Some results for the family of univalent functions related with Limaçon domain

The investigation of univalent functions is one of the fundamental ideas of Geometric function theory (GFT). However, the class of these functions cannot be investigated as a whole for some particular kind of problems. As a result, the study of its subclasses has been receiving numerous attentions. In this direction, subfamilies of the class of univalent functions that map the open unit disc onto the domain bounded by limaçon of Pascal were recently introduced in the literature. Due to the several applications of this domain in Mathematics, Statistics (hypothesis testing problem) and Engineering (rotary fluid processing machines such as pumps, compressors, motors and engines.), continuous investigation of these classes are of interest in this article. To this end, the family of functions for which ς f ′(ς) f (ς) and (ς f ′(ς))′ f ′(ς) map open unit disc onto region bounded by limaçon are studied. Coefficients bounds, Fekete Szegö inequalities and the bounds of the third Hankel determinants are derived. Furthermore, the sharp radius for which the classes are linked to each other and to the notable subclasses of univalent functions are found. Finally, the idea of subordination is utilized to obtain some results for functions belonging to these classes.


Introduction
One of the most fascinating areas of Complex analysis is the study of geometric characterization of univalent functions in the open unit disc U. Because of the challenging problem in studying the class S (1−1) of univalent functions in U as a whole, several subclasses of it emerged. The most studied of these are the classes C CV , S S T , C CV (β) and S S T (β) (0 ≤ β < 1) of convex functions, starlike functions and, convex and starlike functions of order β, respectively. Since the image domains of U plays a significant role in their geometric characterization, various subclasses of S (1−1) have been receiving attention in different directions and perspectives (see [4,6,20,23,28,32,33,[36][37][38]). For this reason, Ma and Minda [17] gave a unified treatment of both S S T and C CV . They considered the class Ψ of analytic univalent functions ψ(ς) with Reψ(ς) > 0 and for which ψ(U) is symmetric with respect to the real axis and starlike with respect to ψ(0) such that ψ (0) > 0. They initiated the following classes of functions that generalized and unified many renowned subclasses of S (1−1) : and where A is the class of analytic functions f (ς) of the form then C CV (ψ) = UCV is the Goodman class of uniformly convex functions [8,30], which was later modified and examined by Kanas and Wisniowska [13,14]. Similarly, S T hpl (s) = S * 1 (1−ς) s , CV hpl (s) = C 1 (1−ς) s (0 < s ≤ 1), are made-known by Kanas and Ebadian [15,16], respectively. These consist of functions f ∈ A such that ς f (ς)/ f (ς) and (ς f (ς)) / f (ς) lie in the domain bounded by the right branch of a hyperbola More special families of Ma and Minda classes can be found in [3, 9, 10, 24-26, 31, 34, 39].
Recently, Kanas et al [18] introduced novel subclasses S T L (s) and CV L (s) of S S T and C CV , respectively. Geometrically, they consist of functions f (ς) ∈ A such that ς f (ς)/ f (ς) and (ς f (ς)) / f (ς) lie in the region bounded by the limaçon defined as as shown in Figure 1 for different values of s. s = 0.35, 0.5, 0.6, 0.71, 0.75 and 1 corresponds to blue, red, green, gray, yellow and black. Some novel properties of these classes were derived in [18]. Motivated by this present work and other aforementioned articles, the goal in this paper is to continue with the investigation of some interesting properties of the classes S T L (s) and CV L (s). To this end, the sharp bounds of the Hankel determinant, subordination conditions as well as some radius results for these novel classes are investigated.

Materials and method
To put our investigations in a clear perspective, some preliminaries and definitions are presented as follows: Denoted by W is the class of analytic functions such that w(0) = 0 and |w(ς)| < 1. These functions are known as Schwarz functions. If f (ς) and g(ς) are analytic functions in U, then f (ς) is subordinate to g(ς) (written as f (ς) ≺ g(ς)) if there exists a Schwarz function w(ς) ∈ W such that f (ς) = g(w(ς)), ς ∈ U. Janowski [12] introduced the class or equivalently, satisfying the inequality As a special cases, P(1, −1) ≡ P and P(1 − 2β, −1) ≡ P(β) (0 ≤ β < 1) are the classes of functions of positive real part and that whose real part is greater than β, respectively (see [7]).
This determinant has been studied by many researchers. In particular Babalola [2] obtained the sharp bounds of H 3 (1) for the classes S S T and C CV . By this definition, H 3 (1) is given as: and the by triangle inequality, Clearly, one can see that H 2 (1) = |δ 3 − δ 2 2 | is a particular instance of the well-known Fekete Szegö functional |δ 3 − µδ 2 2 |, where µ is a real number.
Also, f ∈ CV L (s) if and only if Moreover, the integral representation of functions f ∈ S T L (s) is given as while that of g ∈ CV L (s) is given as Furthermore, the extremal functions for each of the classes are given by and for K s,n (ς) ∈ CV L (s), (2.7)

A set of lemmas
Lemma 3.1.
Throughout this work f (ς) is taken to be of the form (1.1) while w(ς) is of the form (2.1). In the next sections, the main results are presented.

Coefficient results
In this section, we assume 0 < s ≤ 1 2 . First, we establish a few auxiliary results whose applications will be needed hereafter.

(4.4)
It is obvious from Lemma 3.4 and the fact that L s (ς) is convex for 0 < s ≤ 1 2 that |c n | ≤ 2s, n ≥ 1. (4.5) Using this result in (4.4), we obtain We need to show (4.1) by Mathematical induction. For this reason, assume (4.1) is true and proceed to prove From (4.6), and finally, Therefore, Hence, by Mathematical induction, we have the desired result.
In view of Theorem 4.1 and the definition of functions in CV L (s), we are led to the following result. Then The bound 19s 4 12 is sharp for the function Proof.
where w ∈ W. Comparing coefficients of ς, ς 2 and ς 3 in (4.8), we arrive at By Lemma 3.1, we obtain Let x = w 1 , ξ = y with 0 ≤ x ≤ 1 and |y| ≤ 1. Then the triangle inequality gives This means that F (x, |y|) is increasing on the interval [0, 1]. So, Then The bound s 2 9 is sharp for the function Proof. From the definition of f ∈ CV L (s) and (4.9), it is easy to see that The rest of the proof follows as in Theorem 4.3.
Lemma 4.5. Let f ∈ S T L (s). Then The bound 2s 3 is best possible for the function Proof. From (4.9), a computation gives Employing Lemma 3.2, we write the expression for w 3 , and applying the triangle inequality together with Lemma 3.1, we obtain where we have taken w 1 = x, ξ = y with 0 ≤ x ≤ 1 and |y| ≤ 1. Let H(x, |y|) represents the right side of (4.12). Then Thus, where It is clear that H(x) attains its maximum value at x = 7s 2 . Thus, H(x) ≤ H 7s 2 = 2s 3 . Consequently, Proof. Using the definition of f ∈ CV L (s) and (4.9), we find Let w 1 = x (0 ≤ x < 1) and ξ = y with |y| ≤ 1. Then applying Lemma 3.2 and following the procedure of proof as in Theorem 4.5, we arrive at the desired result.

2
, we obtain the required result.
For µ = 1 in Theorem 5.1, we deduce the following sharp result.

2
, we obtain the desired result.
For µ = 1 in Theorem 5.2, we deduce the following sharp result. 6. S T L (s) and CV L (s) radii Theorem 6.1. The CV L (s)-radius for the class S S T (β) (where β = (1 − s) 2 ) is given by This radius is sharp for the functions given by This means that there exists w ∈ W such that Then by Schwarz lemma, It follows from logarithmic differentiation of (6.3) that It is known from [27] that for p ∈ P(β), .
We want to prove that if (6.7) is satisfied. To show that the radius cannot be improved, we consider the function Then for ς = 2s−s 2 2+2s−s 2 , which shows that equality is attained for (6.7).
Corollary 6.1. The S T L (s)-radius and CV L (s)-radius for the classes of starlike and convex functions are given by (6.7).
Following the discussion demonstrated by Sharma et al in [35] for Theorem 3, we present the following results.

or, equivalently, if and only if
Then f ∈ S T L (s) and f ∈ CV L (s), respectively if and only if conditions (7.4) or (7.5) is satisfied Applying Corollary 7.1 along with the integral representation for the classes S T L (s) and CV L (s), respectively, we present the following examples.
Proof. Following the same arguments as in the proof of Theorem 8.1, we arrive at where to show that To achieve this, it is enough to show that the domain bounded by the limaçon is inside the region bounded by the curve h(e iθ ) (θ ∈ [0, 2π)). As a result, we need to find ρ for which .
and p(ς) be analytic in U with p(0) = 1 such that Proof. Let q(ς) = 1+A ς 1+B ς . We have that q(ς) is convex univalent in U. Therefore, following the method of proof in Theorem 8.1, we arrive at where to show that For this, we need to establish that the region bounded by the limaçon lies inside the domain bounded by the curve h(e iθ ) (θ ∈ [0, 2π). A simple observation of (2.5) suggests it suffices to show (8.6) holds. This completes the proof.
It is easy to see that the transformation h(ς) = 1 + ρ ςq (ς) q(ς) maps U onto the disc D(a, r), where We choose to omit the proof of the next theorem since it follows the same argument as in Theorem 8.3. (i) If we put p(ς) = ς f (ς)/ f (ς) and p(ς) = (ς f (ς)) / f (ς) in Theorem 8.1-8.5, we obtain the conditions on ρ for which the respective subordination conditions (8.1), (8.3) and (8.7) imply f ∈ S T L (s) and f ∈ CV L (s). (ii) We note that our condition 0 < s ≤ 1 2 cannot be relaxed in Theorem 8.1 and Theorem 8.2. Otherwise, starlikeness of Q will not be achieved. As such, the proof of the theorems will be extremely difficult to obtain via Lemma 3.3.

Conclusion
The Ma and Minda classes of functions are the comprehensive generalization of the classes S S T and C CV . These classes are vital in (GFT) because of their importance in science and technology. To this end, continuous studies of their subfamily, which are related to Limaçon domain were investigated. Coefficients bounds, Fekete Szegö inequality as well as the upper bounds of the third Hankel determinants for these subclasses were derived. Finally, the techniques of differential subordination were also used to obtain some restrictions for which analytic functions belonged to these families. In addition, to have more new theorems under present examinations, new generalization and applications can be explored with some positive and novel outcomes in various fields of science, especially, in applied mathematics. These new surveys will be presented in future research work being processed by authors of the present paper.