzero for satisfying (65). Let a function be taken as in (66). Then, one can notice that , , for and is holomorphic. Since from (69), the conclusion (66) follows, which complete the proof.

Theorem 11. Let with . If satisfies then

Proof. Let Let and Note that and and are holomorphic in Thus, the function defined by (67), i.e., the identity function, is univalent starlike. Hence, for a function we obtain Thus, for any function with such that i.e., for by applying Lemma 9, we deduce that from the subordination it follows the subordination ☐

Let now take with and for satisfying (65). Define a function as in (72). We see that

for and is holomorphic. Since from (74), (71) follows which completes the proof.

Data Availability

No data sets were used.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The work of Dr. S. Sivasubramanian is supported by a grant from the Science and Engineering Research Board, Government of India under Mathematical Research Impact Centric Support of Department of Science and Technology (DST) vide ref: MTR/2017/000607.