Subclasses of Starlike and Convex Functions Associated with the Limaçon Domain

: Let ST L ( s ) and CV L ( s ) denote the family of analytic and normalized functions f in the unit disk D : = { z : | z | < 1 } , such that the quantity z f (cid:48) ( z ) / f ( z ) or 1 + z f (cid:48)(cid:48) ( z ) / f (cid:48) ( z ) respectively are lying in the region bounded by the limaçon (cid:2) ( u − 1 ) 2 + v 2 − s 4 (cid:3) 2 = 4 s 2 (cid:104)(cid:0) u − 1 + s 2 (cid:1) 2 + v 2 (cid:105) , where 0 < s ≤ 1/ √ 2. The limaçon of Pascal is a curve that possesses properties which qualify it for the several applications in mathematics, statistics (hypothesis testing problem) but also in mechanics (ﬂuid processing applications, known limaçon technology is employed to extract electrical power from low-grade heat, etc.). In this paper we present some results concerning the behavior of f on the classes ST L ( s ) or CV L ( s ) . Some appropriate examples are given.


An Analytic Representation of a Limaçon of Pascal
A limaçon, known also as a limaçon of Pascal is a curve that in polar coordinates has the form where a, b are positive real numbers and θ ∈ 0, 2π). This is also called the limaçon of Pascal. The word "limaçon" comes from the Latin "limax", meaning "snail". Converting to Cartesian coordinates the Equation (1) becomes that has the following parametric form x = (b + a cos θ) cos θ, y = (b + a cos θ) sin θ.
If b ≥ 2a, a limaçon is convex, and if 2a > b > a has an indentation bounded by two inflection points. If b = a, the limaçon degenerates to a cardioid. If b < a, the limaçon has an inner loop, and when b = a/2, it is a trisectrix (but not the Maclaurin trisectrix). In Figure 1, we have plotted the limaçon r = b + a cos θ for some different values of a and b.
An analytic description of a limaçon is given by that maps the unit disk D = {z ∈ C : |z| < 1} of the complex plane C, onto a domain bounded by a limaçon defined by ∂D(s) = u + iv ∈ C : (u − 1) 2 + v 2 − s 4 2 = 4s 2 u − 1 + s 2 2 where s ∈ −1, 1 \ {0} (The Figure 2 shows an example of a image of D by the function L s for different values of s). Indeed, setting z = e iθ with 0 ≤ θ < 2π, we obtain L s (e iθ ) = 1 + s e iθ 2 = (1 + 2s cos θ + s 2 cos 2θ) + i(2s sin θ + s 2 sin 2θ) Let us denote u = u(θ) = L s e iθ and v = v(θ) = L s e iθ . Then Taking a parametrization we can find that the image of unit circle |z| = 1 under L s (·) is a curve given by that is the limaçon of Pascal. Furthermore, L s is an analytic and does not have any poles in D since It is easy to check that the real and an imaginary part of L s (e iθ ) is bounded. Then {L s (D} and {L s (D)} attains its minimum and maximum on ∂D. Indeed, by Equation (4) we have L s e iθ = 1 − s 2 + 2s cos θ + 2s 2 cos 2 θ =: g(θ).
Then F (θ) = 2s(s cos 2 θ + cos θ − s) = 0 if and only if Using an elementary computation we can find then that The above discussion can be summarized as follows (cf. Figure 3).

Definitions and Preliminaries
Let A denote the class of functions f (z) of the form: The special cases occur for β = 0, and then we get the classical classes of starlike and convex univalent functions, denoted ST := ST (0) and CV := CV (0), respectively. Let f and g be analytic in D. Then the function f is said to subordinate to g in D written by f (z) ≺ g(z), if there exists a self-map function ω(z) which is analytic in D with ω(0) = 0 and |ω(z)| < 1; (z ∈ D), and such that f (z) = g(ω(z)); (z ∈ D). If g is univalent in D, then f ≺ g if and only if f (0) = g(0) and f (D) ⊂ g(D) [2].
Let the classes G and N be defined by respectively. Then, it follows from [3] G and N are the families of univalent function, convex and starlike in one direction, respectively. Let P * be the class of analytic univalent function ψ with positive real part in D, ψ (0) > 0 and ψ(D) with respect to ψ(0) = 1 and symmetric with respect to real axis. Ma and Minda [4] gave a unified representation of different subclasses of starlike and convex functions using subordination to some function ψ ∈ P * . The superordinate function ψ is assumed to be univalent. In this way the classes ST (ψ) and CV (ψ) has been defined Specialization of the function ψ leads to a number of well-known function classes. For instance, For various choices of ψ and a detailed discussion about classes we refer to the papers [5][6][7][8][9].

Definition 1 ([10]).
Let ψ : C 2 × D → C and the function h(z) be univalent in D. If the function p is analytic in D and satisfies the following first-order differential subordination then p(z) is called a solution of the differential subordination. A function q ∈ A is said to be a dominant of the differential subordination Equation (8) if p ≺ q for all p satisfying Equation (8). An univalent dominant that satisfiesq ≺ q for all dominants q of Equation (8), is said to be best dominant of the differential subordination.

Lemma 1 ([10]
). Let q be univalent in D, and let Φ be analytic in a domain D containing q(D). If zq (z)Φ(q(z)) is starlike, then and q is the best dominant.
This paper aims to investigate the geometric properties of functions in the classes ST L (s) and CV L (s). In addition, we necessary and sufficient conditions for certain particular members of A to be in the classes ST L (s) and CV L (s).

The Classes ST L (s) and CV L (s) and Its Properties
In the following section, we obtain certain inclusion relations and extremal functions for functions in the classes ST L (s) and CV L (s). Proof. A straightforward calculation shows that g ≡ (L s − 1)/(2s) satisfies In order to prove the second part of lemma, denote for θ ∈ [0, 2π) the function Q(θ) := L s re iθ = 1 + s 2 r 2 + 2sr cos θ, for some 0 < r < 1 and s > 0. It is easy to see that Q attains its minimum at θ = π and maximum at θ = 0, and for s < 0 attains its minimum at θ = 0 and maximum at θ = π.
From Lemma 2 it can be seen that the smallest disk with center (1, 0) that contains L s (z) and the largest disk with center at (1, 0) contained in L s (z) are the following (see Figure 4) Taking into account the properties of a function L s given in Theorem 1 and Lemma 2, we see that for 0 < s ≤ 1/ √ 2, the function L s ∈ P * (see also Figure 5). Additionally, those properties allow to formulate the following definition.
It is clear that P (L s ) is a subfamily of the well-known Carathéodory class P = P ((1 + z)/(1 − z)) of normalized functions in D with positive real part.
On the basis of the relationship between subclasses of the Carathéodory class and the notion of classical starlikeness and convexity we also define the following classes.
These functions are extremal for several problems in the class ST L (s) (see Figure 6). For instance, we have For a function h ∈ A, we have the equivalence: h ∈ CV L (s) if and only if zh (z)/h (z) ≺ L s (z). This gives the structural formula for functions in CV L (s). A function h is in the class CV L (s) if and only if there exists an analytic function p with p ∈ P (L s ), such that This above integral representation supply many examples of functions in class CV L (s). Let p(z) = L s (z n ) ∈ CV L (s), then the functions (see Figure 7) K s,n (z) = z 0 exp 2s n t n + s 2 2n t 2n dt = z + 2s n(n + 1) for some n ≥ 1 are extremal functions for several problems in the class CV L (s). For n = 1 we have From Equation (9), a function f ∈ A is in ST L (s) if and only if Thus we have the following result.

Conclusions
The paper presents exhaustive characteristics of the curve called limaçon of Pascal, taking into account various parameters. Families of convex and starlike functions associated with the limaçon of Pascal, for which standard functionals are located in the domains bounded by the limaçon curve. Examples and properties of extremal functions in defined families were also presented.