On the behaviour of analytic representation of the generalized Pascal snail

We find an unifying approach to the analytic representation of the domain bounded by a generalized Pascal snail. Special cases as the Pascal snail, Both leminiscate, conchoid of the Sluze and a disc are included. The behaviour of functions related to generalized Pascal snail is studied.

The wide applications of the Pascal snail have been known since their description; the newest ones rely on the application to figure the path of airflow around object like plane wings, in the design of race and train tracks but also in cryptography for selecting the points of the curve (ellipse, leminiscate, etc.) over the prime fields. Also, the leminiscates are used in the construction of grids on irregular regions in the development of software for numerically solving partial differential equations. Very recently a method based on leminiscates is applied for meander like regions and rely on covering the region with sectors bounded by two confocal leminiscate and two arcs orthogonal to the Pascal snail (cf. [7]). In this paper we will deal with the Pascal snail (1.5) or (1.6) and its analytical representation. Also, we will discuss the special cases of (1.5) or (1.6) which give some interesting curves.
Let us consider individual cases separately. By a symmetry, from now on we make the assumption: β ≥ α, unless otherwise stated.

Halfplane
For the case when α = 0 and β = 1 the domain L(D) is the halfplane w > γ − 1. The case β = 0, α = −1 gives a halfplane w < 1 − γ which is not the ones of interest to us.

Pascal snail regions
In the case β = α ∈ (−1, 1)\{0} the function L α,α,γ becomes with 0 ≤ γ < 1, that maps the unit disk onto simply connected and bounded region, which can be described as The equation (1.8) can be rewritten in a polar equation The boundary curve, known as Pascal snail (limaçon of Pascal), is a bicircular rational plane algebraic curve of degree 4 which belongs to the family of curves called centered trochoids or epitrochoids (cf. Fig. 3. Certainly L 0,0,γ (D) is a disk). Pascal snail is the inversion of conic sections with respect to a focus.

Hippopede. Leminiscate of Booth
Here we let β = −α, α ∈ (−1, 0). In this case L α,−α,γ is of the form (1.12) We remind that the hippopede is the bicircular rational algebraic curve of degree 4, symmetric with respect to both axes. Any hippopede is the intersection of a torus with one of its tangent planes that is parallel to its axis of rotational symmetry. When c > d > 0 (that is α = 0) such a curve is known as an oval or leminiscate of Booth, see Fig. 6. Since the case d = −c does not hold, the leminiscate (1.12) do not reduce to the leminiscate of Bernoulli. We note that for c/ 2 the hippopede has a flattened segment of the boundary. Summarizing, the domain bounded by the hippopede is bounded and convex for α ∈ − 3 − 2 √ 2, 0 , and concave for α ∈ −1, − 3 − 2 √ 2 .

Remaining cases
In the remaining range of parameters, i.e. −1 < α < β < 1, not considered in previous subsections, the curve L α,β,γ (e it ) is the generalized Pascal snail, that has the form (1.14) We note, that the curve given by (1.13) has similar properties to the Pascal snail, but symmetric only with respect to real axis. It has either horizontal eight-like shape, Fig. 7 The range of the parameters α, β bean-shape, pear-shape or is convex. From this reason the region bounded by (1.13) is convex, or concave. As we can see in the Theorem 1.1 the minimum and maximum of real part are not always achieved on the real axis. Taking into account the geometrical properties of set L(D), we get the following.
The sets B 1 , B 2 are represented on a Fig. 7.
In the special cases we have

Conclusions
In general L(D) is a domain symmetric about the real axis and starlike with respect to origin and such that L(0) = 0, L (0) = 1 > 0. The geometrical properties of the regions L(D) provides a natural bridge between the convex and concave domains. We also note that such domains were discussed in relation of generalized typically-real functions and generalized Chebyshev polynomials of the second kind [4,5]. From Theorem 1.1 we conclude the following Corollary.
and 0 ≤ γ < 1, and let L α,β,γ be the function defined by (1.1). Then, for z ∈ D we have In the sequel we will use the following lemma.

Lemma 1.3 [1]
Let z is a complex number with positive real part. Then for any real number t such that t ∈ [0, 1], we have z t ≥ ( z) t .

Subclass of the Carathèodory class related to the generalized Pascal snail
Denote by P the Carathèodory class of functions i.e. P = {p : The fundamental importance of P in geometric functions theory relies on the construction of several related families of analytic functions and is well known. Hence, various subclasses of P were defined and studied. Classical cases are related to the halfplane and angular domain i.e. P(α) that denotes a subclass of P consisting of functions with real part greater than α (0 ≤ α < 1), and P γ the class with argument between −γ π/2 and γ π/2 (0 < γ ≤ 1). Also, several subfamilies of P were determined by the fact that some functionals are contained in convex subdomains of right halfplane. Therefore any subfamily of halfplane domains were considered in the context to a subfamily of P. Hence a definition of the domains related to the Pascal snail was a motivation to the definition of some subclass of P associated with such domains. To do this we first translate a domain L α,β,γ (D) with a vector (1, 0) in order to obtain a domain D α,β,γ contained in a right halfplane such that 1 ∈ D α,β,γ . The boundary of the domain D α,β,γ is then described as follows: ∂D α,β,γ = u + iv : We note that D α,β,γ is contained in a halfplane w > 1 + L 0 , where L 0 is given in Corollary 1.2. Anyway, there is substantial difference between D α,β,γ and a halfplane because D α,β,γ is not always a convex domain. However, when α = 0 and β → 1 − then D α,β,γ tends to a halfplane w > γ − 1. Thus D α,β,γ provides a natural bridge between the convex and the concave domains. Now, we define a function T α,β,γ as and where L 0 (α, β, γ ) and M 0 (α, β, γ ) are given in Corollary 1.2.
Now, we are ready to construct a class P snail (α, β, γ ) as follows

The classes ST snail (˛,ˇ, ), CV snail (˛,ˇ, ) and their properties
In this Section we give a concise presentation of some families of analytic functions related to the generalized Pascal snail T α,β,γ . We will study some subclasses of S with functions analytic and univalent in D of the form We also recall a class ST (β) ⊂ S, called starlike functions of order 0 ≤ β < 1, that consist of functions f satisfying a condition and a class CV(β), called convex functions of order 0 ≤ β < 1, with analytic condition 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 of the unit disk ω, analytic in D with ω(0) = 0 and such that f (z) = g(ω(z)). If g is univalent in D, then f ≺ g if and only if f (0) = g(0) and f (D) ⊂ g (D).
For the case αβ < 0, the critical points are θ = 0, θ = π , and the solutions of the equation We consider three separate cases, the first is α + β > 0. Then for critical points θ = 0 and the solutions of the equation (3.3) we have The second case is α + β < 0. Then for the critical points θ = π and the solutions of the equation (3.3), we obtain Finally for the case α + β = 0, the critical points are θ = 0, θ = π/2, θ = π, θ = 3π/2 and θ = 2π . For such θ we conclude ≥ 0 and this is equivalent to the range −2 + √ 3 ≤ α ≤ 2 − √ 3. Now, we define a family of functions related to the Pascal snail T α,β,γ and present various relations of that family with the previously known classes.
In the case α = β the function G(t) has the form whose behavior is similar to the behavior of G(t) for α = β. The same situation holds for H (t), α = β.
The function F defined by where α,β,γ given by (3.12), is analytic in D, Taking into account Lemma 3.1, we deduce that the function F is convex in D. Applying [14], we conclude that , and by (3.12), the required result follows.