Certain subclass of analytic functions based on q -derivative operator associated with the generalized Pascal snail and its applications

: By the principle of di ﬀ erential subordination and the q -derivative operator, we introduce the q -analog SP qsnail ( λ ; α, β, γ ) of certain class of analytic functions associated with the generalized Pascal snail. Firstly, we obtain the coe ﬃ cient estimates and Fekete-Szeg¨o functional inequalities for this class. Meanwhile, we also estimate the corresponding symmetric Toeplitz determinant. Secondly, for all the above results we provide the corresponding results for the reduced classes SP qsnail ( α, β, γ ) and RP qsnail ( α, β, γ ). Thirdly, we characterize the Bohr radius problems for the function class SP qsnail ( α, β, γ ). Lastly, we establish certain results for some new subclasses of functions deﬁned by the neutrosophic Poisson distribution series.


Introduction
Recall that the complex valued mapping L α,β,γ : D → C is defined by for 0 ≤ α, β ≤ 1 (αβ ±1) and 0 ≤ γ < 1, whose image is in the domain ∆(α, β, γ) with the boundary ∂∆(α, β, γ) = w = u + vi ∈ C : that is called the generalized Pascal snail [16] (see Figures 1 and 2). It is well known that Pascal snail is the inversion of conic sections with respect to a focus. Note that M 1 = 2 − 2γ and M 2 = (2 − 2γ)(α + β). Define by A the class of analytic functions f which are expanded with the Taylor-Maclaurin's series f (z) = z + ∞ n=2 a n z n (1.4) in the open unit disk D = {z ∈ C :| z |< 1}. Further, if f ∈ A is univalent in D, then we denote such class of functions by S. For two analytic functions F and G in D, if there has a Schwarz function ω ∈ Ω with ω(0) = 0 and | ω(z) |< 1 for z ∈ D, such that F(z) = G(ω(z)), then F is subordinate to G in D, i.e. F ≺ G. In addition, if G ∈ S, then there exists the following equivalent relation [20]: Besides, if ω(z) = z, then F is majorized by G in D, i.e. F G. Lately, the study of the q-calculus has riveted the rigorous consecration of researchers. The great attention is due to its gains in many areas of mathematics and physics. The significance of the qderivative operator D q is quite evident by its applications in the study of several subclasses of analytic functions. Initially, in 1990, Ismail et al. [14] gave the idea of q-starlike functions. Nevertheless, a firm base of the usage of the q-calculus in the context of geometric function theory was efficiently established (refer to the work by Purohit and Raina [24]), and the use of the generalized basic (or q-) hypergeometric functions in geometric function theory was made by Srivastava (see [25] for more details). After that, the extraordinary studies have been done by many mathematicians, which offer a significant part in the encroachment of geometric function theory (see [27][28][29][30][31]33]).
For f ∈ A, its q-derivative or the q-difference D q f (z) is given by [n] q a n z n−1 , 0 < q < 1, where the q-derivative operator D q f (z) (refer to [15]) of f is defined by provided that f (0) exists, and the q-number [n] q is exactly [ς] q when ς = n ∈ N, with Note that D q f (z) → f (z) when q → 1 − , where f is the ordinary derivative of f . For f ∈ A, we recall the symmetric determinant matrix T j (n) given in [9] as below: T j (n) = a n a n+1 · · · a n+ j−1 a n+1 a n · · · a n+ j . . . . . . . . . . . . a n+ j−1 a n+ j · · · a n .
Here, we also point out that Let the image of the analytic function g(z) = ∞ n=0 b n z n in D belong to Ω ⊆ D. In 1914, Harald Bohr [10] showed that the inequality ∞ n=0 |b n z n | ≤ 1 in the disc D δ = {z||z| ≤ δ} with δ ≥ 1 6 . Because of the works of Weiner, Riesz and Schur, we know that r * = 1 3 is best number and called Bohr radius that named after Niels Bohr, the founder of quantum theory [23]. Further, the corresponding inequality (the called Bohr inequality) can be denoted by for |z| < r Ω with respect to the Euclidean distance d. The largest radius r Ω is called the Bohr radius for the corresponding class. For more details for the Bohr radius and Bohr inequality, we can refer to [2,6,7,12,17].
Recently, Kanas and Masih [16] considered the analytic representation of the domain by a generalized Pascal snail. Besides, Allu and Halder [5] investigated Bohr radius for certain classes of starlike and convex univalent functions (also refer to [8,11]). Thomas et al. [9,34] studied the symmetric Toeplitz determinants for starlike and close-to-convex functions. Stimulated by recent studies [1,3,4,18,25,26], we introduce and study certain new subclasses of analytic functions associated with the generalized Pascal snail involving q-derivative operator, and obtain the corresponding upper estimates of the initial coefficients a 2 and a 3 of Taylors series and Fekete-Szegö functional inequalities for functions of the new subclasses given in Definition 1.1. In addition, we characterize the Bohr radius problems for certain reduced version of this class. Srivastava and Porwal [32] ever investigated the coefficient inequalities of Poisson distribution series in conic domain related to uniformly convex, k-spiralike and starlike functions. Along this line, Oladipo [21] estimated the bound on the first few coefficients and classical Fekete-Szegö problem for Poisson and neutrosophic Poisson distribution series connected with Chebyshev polynomials. As an application, all our results are almost generalized into the new class related with the neutrosophic Poisson distribution series. Now, by term of the unified subordination technique by Ma-Mind [19], we introduce the following subclass of analytic and univalent functions associated with the generalized Pascal snail and qderivative operator. Definition 1.1. Let L α,β,γ be given by (1.1). A function f ∈ A is said to be in the class SP q snail (λ; α, β, γ) if the following subordination: holds for z ∈ D, where 0 ≤ λ ≤ 1.
Note that by specializing the parameter λ, we get the reduced classes below: Denote by P the class of all analytic and univalent functions h(z) of the following form: satisfying [h(z)] > 0 and h(0) = 1. To proceed our results, we are ready for some indispensable Lemmas given below.
Then the sharp estimates | c n |≤ 2, n ∈ N, are true. In particular, the equality holds for all n for the following function: If h(z) ∈ P, then, for any complex κ ∈ C, Assume that the function h(z) ∈ P and κ ∈ R. Then For κ < 0 or κ > 1, the inequality holds literally if and only if h(z) = 1+z 1−z or one of its rotations. If 0 < κ < 1, the inequality holds literally if and only if h(z) = 1+z 2 1−z 2 or one of its rotations. In particular, if κ = 0, then the sharp result holds for the following function: or one of its rotations. If κ = 1, then the sharp result holds for the following function: or one of its rotations. If 0 < κ < 1, then the upper bound is sharp as follows: and

Coefficient bounds and Fekete-Szegö inequalities for
Denote the function h ∈ P by Then, from (1.6) we derive that such that u(z) ∈ Ω. By (1.1) and (2.1), we imply that Throughout our study unless otherwise stated we note that Now, we characterize the functional estimates for the class SP q snail (λ; α, β, γ) and establish the next theorems for the coefficient bounds and the corresponding Feteke-Szegö problems.
By taking λ = 0 in Theorem 2.1, we deduce the corollary below.
Hence, we apply Lemma 1.2 to Eq (2.10) and show that Theorem 2.2 holds true.
4. If f (z) given by (1.4) belongs to the class RP q snail (α, β, γ), then If we let µ ∈ R, then we are based on the proof of Theorem 2.2 and Lemma 1.3 to establish the Fekete-Szegö functional inequality for SP q snail (λ; α, β, γ). Theorem 2.3. For µ ∈ R, if f (z) ∈ A belongs to the class SP q snail (λ; α, β, γ), then where ρ is the same as in Theorem 2.2, In addition, we fix Then, each of the following inequalities holds: Proof. For µ ∈ R and ρ in Theorem 2.2, if we let ρ ≤ 0, then we know that Similarly, we get that for ρ ≥ 1, and for ρ = 1 2 , Therefore, together with Lemma 1.3 and Eqs (2.7) and (2.10), we can prove that Theorem 2.3 holds true.
Corollary 2.5. For µ ∈ R, if f (z) ∈ A belongs to the class SP q snail (α, β, γ), then where ρ is the same as in Corollary 2.3, In addition, we fix Then, each of the following inequalities holds: Corollary 2.6. For µ ∈ R, if f (z) ∈ A belongs to the class RP q snail (α, β, γ), then where ρ is the same as in Corollary 2.4, In addition, we put Then, each of the following inequalities holds: 3. Symmetric Toeplitz determinants for SP q snail (λ; α, β, γ) Now we pay attention to the symmetric Toeplitz determinants T 2 (2) and T 3 (1) for the class SP q snail (λ; α, β, γ). From (2.7) and (2.8), in view of Lemma 1.1, we easily obtain the theorem below. Theorem 3.1. If f (z) given by (1.4) belongs to the class SP q snail (λ; α, β, γ), then Corollary 3.1. If f (z) given by (1.4) belongs to the class SP q snail (α, β, γ), then Corollary 3.2. If f (z) given by (1.4) belongs to the class RP q snail (α, β, γ), then Based on the proof of Theorem 2.2 and Lemma 1.2, we consider the symmetric determinant T 3 (1) for SP q snail (λ; α, β, γ) and establish the next theorem. Theorem 3.2. If f (z) ∈ A belongs to the class SP q snail (λ; α, β, γ), then Proof. According to the definition of symmetric Toeplitz determinant, we remark that Therefore, with (2.8) we apply Theorem 2.2 into the inequality (3.1) to ensure Theorem 3.2 is true.
Corollary 3.5. If f (z) ∈ A belongs to the class SP snail (α, β, γ), where ρ is the same as in Corollary 2.3, and Corollary 3.6. If f (z) ∈ A belongs to the class RP q snail (α, β, γ), then where ρ is the same as in Corollary 2.4.

4.
Bohr radius problems for SP snail (α, β, γ) Next we study the Bohr radius problems for SP snail (α, β, γ). Here, we following the methods of Allu and Halder [5]. Define the following function ∈ S by Note that is the same role as Kobe function for the class SP snail (α, β, γ). Now, without proof we state our results as follows.
In this case, the class SP q snail (α, β, γ) is said to satisfy the Bohr phenomenon.

Application to functions defined by neutrosophic Poisson distribution series
From now on, by letting ℘ N (z) as the neutrosophic Poisson distribution series, we study the following problems. As is well known that the classical probability distributions only deal with specified data and specified parameter values, while the neutrosophic probability distribution is deeply concerned with some more general and clear ones. In fact, neutrosophic Poisson distribution of a discrete variable ξ is a classical Poisson distribution of x with the imprecise parameter value. A variable ξ is said to have the neutrosophic Poisson distribution if its probability with the value k ∈ N * = N∪{0} is where the distribution parameter m N is the expected value and the variance, that is to say, NE(x) = NV(x) = m N for the neutrosophic statistical number N = d + I (refer to [22] and the references cited). Define a power series whose coefficients are probabilities of neutrosophic Poisson distribution by For f ∈ A, we take the convolution operator * to introduce the linear operator N : E(m N , n)a n z n , Specially Referring to Definition 1.1, now we introduce the new class associated with the neutrosophic Poisson distribution series.
Definition 5.1. Let L α,β,γ be given by (1.1). For 0 ≤ λ ≤ 1, a function f ∈ A is said to be in the class NSP q snail (λ; α, β, γ) if the following subordination holds for z ∈ D, where N f (z) is given by (5.1).
For µ ∈ R, if f (z) ∈ A belongs to the class NSP q snail (λ; α, β, γ), then where is the same as in Theorem 5.1, In addition, we fix Then, each of the following inequalities holds: Similarly, by applying Theorems 3.1-3.3, we can establish the theorems below. .

Conclusions
By involving the generalized Pascal snail and q-derivative operator, certain new subclass of analytic and univalent functions can be defined to improve the classical starlike functions. In our main results, for this class we obtain the corresponding the Fekete-Szegö functional inequalities and the symmetric Toeplitz determinants as well as the bound estimates of the coefficients a 2 and a 3 . In addition, we characterize the Bohr radius problems for the reduced version of this class. Moreover, the above results are applied to the neutrosophic Poisson distribution series. Besides, some other problems like Hankel determinant, partial sum inequalities, and many more can be discussed for this class as the future work. In fact, we also replace the generalized Pascal snail by the other Limaçons. In the neutrosophic logic sense, other types of probability distributions, for example, exponential distributions, Bernoulli distributions and uniform distributions, can be studied in various classes of analytic functions.