Repeated-root constacyclic codes of length $p_1p_2^t p^s$ and their dual codes

1.   Introduction and preliminary definitions

In his survey-cum-expository review article, Srivastava ^[1] presented and motivated about brief expository overview of the classical $q$ -analysis versus the so-called $(p, q)$-analysis with an obviously redundant additional parameter $p$. We also briefly consider several other families of such extensively and widely-investigated linear convolution operators as (for example) the Dziok-Srivastava, Srivastava-Wright and Srivastava-Attiya linear convolution operators, together with their extended and generalized versions. The theory of $(p, q)$-analysis has important role in many areas of mathematics and physics. Our usages here of the $q$-calculus and the fractional $q$-calculus in geometric function theory of complex analysis are believed to encourage and motivate significant further developments on these and other related topics (see Srivastava and Karlsson [2,pp. 350-351], Srivastava ^[3,4]). Our main objective in this survey-cum-expository article is based chiefly upon the fact that the recent and future usages of the classical $q$-calculus and the fractional $q$-calculus in geometric function theory of complex analysis have the potential to encourage and motivate significant further researches on many of these and other related subjects. Jackson ^[5,6] was the first that gave some application of $q$ -calculus and introduced the $q$-analogue of derivative and integral operator (see also ^[7,8]), we apply the concept of $q$ -convolution in order to introduce and study the general Taylor-Maclaurin coefficient estimates for functions belonging to a new class of normalized analytic in the open unit disk, which we have defined here.

Let $\mathcal{A}$ denote the class of analytic functions of the form

and let $\mathcal{S}\subset \mathcal{A}$ consisting on functions that are univalent in $\Delta$. If the function $h\in \mathcal{A}$ is given by

The Hadamard product (or convolution) of $f$ and $h$, given by $\left(1.1\right)$ and $\left(1.2\right)$, respectively, is defined by

If $f$ and $F$ are analytic functions in $\Delta$, we say that $f$ is subordinate to $F$, written as $f(z)\prec F(z)$, if there exists a Schwarz function $s$, which is analytic in $\Delta$, with $s(0) = 0$, and $\left\vert s(z)\right\vert < 1$ for all $z\in \Delta$, such that $f(z) = F(s(z))$, $z\in \Delta$. Furthermore, if the function $F$ is univalent in $\Delta$, then we have the following equivalence (^[9,10])

The Koebe one-quarter theorem (see ^[11]) prove that the image of $\Delta$ under every univalent function $f\in \mathcal{S}$ contains the disk of radius $\dfrac{1}{4}$. Therefore, every function $f\in \mathcal{S}$ has an inverse $f^{-1}$ that satisfies

where

A function $f\in \mathcal{A}$ is said to be bi-univalent in $\Delta$ if both $f$ and $f^{-1}$ are univalent in $\Delta$. Let $\Sigma$ represent the class of bi-univalent functions in $\Delta$ given by (1.1). The class of analytic bi-univalent functions was first familiarised by Lewin ^[12], where it was shown that $\left\vert a_{2}\right\vert < 1.51$. Brannan and Clunie ^[13] enhanced Lewin's result to $\left\vert a_{2}\right\vert < \sqrt{2}$ and later Netanyahu ^[14] proved that $\left\vert a_{2}\right\vert < \frac{4}{3}.\ $

Note that the functions

with their corresponding inverses

are elements of $\Sigma$ (see ^[15,16]). For a brief history and exciting examples in the class $\Sigma$ (see ^[17]). Brannan and Taha ^[18] (see also ^[16]) presented certain subclasses of the bi-univalent functions class $\Sigma$ similar to the familiar subclasses $S^{\ast }\left(\alpha \right)$ and $K\left(\alpha \right)$ of starlike and convex functions of order $\alpha$ $\left(0\leq \alpha < 1\right)$, respectively (see ^[17,19,20]). Ensuing Brannan and Taha ^[18], a function $f\in \mathcal{A}$ is said to be in the class $S_{\Sigma }^{\ast }\left(\alpha \right)$ of bi-starlike functions of order $\alpha$ $\left(0 < \alpha \leq 1\right)$, if each of the following conditions are satisfied:

and

where the function $g$ is the analytic extension of $f^{-1}$ to $\Delta$, given by

A function $f\in \mathcal{A}$ is said to be in the class $K_{\Sigma }\left(\alpha \right)$ of bi-convex functions of order $\alpha$ $\left(0 < \alpha \leq 1\right)$, if each of the following conditions are satisfied:

and

The classes $S_{\Sigma}^{\ast}\left(\alpha\right)$ and $K_{\Sigma}\left(\alpha\right)$ of bi-starlike functions of order $\alpha$ and bi-convex functions of order $\alpha$ $\left(0 < \alpha\leq 1\right)$, corresponding to the function classes $S^{\ast}\left(\alpha\right)$ and $K\left(\alpha\right)$, were also introduced analogously. For each of the function classes $S_{\Sigma}^{\ast}\left(\alpha\right)$ and $K_{\Sigma}\left(\alpha\right)$, they found non-sharp estimates on the first two Taylor-Maclaurin coefficients $\left\vert a_{2}\right\vert$ and $\left\vert a_{3}\right\vert$ (^[16,18]).

1.1.   Faber polynomial expansion of functions $f\in \mathcal{A}$

The Faber polynomials introduced by Faber ^[21] play an important role in various areas of mathematical sciences, especially in Geometric Function Theory of Complex Analysis (see, for details, ^[22]). In 2013, Hamidi and Jahangiri ^[23,24,25] took a new approach to show that the initial coefficients of classes of bi- starlike functions e as well as provide an estimate for the general coefficients of such functions subject to a given gap series condition.Recently, their idea of application of Faber polynomials triggered a number of related publications by several authors (see, for example, ^[26,27,28] and also references cited threin) investigated some interesting and useful properties for analytic functions. Using the Faber polynomial expansion of functions $f\in \mathcal{A}$ has the form (1.1), the coefficients of its inverse map may be expressed as

where

such that $U_{i}\ $with $7\leq i\leq m$ is a homogeneous polynomial in the variables $a_{2}, a_{3}, ..., a_{m},$ In particular, the first three terms of $\mathcal{K}_{m-1}^{-m}$ are

In general, an expansion of $\mathcal{K}_{m}^{-n}$ $(n\in \mathbb{N})$ is (see ^{[29,30,31,32,33]})

where $\mathcal{D}_{m}^{n} = \mathcal{D}_{m}^{n}(a_{2}, a_{3}, ...)$ and

while $a_{1} = 1$ and the sum is taken over all non-negative integers $i_{1}...i_{m}$ satisfying

Evidently

1.2.   Quantum calculus operator

Srivastava ^[1] made use of several operators of $q$-calculus and fractional $q$-calculus and recollecting the definition and representations. The $q$-shifted factorial is defined for $\kappa, q\in \mathbb{C}$ and $n\in \mathbb{N}_{0} = \mathbb{N}\cup \{0\}$ as follows

By using the $q$-Gamma function $\Gamma _{q}(z),$ we get

where (see ^[34])

Also, we note that

and, the $q$-Gamma function $\Gamma _{q}(z)$ is known

where $\left[m\right] _{q}$ symbolizes the basic $q$-number defined as follows

Using the definition formula (1.7) we have the next two products:

${\rm{(i)}}$ For any non-negative integer $m$, the $q$-shifted factorial is given by

${\rm{(ii)}}$ For any positive number $r$, the $q$-generalized Pochhammer symbol is defined by

It is known in terms of the classical (Euler's) Gamma function $\Gamma \left(z\right)$, that

Also, we observe that

where $\left(\kappa \right) _{m}$ is the familiar Pochhammer symbol defined by

For $0 < q < 1$, the $q$-derivative operator (or, equivalently, the $q$- difference operator) El-Deeb et al. ^[35] defined $D_{q}\ $for $f\ast h$ given by $\left(1.3\right)$ is defined by (see ^[5,6])

where, as in the definition (1.7)

For $\kappa > -1$ and $0 < q < 1$, El-Deeb et al. ^[35] (see also) defined the linear operator $\mathcal{H}_{h}^{\kappa, q}:\mathcal{A} \rightarrow \mathcal{A}$ by

where the function $\mathcal{M}_{q, \kappa +1}$ is given by

A simple computation shows that

From the definition relation (1.9), we can easily verify that the next relations hold for all $f\in \mathcal{A}$:

Remark 1. Taking precise cases for the coefficients $b_{m}$ we attain the next special cases for the operator $\mathcal{H}_{h}^{\kappa, q}$:

(ⅰ) For $b_{m} = 1,$ we obtain the operator $\mathcal{I}_{q}^{\kappa }$ defined by Srivastava ^[32] and Arif et al. ^[36] as follows

(ⅱ) For $b_{m} = \dfrac{(-1)^{m-1}\Gamma (\upsilon +1)}{4^{m-1}(m-1)!\Gamma (m+\upsilon)}$, $\upsilon > 0$, we obtain the operator $\mathcal{N} _{\upsilon, q}^{\kappa }$ defined by El-Deeb and Bulboacă ^[37] and El-Deeb ^[38] as follows

where

(ⅲ) For $b_{m} = \left(\dfrac{n+1}{n+m}\right) ^{\alpha }$, $\alpha > 0$, $n\geq 0$, we obtain the operator $\mathcal{M}_{n, q}^{\kappa, \alpha }$ defined by El-Deeb and Bulboacă ^[39] and Srivastava and El-Deeb ^[40] as follows

(ⅳ) For $b_{m} = \dfrac{\rho ^{m-1}}{(m-1)!}e^{-\rho }$, $\rho > 0$, we obtain the $q$-analogue of Poisson operator defined by El-Deeb et al. ^[35] (see ^[41]) as follows

(ⅴ) For $b_{m} = \left[\dfrac{1+\ell +\mu (m-1)}{1+\ell }\right] ^{n}$, $n\in \mathbb{Z}$, $\ell \geq 0$, $\mu \geq 0$, we obtain the $q$-analogue of Prajapat operator defined by El-Deeb et al. ^[35] (see also ^[42]) as follows

(ⅵ) For $b_{m} = \left(\begin{array}{c} n+m-2 \\ m-1 \end{array} \right) \theta ^{m-1}\left(1-\theta \right) ^{n}\, \ n\in \mathbb{N}, \ 0\leq \theta \leq 1, \ $we obtain the $q$-analogue of the Pascal distribution operator defined by Srivastava and El-Deeb ^[28] (see also ^[35,43,44]) as follows

The purpose of the paper is to present a new subclass of functions $\mathcal{ L}_{\Sigma }^{q, \kappa }\left(\eta; h;\Phi \right)$ of the class $\Sigma$, that generalize the previous defined classes. This subclass is defined with the aid of a general $\mathcal{H}_{h}^{\kappa, q}$ linear operator defined by convolution products composed with the aid of $q$-derivative operator. This new class extend and generalize many preceding operators as it was presented in Remark 1, and the main goal of the paper is find estimates on the coefficients $\left\vert a_{2}\right\vert$, $\left\vert a_{3}\right\vert$, and for the Fekete-Szegö functional for functions in these new subclasses. These classes will be introduced by using the subordination and the results are obtained by employing the techniques used earlier by Srivastava et al. ^[16]. This last work represents one of the most important study of the bi-univalent functions, and inspired many investigations in this area including the present paper, while many other recent papers deals with problems initiated in this work, like ^{[33,44,45,46,47,48]}, and many others. Inspired by the work of Silverman and Silvia ^[49] (also see^[50]) and recent study by Srivastava et al ^[51], in this article, we define the following new subclass of bi-univalent functions $\mathcal{M}_{\Sigma }^{q, \kappa }\left(\varpi, \vartheta, h\right)$ as follows:

Definition 1. Let $\varpi \in(-\pi, \pi]$ and let the function $f\in \Sigma$ be of the form (1.1) and $h$ is given by (1.2), the function $f$ is said to be in the class $\mathcal{ M}_{\Sigma }^{q, \kappa }\left(\varpi, \vartheta, h\right)$ if the following conditions are satisfied:

and

with $\kappa > -1$, $0 < q < 1$, $0\leq \vartheta < 1\ $and $z, w\in \Delta$, where the function $g$ is the analytic extension of $f^{-1}$ to $\Delta$, and is given by (1.4).

Definition 2. Let $\varpi = 0$ and let the function $f\in \Sigma$ be of the form (1.1) and $h$ is given by (1.2), the function $f$ is said to be in the class $\mathcal{M}_{\Sigma }^{q, \kappa }\left(\vartheta, h\right)$ if the following conditions are satisfied:

and

with $\kappa > -1$, $0 < q < 1$, $0\leq \vartheta < 1\ $and $z, w\in \Delta$, where the function $g$ is the analytic extension of $f^{-1}$ to $\Delta$, and is given by (1.4).

Definition 3. Let $\varpi = \pi$ and let the function $f\in \Sigma$ be of the form (1.1) and $h$ is given by (1.2), the function $f$ is said to be in the class $\mathcal{HM}_{\Sigma }^{q, \kappa }\left(\vartheta, h\right)$ if the following conditions are satisfied:

with $\kappa > -1$, $0 < q < 1$, $0\leq \vartheta < 1\ $and $z, w\in \Delta$, where the function $g$ is the analytic extension of $f^{-1}$ to $\Delta$, and is given by (1.4).

Remark 2. (ⅰ) Putting $q\rightarrow 1^{-}$ we obtain that $\lim\limits_{q\rightarrow 1^{-}}\mathcal{M}_{\Sigma }^{q, \kappa }\left(\varpi, \vartheta; h\right) = :\mathcal{G}_{\Sigma }^{\kappa }\left(\varpi, \vartheta; h\right)$, where $\mathcal{G}_{\Sigma }^{\kappa }\left(\varpi, \vartheta; h\right)$ represents the functions $f\in \Sigma$ that satisfy (1.18) and (1.19) for $\mathcal{H}_{h}^{\kappa, q}$ replaced with $\mathcal{I}_{h}^{\kappa }$ (1.10).

(ⅱ) Fixing $b_{m} = \dfrac{(-1)^{m-1}\Gamma (\upsilon +1)}{ 4^{m-1}(m-1)!\Gamma (m+\upsilon)}$, $\upsilon > 0$, we obtain the class $\mathcal{B}_{\Sigma }^{q, \kappa }\left(\varpi, \vartheta, \upsilon \right)$, that represents the functions $f\in \Sigma$ that satisfy (1.18) and (1.19) for $\mathcal{H}_{h}^{\kappa, q}$ replaced with $\mathcal{N}_{\upsilon, q}^{\kappa }$ (1.12).

(ⅲ) Taking $b_{m} = \left(\dfrac{n+1}{n+m}\right) ^{\alpha }$, $\alpha > 0$, $n\geq 0$, we obtain the class $\mathcal{L}_{\Sigma }^{q, \kappa }\left(\varpi, \vartheta, n, \alpha \right)$, that represents the functions $f\in \Sigma$ that satisfy (1.18) and (1.19) for $\mathcal{H} _{h}^{\kappa, q}$ replaced with $\mathcal{M}_{n, q}^{\kappa, \alpha }$ (1.14).

(ⅳ) Fixing $b_{m} = \dfrac{\rho ^{m-1}}{(m-1)!}e^{-\rho }$, $\rho > 0$, we obtain the class $\mathcal{M}_{\Sigma }^{q, \kappa }\left(\varpi, \vartheta, \rho \right)$, that represents the functions $f\in \Sigma$ that satisfy (1.18) and (1.19) for $\mathcal{H}_{h}^{\kappa, q}$ replaced with $\mathcal{I}_{q}^{\kappa, \rho }$ (1.15).

(ⅴ) Choosing $b_{m} = \left[\dfrac{1+\ell +\mu (m-1)}{1+\ell }\right] ^{n}$, $n\in \mathbb{Z}$, $\ell \geq 0$, $\mu \geq 0$, we obtain the class $\mathcal{ M}_{\Sigma }^{q, \kappa }\left(\varpi, \vartheta, n, \ell, \mu \right)$, that represents the functions $f\in \Sigma$ that satisfy (1.18) and (1.19) for $\mathcal{H}_{h}^{\kappa, q}\ $replaced with $\mathcal{J}_{q, \ell, \mu }^{\kappa, n}$ (1.16).

2.   Coefficient bounds for $f\in\mathcal{M}_{\Sigma }^{q, \kappa }\left(\varpi, \vartheta; h\right)$

Throughout this paper, we assume that

Recall the following Lemma which will be needed to prove our results.

Lemma 1. (Caratheodory Lemma ^[11]) If $\phi \in \mathcal{P}$ and $\phi (z) = 1+\sum_{n = 1}^{\infty }c_{n}z^{n}$ then $\left\vert c_{n}\right\vert \leq 2$ for each $n$, this inequality is sharp for all $n$ where $\mathcal{P}$ is the family of all functions $\phi$ analytic and having positive real part in $\Delta$ with $\phi (0) = 1$.

We firstly introduce a bound for the general coefficients of functions belong to the class $\mathcal{M}_{\Sigma }^{q, \kappa }\left(\varpi, \vartheta; h\right).$

Theorem 2. Let the function $f$ given by $(1.1)$ belongs to the class $\mathcal{M}_{\Sigma }^{q, \kappa }\left(\varpi, \vartheta; h\right)$. If $a_{k} = 0$ for $2\leq k\leq m-1$, then

Proof. If $f\in \mathcal{M}_{\Sigma }^{q, \kappa }\left(\varpi, \vartheta; h\right)$, from (1.18), (1.19), we have

and

Since

we know that there are two positive real part functions:

and

where

so that

and

Using (2.1) and comparing the corresponding coefficients in (2.3), we obtain

and similarly, by using (2.2) in the equality (2.4), we have

under the assumption $a_{k} = 0$ for $0\leq k\leq m-1$, we obtain $A_{m} = -a_{m}$ and so

and

Taking the absolute values of (2.7) and (2.8), we conclude that

Applying the Caratheodory Lemma 1, we obtain

which completes the proof of Theorem.

Theorem 3. Let the function $f$ given by (1.1) belongs to the class $\mathcal{M}_{\Sigma }^{q, \kappa }\left(\varpi, \vartheta; h\right)$, then

and

Proof. Fixing $m = 2\ $and $m = 3$ in (2.5), (2.6), we have

and

From (2.12) and (2.14), by using the Caratheodory Lemma1, we obtain

Also, from (2.13) and (2.15), we have

and by using the Caratheodory Lemma 1, we obtain

From (2.16) and (2.18), we obtain the desired estimate on the coefficient as asserted in (2.9).

To find the bound on the coefficient $\left\vert a_{3}\right\vert,$ we subtract (2.15) from (2.13). we get

or

substituting the value of $a_{2}^{2}\ $from (2.12) into (2.19), we obtain

Using the Caratheodory Lemma 1, we find that

and from (2.13), we have

Appling the Caratheodory Lemma 1, we obtain

Combining (2.20) and (2.21), we have the desired estimate on the coefficient $\left\vert a_{3}\right\vert$ as asserted in (2.10).

Finally, from (2.15), we deduce that

Thus the proof of Theorem 3 was completed.

3.   Fekete-Szegö inequality for $f\in\mathcal{M}_{\Sigma }^{q, \kappa }\left(\varpi, \vartheta; h\right)$

Fekete and Szegö ^[52] introduced the generalized functional $|a_{3}-\aleph a_{2}^{2}|,$ where $\aleph$ is some real number. Due to Zaprawa ^[53], (also see ^[54]) in the following theorem we determine the Fekete-Szegö functional for $f\in \mathcal{M}_{\Sigma }^{q, \kappa }\left(\varpi, \vartheta; h\right)$.

Theorem 4. Let the function $f$ given by (1.1) belongs to the class $\mathcal{M}_{\Sigma }^{q, \kappa }\left(\varpi, \vartheta; h\right)$ and $\aleph\in\mathbb{R}$. Then we have

Proof. From (2.17) and (2.19)we obtain

So we have

Then, by taking modulus of (3.1), we conclude that

Taking $\aleph = 1$, we have the following result.

4.   Conclusions and observations

In the current paper, we mainly get upper bounds of the initial Taylors coefficients of bi-univalent functions related with $q-$ calculus operator. By fixing $b_m$ as demonstrated in Remark 1, one can effortlessly deduce results correspondents to Theorems 2 and 3 associated with various operators listed in Remark 1. Further allowing $q\rightarrow 1^{-}$ as itemized in Remark 2 we can outspread the results for new subclasses stated in Remark 2. Moreover by fixing $\varpi = 0$ and $\varpi = \pi$ in Theorems 2 and 3, we can easily state the results for $f\in\mathcal{M}_{\Sigma }^{q, \kappa }\left(\vartheta; h\right)$ and $f\in\mathcal{HM}_{\Sigma }^{q, \kappa }\left(\vartheta; h\right)$. Further by suitably fixing the parameters in Theorem 4, we can deduce Fekete-Szegö functional for these function classes. By using the subordination technique, we can extend the study by defining a new class

where $\Psi(z)$ the function $\Psi$ is an analytic univalent function such that $\Re \; \left(\Psi\right) > 0 \; \; \mathrm{in }\; \; \Delta$ with $\Psi(0) = 1, \; \Psi^{\prime }(0) > 0$ and $\Psi$ maps $\Delta$ onto a region starlike with respect to 1 and symmetric with respect to the real axis and is given by $\Psi(z) = z+B_1z+B_2z^2+B_3z^3+\cdots, (B_1 > 0).$ Also, motivating further researches on the subject-matter of this, we have chosen to draw the attention of the interested readers toward a considerably large number of related recent publications (see, for example, ^[1,2,4]). and developments in the area of mathematical analysis. In conclusion, we choose to reiterate an important observation, which was presented in the recently-published review-cum-expository review article by Srivastava (^[1], p. 340), who pointed out the fact that the results for the above-mentioned or new $q-$ analogues can easily (and possibly trivially) be translated into the corresponding results for the so-called $(p; q)-$analogues(with $0 < |q| < p \leq 1$)by applying some obvious parametric and argument variations with the additional parameter $p$ being redundant.

Acknowledgments

The researcher(s) would like to thank the Deanship of Scientific Research, Qassim University for funding the publication of this project.The authors are grateful to the reviewers for their valuable remarks, comments, and advices that help us to improve the quality of the paper.

Conflict of interest

The authors declare that they have no competing interests.