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

1.   Introduction

Recall that the complex valued mapping $\mathcal{L}_{\alpha, \beta, \gamma}: \mathbb{D}\rightarrow \mathbb{C}$ is defined by

where

for $0\leq\alpha, \beta\leq1\; (\alpha\beta\neq\pm1)$ and $0\leq\gamma < 1$, whose image is in the domain $\Delta(\alpha, \beta, \gamma)$ with the boundary

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 $M_{1} = 2-2\gamma$ and $M_{2} = (2-2\gamma)(\alpha+\beta).$

Define by $\mathcal{A}$ the class of analytic functions $f$ which are expanded with the Taylor-Maclaurin's series

in the open unit disk $\mathbb{D} = \{z\in\mathbb{C}: \mid z\mid < 1\}$. Further, if $f\in\mathcal{A}$ is univalent in $\mathbb{D}$, then we denote such class of functions by $\mathcal{S}$. For two analytic functions $F$ and $G$ in $\mathbb{D}$, if there has a Schwarz function $\omega\in\Omega$ with $\omega(0) = 0$ and $\mid \omega(z)\mid < 1$ for $z\in\mathbb{D},$ such that $F(z) = G(\omega(z))$, then $F$ is subordinate to $G$ in $\mathbb{D}$, i.e. $F\prec G$. In addition, if $G\in\mathcal{S}$, then there exists the following equivalent relation ^[20]:

Besides, if $\omega(z) = z$, then $F$ is majorized by $G$ in $\mathbb{D}$, i.e. $F\ll G$.

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 $q$-derivative operator $\mathcal{D}_{q}$ 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 $f\in\mathcal{A}$, its $q$-derivative or the $q$-difference $\mathcal{D}_{q}f(z)$ is given by

where the $q$-derivative operator $\mathcal{D}_{q}f(z)$ (refer to ^[15]) of $f$ is defined by

provided that $f'(0)$ exists, and the $q$-number $[n]_{q}$ is exactly $[\varsigma]_{q}$ when $\varsigma = n\in\mathbb{N}$, with

Note that $\mathcal{D}_{q}f(z)\rightarrow f'(z)$ when $q\rightarrow1^{-}$, where $f'$ is the ordinary derivative of $f$.

For $f\in\mathcal{A}$, we recall the symmetric determinant matrix $\mathcal{T}_{j}(n)$ given in ^[9] as below:

Here, we also point out that

and

Let the image of the analytic function $g(z) = \sum^{\infty}_{n = 0}b_{n}z^{n}$ in $\mathbb{D}$ belong to $\Omega\subseteq\mathbb{D}$. In 1914, Harald Bohr ^[10] showed that the inequality $\sum^{\infty}_{n = 0}\vert b_{n}z^{n}\vert\leq1$ in the disc $\mathbb{D}_{\delta} = \{z| \vert z\vert\leq \delta\}$ with $\delta\geq\frac{1}{6}$. Because of the works of Weiner, Riesz and Schur, we know that $r^{*} = \frac{1}{3}$ 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 $\vert z\vert < r_{\Omega}$ with respect to the Euclidean distance $d$. The largest radius $r_{\Omega}$ 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. 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 $q$-derivative operator.

Definition 1.1. Let $\mathcal{L}_{\alpha, \beta, \gamma}$ be given by (1.1). A function $f\in\mathcal{A}$ is said to be in the class $\mathcal{SP}^{q}_{snail}(\lambda; \alpha, \beta, \gamma)$ if the following subordination:

holds for $z\in\mathbb{D}$, where $0\leq\lambda\leq1$.

Note that by specializing the parameter $\lambda$, we get the reduced classes below:

$(1)$ $\mathcal{SP}^{q}_{snail}(0;\alpha, \beta, \gamma)\equiv\mathcal{SP}^{q}_{snail}(\alpha, \beta, \gamma) = \{f\in \mathcal{A}: \frac{z\mathcal{D}_{q}f(z)}{f(z)}\prec1+\mathcal{L}_{\alpha, \beta, \gamma}(z)\}$,

$(2)$ $\mathcal{SP}^{q}_{snail}(1;\alpha, \beta, \gamma)\equiv\mathcal{RP}^{q}_{snail}(\alpha, \beta, \gamma) = \{f\in \mathcal{A}: \mathcal{D}_{q}f(z)\prec1+\mathcal{L}_{\alpha, \beta, \gamma}(z)\}$.

Denote by $\mathcal{P}$ the class of all analytic and univalent functions $h(z)$ of the following form:

satisfying $\Re\; [h(z)] > 0$ and $h(0) = 1$. To proceed our results, we are ready for some indispensable Lemmas given below.

Lemma 1.1. ^[13] Let $h(z)\in\mathcal{P}$. Then the sharp estimates

are true. In particular, theequality holds for all $n$ for the following function:

Lemma 1.2. ^[19] If $h(z)\in\mathcal{P}$, then, for any complex $\kappa \in \mathbb{C}$,

and the sharp result holds for the functions

Lemma 1.3. ^[19] Assume that the function $h(z)\in\mathcal{P}$ and $\kappa\in\mathbb{R}$. Then

For $\kappa < 0$ or $\kappa > 1$, the inequality holds literally ifand only if $h(z) = \frac{1+z}{1-z}$ or one of its rotations. If $0 < \kappa < 1$, the inequality holds literally if and only if $h(z) = \frac{1+z^{2}}{1-z^{2}}$ or one of its rotations. Inparticular, if $\kappa = 0$, then the sharp result holds for thefollowing function:

or one of its rotations. If $\kappa = 1$, then the sharp resultholds for the following function:

or one of its rotations. If $0 < \kappa < 1$, then the upper bound issharp as follows:

and

2.   Coefficient bounds and Fekete-Szegö inequalities for $f\in{\mathcal{SP}^{q}_{snail}(\lambda; \alpha, \beta, \gamma)}$

Denote the function $h\in\mathcal{P}$ by

Then, from (1.6) we derive that

such that $u(z)\in\Omega$. 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 $\mathcal{SP}^{q}_{snail}(\lambda; \alpha, \beta, \gamma)$ and establish the next theorems for the coefficient bounds and the corresponding Feteke-Szegö problems.

Theorem 2.1. If $f(z)$ given by (1.4) belongs to the class $\mathcal{SP}^{q}_{snail}(\lambda; \alpha, \beta, \gamma)$, then

and

Proof. Assume that $f\in\mathcal{SP}^{q}_{snail}(\lambda; \alpha, \beta, \gamma)$. Then, there exists a Schwarz function $u(z)\in\Omega$ so that

Since

for $f\in\mathcal{A}$, combing (2.5) and (2.6), with (2.2) we get that

and

such that

and

By Lemma 1.1 we assert that Theorem 2.1 holds true.

By taking $\lambda = 0$ in Theorem 2.1, we deduce the corollary below.

Corollary 2.1. If $f(z)$ given by (1.4) belongs to the class $\mathcal{SP}^{q}_{snail}(\alpha, \beta, \gamma)$, then

and

By taking $\lambda = 1$ in Theorem 2.1, we deduce the corollary below.

Corollary 2.2. If $f(z)$ given by (1.4) belongs to the class $\mathcal{RP}^{q}_{snail}(\alpha, \beta, \gamma)$, then

and

Theorem 2.2. If $f(z)$ given by (1.4) belongs to the class $\mathcal{SP}^{q}_{snail}(\lambda; \alpha, \beta, \gamma)$, then

holds for $\mu\in\mathbb{C}$, where

Proof. For $\mu\in\mathbb{C}$, by using (2.7) and (2.8), we infer that

where

Hence, we apply Lemma 1.2 to Eq (2.10) and show that Theorem 2.2 holds true.

Corollary 2.3. If $f(z)$ given by (1.4) belongs to the class $\mathcal{SP}^{q}_{snail}(\alpha, \beta, \gamma)$, then

holds for $\mu\in\mathbb{C}$, where

Corollary 2.4. If $f(z)$ given by (1.4) belongs to the class $\mathcal{RP}^{q}_{snail}(\alpha, \beta, \gamma)$, then

holds for $\mu\in\mathbb{C}$, where

If we let $\mu\in\mathbb{R}$, then we are based on the proof of Theorem 2.2 and Lemma 1.3 to establish the Fekete-Szegö functional inequality for $\mathcal{SP}^{q}_{snail}(\lambda; \alpha, \beta, \gamma)$.

Theorem 2.3. For $\mu\in\mathbb{R}$, if $f(z)\in\mathcal{A}$ belongs to theclass $\mathcal{SP}^{q}_{snail}(\lambda; \alpha, \beta, \gamma)$, then

where $\rho$ is the same as in Theorem 2.2,

and

In addition, we fix

Then, each of the following inequalities holds:

(A) For $\mu\in[\aleph_{1}, \aleph_{3}]$,

(B) For $\mu\in[\aleph_{3}, \aleph_{2}]$,

Proof. For $\mu\in\mathbb{R}$ and $\rho$ in Theorem 2.2, if we let $\rho\leq0$, then we know that

and obtain

Similarly, we get that for $\rho\geq1$,

and for $\rho = \frac{1}{2}$,

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 $\mu\in\mathbb{R}$, if $f(z)\in\mathcal{A}$ belongs to theclass $\mathcal{SP}^{q}_{snail}(\alpha, \beta, \gamma)$, then

where $\rho$ is the same as in Corollary 2.3,

and

In addition, we fix

Then, each of the following inequalities holds:

(A) For $\mu\in[\aleph_{1}, \aleph_{3}]$,

(B) For $\mu\in[\aleph_{3}, \aleph_{2}]$,

Corollary 2.6. For $\mu\in\mathbb{R}$, if $f(z)\in\mathcal{A}$ belongs to theclass $\mathcal{RP}^{q}_{snail}(\alpha, \beta, \gamma)$, then

where $\rho$ is the same as in Corollary 2.4,

In addition, we put

Then, each of the following inequalities holds:

(A) For $\mu\in[\aleph_{1}, \aleph_{3}]$,

(B) For $\mu\in[\aleph_{3}, \aleph_{2}]$,

3.   Symmetric Toeplitz determinants for $\mathcal{SP}^{q}_{snail}(\lambda; \alpha, \beta, \gamma)$

Now we pay attention to the symmetric Toeplitz determinants $T_{2}(2)$ and $T_{3}(1)$ for the class $\mathcal{SP}^{q}_{snail}(\lambda; \alpha, \beta, \gamma)$. 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 $\mathcal{SP}^{q}_{snail}(\lambda; \alpha, \beta, \gamma)$, then

Corollary 3.1. If $f(z)$ given by (1.4) belongs to the class $\mathcal{SP}^{q}_{snail}(\alpha, \beta, \gamma)$, then

Corollary 3.2. If $f(z)$ given by (1.4) belongs to the class $\mathcal{RP}^{q}_{snail}(\alpha, \beta, \gamma)$, then

Based on the proof of Theorem 2.2 and Lemma 1.2, we consider the symmetric determinant $T_{3}(1)$ for $\mathcal{SP}^{q}_{snail}(\lambda; \alpha, \beta, \gamma)$ and establish the next theorem.

Theorem 3.2. If $f(z)\in\mathcal{A}$ belongs to the class $\mathcal{SP}^{q}_{snail}(\lambda; \alpha, \beta, \gamma)$, then

holds, where

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 3.3. If $f(z)\in\mathcal{A}$ belongs to the class $\mathcal{SP}^{q}_{snail}(\alpha, \beta, \gamma)$, then

holds, where

Corollary 3.4. If $f(z)\in\mathcal{A}$ belongs to the class $\mathcal{RP}^{q}_{snail}(\alpha, \beta, \gamma)$, then

holds, where

Similarly, from Theorem 2.3 and Lemma 1.3, we also estimate the symmetric determinant $T_{3}(1)$ for $\mathcal{SP}^{q}_{snail}(\lambda; \alpha, \beta, \gamma)$.

Theorem 3.3. If $f(z)\in\mathcal{A}$ belongs to the class $\mathcal{SP}^{q}_{snail}(\lambda; \alpha, \beta, \gamma)$, then

where $\rho$ is the same as in Theorem 2.2, and

Proof. Assume that $\mu = 2$ in Theorem 2.3. Then, we derive that

where $\aleph_{i}\; (i = 1, 2)$ are the same as in Theorem 2.3. If $2\leq\aleph_{1}$, then we infer that

Similarly, we see that $\aleph_{1}\leq2\leq\aleph_{2}$ and $2\geq\aleph_{2}$ are equivalent to $M_{2}-M_{1}\leq\Xi\leq M_{2}+M_{1}$ and $\Xi\geq M_{2}+M_{1}$, respectively. Moreover, together with (2.8) and the inequality (3.1) we complete the proof of Theorem 3.3.

Corollary 3.5. If $f(z)\in\mathcal{A}$ belongs to the class $\mathcal{SP}_{snail}(\alpha, \beta, \gamma)$,

where $\rho$ is the same as in Corollary 2.3, and

Corollary 3.6. If $f(z)\in\mathcal{A}$ belongs to the class $\mathcal{RP}^{q}_{snail}(\alpha, \beta, \gamma)$, then

where $\rho$ is the same as in Corollary 2.4.

4.   Bohr radius problems for $\mathcal{SP}_{snail}(\alpha, \beta, \gamma)$

Next we study the Bohr radius problems for $\mathcal{SP}_{snail}(\alpha, \beta, \gamma)$. Here, we following the methods of Allu and Halder ^[5]. Define the following function $\hbar\in\mathcal{S}$ by

Note that $\hbar$ is the same role as Kobe function for the class $\mathcal{SP}_{snail}(\alpha, \beta, \gamma)$. Now, without proof we state our results as follows.

Theorem 4.1. If $f(z)$ given by (1.4) belongs to the class $\mathcal{SP}^{q}_{snail}(\alpha, \beta, \gamma)$ and $1+\mathcal{L}_{\alpha, \beta, \gamma}(z)$ is in the Hardy class $\mathcal{H}^{2}$ of analytic functions in $\mathbb{D}$, then

for $\vert z\vert < \max\{r^{*}, 1/3\}$, where $r^{*}$ is thesmallest positive solution of

in $(0, 1)$, and $\hbar(z)$ is defined by (4.1) with

In this case, the class $\mathcal{SP}^{q}_{snail}(\alpha, \beta, \gamma)$ is said to satisfythe Bohr phenomenon.

5.   Application to functions defined by neutrosophic Poisson distribution series

From now on, by letting $\wp_{N}(z)$ 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 $\xi$ is a classical Poisson distribution of $x$ with the imprecise parameter value. A variable $\xi$ is said to have the neutrosophic Poisson distribution if its probability with the value $k\in\mathbb{N}^{*} = \mathbb{N}\cup\{0\}$ 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 $f\in\mathcal{A}$, we take the convolution operator $\ast$ to introduce the linear operator $\mathfrak{N}:\mathcal{A}\rightarrow \mathcal{A}$ defined by

where

Specially

Referring to Definition 1.1, now we introduce the new class associated with the neutrosophic Poisson distribution series.

Definition 5.1. Let $\mathcal{L}_{\alpha, \beta, \gamma}$ be given by (1.1). For $0\leq\lambda\leq1,$ a function $f\in\mathcal{A}$ is said to be in the class $\mathcal{NSP}^{q}_{snail}(\lambda; \alpha, \beta, \gamma)$ if the following subordination

holds for $z\in\mathbb{D}$, where $\mathfrak{N} f(z)$ is given by (5.1).

As the similar as Definition 1.1, we denote that

and

By applying Theorems 2.1–2.3, we can deduce the theorems below.

Theorem 5.1. If $f(z)$ given by (1.4) belongs to the class $\mathcal{NSP}^{q}_{snail}(\lambda; \alpha, \beta, \gamma)$, then

and

holds for $\mu\in\mathbb{C}$, where

Theorem 5.2. For $\mu\in\mathbb{R}$, if $f(z)\in\mathcal{A}$ belongs to theclass $\mathcal{NSP}^{q}_{snail}(\lambda; \alpha, \beta, \gamma)$, then

where $\varrho$ is the same as in Theorem 5.1,

and

In addition, we fix

Then, each of the following inequalities holds:

(A) For $\mu\in[\Upsilon_{1}, \Upsilon_{3}]$,

(B) For $\mu\in[\Upsilon_{3}, \Upsilon_{2}]$,

Similarly, by applying Theorems 3.1–3.3, we can establish the theorems below.

Theorem 5.3. If $f(z)$ given by (1.4) belongs to the class $\mathcal{NSP}^{q}_{snail}(\lambda; \alpha, \beta, \gamma)$, then

Theorem 5.4. If $f(z)\in\mathcal{A}$ belongs to the class $\mathcal{NSP}^{q}_{snail}(\lambda; \alpha, \beta, \gamma)$, then

holds, where

Theorem 5.5. If $f(z)\in\mathcal{A}$ belongs to the class $\mathcal{NSP}^{q}_{snail}(\lambda; \alpha, \beta, \gamma)$, then

where $\varrho$ is the same as in Theorem 5.4, and

6.   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 Lima\c{c}ons. 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.

Acknowledgments

This work was supported by Natural Science Foundation of Ningxia (Grant No. 2020AAC03066) and Natural Science Foundation of China (Grant Nos. 42064004; 11762016; 11571299).

Conflict of interest

The authors declare no conflicts of interest.