Generalized third-kind Chebyshev tau approach for treating the time fractional cable problem

Waleed Mohamed Abd-Elhameed; Omar Mazen Alqubori; Abdulrahman Khalid Al-Harbi; Mohammed H. Alharbi; Ahmed Gamal Atta; Waleed Mohamed Abd-Elhameed; Omar Mazen Alqubori; Abdulrahman Khalid Al-Harbi; Mohammed H. Alharbi; Ahmed Gamal Atta

doi:10.3934/era.2024288

Electronic Research Archive

2024, Volume 32, Issue 11: 6200-6224. doi: 10.3934/era.2024288

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Research article Special Issues

Generalized third-kind Chebyshev tau approach for treating the time fractional cable problem

1.
Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt
2.
Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah, Saudi Arabia
3.
Department of Mathematics, Faculty of Education, Ain Shams University, Roxy 11341, Cairo, Egypt

Received: 11 September 2024 Revised: 30 October 2024 Accepted: 13 November 2024 Published: 18 November 2024

This work introduces a computational method for solving the time-fractional cable equation (TFCE). We utilize the tau method for the numerical treatment of the TFCE, using generalized Chebyshev polynomials of the third kind (GCPs3) as basis functions. The integer and fractional derivatives of the GCPs3 are the essential formulas that serve to transform the TFCE with its underlying conditions into a matrix system. This system can be solved using a suitable algorithm to obtain the desired approximate solutions. The error bound resulting from the approximation by the proposed method is given. The numerical algorithm has been validated against existing methods by presenting numerical examples.

Keywords:

Citation: Waleed Mohamed Abd-Elhameed, Omar Mazen Alqubori, Abdulrahman Khalid Al-Harbi, Mohammed H. Alharbi, Ahmed Gamal Atta. Generalized third-kind Chebyshev tau approach for treating the time fractional cable problem[J]. Electronic Research Archive, 2024, 32(11): 6200-6224. doi: 10.3934/era.2024288

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Abstract

1. Introduction and definition

Let us consider the following second-order linear homogeneous differential equation

$\begin{equation} z^{2}w^{\prime \prime }(z)+zw^{\prime }(z)+\left( z^{2}-u^{2}\right) w(z) = 0 { \ \ \ \ \ \ }\left( u\in \mathbb{C} \right) . \end{equation}$

(1.1)

The differential equation in (1.1) is famous Bessel's differential equation. Its solution is denoted by $J_{u}(z)$ and known as Bessel function. The familiar representation of $J_{u}(z)$ is given by (1.2) and is defined by particular solution of (1.1) as follows:

$\begin{equation} J_{u}(z) = \sum\limits_{n = 0}^{\infty }\frac{\left( -1\right) ^{n}}{n!\Gamma \left( u+n+1\right) }\left( \frac{z}{2}\right) ^{2n+u}\ \ \ \ \ \ \ \left( z\in \mathbb{C} \right) , \end{equation}$

(1.2)

where $\Gamma$ is the familiar Euler Gamma function. For a comprehensive study of Bessel function of first kind, see ^[9,30].

Let $\mathcal{A}$ represents the class of all those functions which are analytic in the open unit disk

$\begin{equation*} E = \left \{ z:z\in \mathbb{C} \ \ \ \text{and }\ \ \ \left \vert z\right \vert \lt 1\right \} \end{equation*}$

and having the series expansion of the form

$\begin{equation} f(z) = z+\sum\limits_{n = 2}^{\infty }a_{n}z^{n}{ \ \ \ \ \ \ \ \ }\left( z\in E\right) . \end{equation}$

(1.3)

Let $\mathcal{S}$ be the subclass of $\mathcal{A}$ consisting the functions that is univalent in $E$ and satisfy the normalized conditions

$\begin{equation*} f\left( 0\right) = 0\ \ \ \ \ \text{and}\ \ \ \ f^{\prime }\left( 0\right) = 1. \end{equation*}$

Let two functions $f$ and $g$ are analytic in $E$ , then $f$ is subordinate to $g$ , (written as $f\prec g),$ if there exists a Schwarz function $h\left(z\right)$ , which is analytic in $E$ with

$\begin{equation*} h\left( 0\right) = 0\ \ \ \ \ \ \ \text{and} \ \ \ \ \ \left \vert h\left( z\right) \right \vert \lt 1, \end{equation*}$

such that

$\begin{equation*} f\left( z\right) = g\left( h\left( z\right) \right) . \end{equation*}$

If $g$ is univalent in $E,$ then

$\begin{equation*} f\left( z\right) \prec g\left( z\right) \Leftrightarrow \text{ }f\left( 0\right) = 0 = g(0)\text{ and }f\left( E\right) \subset g\left( E\right) . \end{equation*}$

For $f\in \mathcal{A},$ given by (1.3) and another function $g,$ given by

$\begin{equation*} g(z) = z+\sum \limits_{n = 2}^{\infty }b_{n}z^{n}, \end{equation*}$

then the Hadamard product (or convolution) of $f(z)$ and $g(z)$ is given by

$\begin{equation*} \left( f\ast g\right) (z) = z+\sum \limits_{n = 2}^{\infty }a_{n}b_{n}z^{n} = \left( g\ast f\right) (z). \end{equation*}$

In ^[34], Robertson introduced the class of starlike $(\mathcal{S}^{\ast })$ and class of convex $(\mathcal{C})$ functions and be defined as:

$\begin{equation*} \mathcal{S}^{\ast } = f\in \mathcal{A}:\Re\left( \frac{zf^{\prime }(z)}{ f(z)}\right) \gt 0 \ \text{and }\mathcal{C = }\ f\in \mathcal{A}:\Re \left( 1+\frac{zf^{\prime \prime }(z)}{f^{\prime }(z)}\right) \gt 0. \end{equation*}$

It can easily seen that

$\begin{equation*} f\in \mathcal{C}{ \ \ }\Longleftrightarrow \ zf^{^{\prime }}\in \mathcal{S}^{\ast }\text{.} \end{equation*}$

After that Srivastava and Owa investigated these subclasses in ^[43].

Let $f\in \mathcal{A}$ and $g\in \mathcal{S}^{\ast },$ is said to be close to convex $(\mathcal{K})$ functions if and only if

$\begin{equation*} \Re\left( \frac{zf^{\prime }(z)}{g(z)}\right) \gt 0. \end{equation*}$

Furthermore, Kanas and Wisniowska in ^[12] introduced subclasses of $k$ -uniformly convex $\left(k-\mathcal{UCV}\right)$ and $\left(k-\mathcal{ST} \right)$ and be defined as:

$\begin{equation*} k-\mathcal{UCV = }\left \{ f\in \mathcal{A}:\ \Re\left( 1+\frac{ zf^{\prime \prime }(z)}{f^{\prime }(z)}\right) \gt k\left \vert \frac{zf^{\prime \prime }(z)}{f^{\prime }(z)}\right \vert ,\text{ }z\in E,\text{ }k\geq 0\right \} \end{equation*}$

and

$\begin{equation*} k-\mathcal{ST = }\left \{ f\in \mathcal{A}:\ \Re\left( \frac{ zf^{\prime }(z)}{f(z)}\right) \gt k\left \vert \frac{zf^{\prime }(z)}{f(z)} -1\right \vert ,\text{ }z\in E,\text{ }k\geq 0\right \} . \end{equation*}$

Note that

$\begin{equation*} f\in k-\mathcal{UCV}\ \ \ \Longleftrightarrow \ zf^{\prime }\in k-\mathcal{ST}\text{.} \end{equation*}$

For the further developments Kanas and Srivastava studied these subclasses $\left(k-\mathcal{UCV}\right)$ and $\left(k-\mathcal{ST}\right)$ of analytic functions in ^[11]. For particular value of $k = 1$ , then $k- \mathcal{UCV = }$ $\mathcal{UCV}$ and $k-\mathcal{ST = S}^{\ast }$

Kanas and Wisniowska ^[13,14] (see also ^[11] and ^[15]) defined these subclasses of analytic functions subject to the conic domain $\Omega _{k},$ where

$\begin{equation*} \Omega _{k} = a+ib:a^{2} \gt k^{2}\left \{ \left( a-1\right) ^{2}+b^{2}\right \} , \text{ }a \gt 0, \ k\geq 0. \end{equation*}$

For $k = 0$ , the domain $\Omega _{k}$ presents the right half plane, for $0 < k < 1$ , the domain $\Omega _{k}$ presents hyperbola, for $k = 1$ its presents parabola and an ellipse for $k > 1$ .

For this conic domain, the following functions play the role of extremal functions.

$\begin{equation} p_{k}\left( z\right) = \left \{ \begin{array}{ll} \phi _{1}(z) & \text{for} \ \ k = 0, \\ \phi _{2}(z) & \text{for} \ \ k = 1, \\ \phi _{3}(z) & \text{for} \ \ 0 \lt k \lt 1, \\ \phi _{4}(z) & \text{for }\ \ k \gt 1, \end{array} \right. \end{equation}$

(1.4)

where

$\begin{equation*} \phi _{1}(z)) = \frac{1+z}{1-z}, \end{equation*}$

$\begin{equation*} \phi _{2}(z) = 1+\frac{2}{\pi ^{2}}\left( \log \frac{1+\sqrt{z}}{1-\sqrt{z}} \right) ^{2}, \end{equation*}$

$\begin{equation*} \phi _{3}(z) = 1+\frac{2}{1-k^{2}}\sinh ^{2}\left \{ \left( \frac{2}{\pi } \arccos k\right) \arctan h\sqrt{z}\right \} , \end{equation*}$

$\begin{equation*} \phi _{4}(z) = 1+\frac{1}{k^{2}-1}\, \sin \left( \frac{\pi }{2R(t)}\, \int_{0}^{\frac{y(z)}{\sqrt{t}}}\, \frac{dx}{\sqrt{1-x^{2}}\sqrt{1-t^{2}x^{2} }}\right) +\frac{1}{k^{2}-1} \end{equation*}$

and

$\begin{equation*} y(z) = \frac{z-\sqrt{t}}{1-\sqrt{t}z}\ \ \ \ \ \ \ \ \left \{ t\in (0,1)\right \} \end{equation*}$

is chosen such that

$\begin{equation*} k = \cosh \,(\pi R^{\prime }(t)/(4R(t))). \end{equation*}$

Here $R(t)$ is Legender's complete elliptic integral of first kind (see ^[13,14]).

Since the $q$ -calculus is being vastly used in different areas of mathematics and physics it is of great interest of researchers. In the study of Geometric Function Theory, the versatile applications of $q$ -derivative operator make it remarkably significant. Initially, in the year 1990, Ismail et al. ^[5] gave the idea of $q$ -starlike functions. Nevertheless, a firm foothold of the usage of the $q$ -calculus in the context of Geometric Function Theory was effectively established, and the use of the generalized basic (or $q$ -) hypergeometric functions in Geometric Function Theory was made by Srivastava (see for detail ^[37]). For the study of various families of analytic and univalent function, the quantum (or $q$ -) calculus has been used as a important tools. Jackson ^[7,8] first defined the $q$ -derivative and integral operator as well as provided some of their applications. The $q$ -Ruscheweyh differential operator was defined by Kanas and Raducanu in ^[10]. Recently, by using the concept of convolution Srivastava ^[40] introduced $q$ -Noor integral operator and studied some of its applications. Many $q$ -differential and $q$ -integral operators can be written in term of convolution, for detail we refer ^{[4,23,36,39,41]} see also ^[16,18]. Moreover, Srivastava et al. (see, for example, ^[35,44,45]) published a set of articles in which they concentrated upon the classes of $q$ -starlike functions related with the Janowski functions from several different aspects. Additionally specking, a recently-published survey-cum-expository review article by Srivastava ^[38] is potentially useful for researchers and scholars working on these topics. In this survey-cum-expository review article ^[38], the mathematical explanation and applications of the fractional $q$ -calculus and the fractional $q$ -derivative operators in Geometric Function Theory was systematically investigated. For other recent investigations involving the $q$ -calculus, one may refer to ^{[1,19,22,24,25,31,32,33]} and ^[17]. We remark in passing that, in the above-cited recently-published survey-cum-expository review article ^[38], the so-called $(p, q)$ -calculus was exposed to be a rather trivial and inconsequential variation of the classical $q$ -calculus, the additional parameter $p$ being redundant or superfluous (see, for details, [, p. 340]). In order to have a better understanding of the present article we provide some notation and concepts of quantum (or $q$ -) calculus used in this article.

Definition 1. (^[10]). Let $q\in (0, 1)$ and define the $q$ -number $[\eta ]_{q}$ as:

$\begin{eqnarray*} \left[ \eta \right] _{q} & = &\frac{1-q^{\eta }}{1-q},{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\eta \in \mathbb{C} , \\ && \\ & = &1+q+...+q^{n-1},\ \ \ \eta = n\in \mathbb{N} , \\ && \\ \left[ 0\right] & = &0,{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\eta = 0. \end{eqnarray*}$

Definition 2. Let $q\in \left(0, 1\right),$ $n\in \mathbb{N}$ and define the $q$ -factorial $[n]_{q}!$

$\begin{equation*} \left[ n\right] _{q}! = \left[ 1\right] _{q}\left[ 2\right] _{q}...\left[ n \right] _{q} \ \text{and} \ \left[ 0\right] _{q}! = 1. \end{equation*}$

Definition 3. The $q$ -generalized Pochhammer symbol $[a]_{n, q}$ be defined as:

$\begin{equation*} \left[ a\right] _{n,q} = \prod \limits_{k = 1}^{n}\left( 1-aq^{k-1}\right) , \ \ n\in \mathbb{N} \end{equation*}$

and

$\begin{equation*} \left[ a\right] _{\infty ,q} = \prod \limits_{k = 1}^{\infty }\left( 1-aq^{k-1}\right) . \end{equation*}$

Definition 4. The $q$ -Gamma function $\Gamma _{q}(n)$ is defined by

$\begin{equation*} \Gamma _{q}(n) = \frac{\left[ q,q\right] _{\infty }}{\left[ q^{n},q\right] _{\infty }}\left( 1-q\right) ^{n-1}. \end{equation*}$

The $q$ -Gamma function $\Gamma _{q}(n)$ satisfies the following functional equation

$\begin{equation*} \Gamma _{q}(n+1) = \left( \frac{1-q^{n}}{1-q}\right) \Gamma _{q}(n). \end{equation*}$

Definition 5. (^[7]). For $f\in \mathcal{A}$ , and the $q$ -derivative operator or $q$ -difference operator be defined as:

$\begin{eqnarray} D_{q}f\left( z\right) & = &\frac{f(z)-f\left( qz\right) }{\left( 1-q\right) z} {\ \ \ \ \ \ \ \ }\left( z\in E\right) , \\ && \\ D_{q}f\left( z\right) & = &1+\sum\limits_{n = 2}^{\infty }\left[ n\right] _{q}a_{n}z^{n-1} \end{eqnarray}$

(1.5)

and

$\begin{equation*} D_{q}z^{n} = \left[ n\right] _{q}z^{n-1}. \end{equation*}$

Definition 6. (^[5]). An analytic function $f\in$ $\mathcal{S}_{q}^{\ast }$ if

$\begin{equation} f(0) = f^{^{\prime }}(0) = 1, \end{equation}$

(1.6)

and

$\begin{equation} \left \vert \frac{zD_{q}f(z)}{f(z)}-\frac{1}{1-q}\right \vert \leq \frac{1}{ 1-q}, \end{equation}$

(1.7)

we can rewrite the conditions (1.7) as follows, (see ^[46]).

$\begin{equation*} \frac{zD_{q}f(z)}{f(z)}\prec \frac{1+z}{1-qz}. \end{equation*}$

Here Serivastava et al. ^[39] (see also ^[42]) defined the following definition by making use of quantum (or $q$ -) calculus, principle of subordination and general conic domain $\Omega _{k, q}$ as:

Definition 7. (^[39]). Let $k\geq 0$ and $q\in \left(0, 1\right)$ . A function $p(z)$ is said to be in the class $k-\mathcal{P}_{q}$ if and only if

$\begin{equation} p(z)\prec p_{k,q}(z) = \frac{2p_{k}(z)}{\left( 1+q\right) +\left( 1-q\right) p_{k}(z)} \end{equation}$

(1.8)

and $p_{k}(z)$ is given by (1.4).

Geometrically, the function $p(z)\in k-\mathcal{P}_{q}$ takes all values from the domain $\Omega _{k, q}$ which is defined as follows:

$\begin{equation*} \Omega _{k,q} = \left \{ w:\Re\left( \frac{\left( 1+q\right) w}{2+\left( q-1\right) w}\right) \gt k\left \vert \frac{\left( 1+q\right) w}{2+\left( q-1\right) w}-1\right \vert \right \} . \end{equation*}$

Remark 1. We see that

$\begin{equation*} k-\mathcal{P}_{q}\subseteq \mathcal{P}\left( \frac{2k}{2k+1+q}\right) \end{equation*}$

and

$\begin{equation*} \Re(p(z)) \gt \Re(p_{k,q}(z)) \gt \frac{2k}{2k+1+q}. \end{equation*}$

For $q\rightarrow 1-$ , then we have

$\begin{equation*} k-\mathcal{P}_{q} = \mathcal{P}\left( \frac{k}{k+1}\right) , \end{equation*}$

where the class $\mathcal{P}\left(\frac{k}{k+1}\right)$ introduced by Kanas and Wisniowska ^[13] and therefore,

$\begin{equation*} \Re(p(z)) \gt \Re(p_{k}(z)) \gt \frac{k}{k+1}. \end{equation*}$

Also for $k = 0$ and $q\rightarrow 1-,$ we have

$\begin{equation*} k-\mathcal{P}_{q} = \mathcal{P} \end{equation*}$

and

$\begin{equation*} \Re(p(z)) \gt 0. \end{equation*}$

Remark 2. For $q\rightarrow 1-,$ then $\Omega _{k, q} = \Omega _{k},$ where domain $\Omega _{k}$ introduced by Kanas and Wisniowska in ^[13].

By Applying $q$ -derivative operator we introduce new subclasses of $q$ -starlike functions, $q$ -convex functions, $q$ -close to convex functions and $q$ -quasi-convex functions as follows:

Definition 8. ^[42] For $f\in \mathcal{A},$ $k\geq 0,$ then $f\in k-\mathcal{ST}_{q}$ if and only if

$\begin{equation} \frac{zD_{q}f\left( z\right) }{f(z)}\prec p_{k,q}\left( z\right) . \end{equation}$

(1.9)

Definition 9. ^[42] For $f\in \mathcal{A},$ $k\geq 0,$ then $f\in k-\mathcal{UCV}_{q}$ if and only if

$\begin{equation*} \frac{D_{q}\left( zD_{q}f\left( z\right) \right) }{D_{q}f(z)}\prec p_{k,q}\left( z\right) . \end{equation*}$

It can easily seen that

$\begin{equation} f\in k-\mathcal{UCV}_{q} \ \ \ \text{ iff} \ zD_{q}f\in k-\mathcal{ST}_{q} \text{.} \end{equation}$

(1.10)

Definition 10. ^[42] For $f\in \mathcal{A},$ $k\geq 0,$ then $f\in k-\mathcal{UCC}_{q}$ if and only if

$\begin{equation*} \frac{zD_{q}f\left( z\right) }{g(z)}\prec p_{k,q}\left( z\right) , \ \text{ for some }g(z)\in k-\mathcal{ST}_{q}. \end{equation*}$

Definition 11. ^[42] For $f\in \mathcal{A},$ $k\geq 0,$ then $f\in k-\mathcal{UQV}_{q}$ if and only if

$\begin{equation*} \frac{D_{q}\left( zD_{q}f\left( z\right) \right) }{D_{q}g(z)}\prec p_{k,q}\left( z\right) , \ \text{for some }g(z)\in k-\mathcal{UCC}_{q}. \end{equation*}$

Remark 3. For $q\rightarrow 1-,$ then all theses newly defined subclasses reduces to the well-known subclasses of analytic functions introduced in ^[29].

The Jackson $q$ -Bessel functions and the Hahn-Exton $q$ -Bessel functions are, respectively, defined by

$\begin{equation*} J_{u}^{1}(z,q) = \frac{\left[ q^{u+1},q\right] _{\infty }}{\left[ q,q\right] _{\infty }}\sum\limits_{n = 1}^{\infty }\frac{\left( -1\right) ^{n}q^{n(n+u)}}{\left[ q,q\right] _{n}\left[ q^{u+1},q\right] _{n}}\left( \frac{z}{2}\right) ^{2n+u} \end{equation*}$

and

$\begin{equation*} J_{u}^{2}(z,q) = \frac{\left[ q^{u+1},q\right] _{\infty }}{\left[ q,q\right] _{\infty }}\sum\limits_{n = 1}^{\infty }\frac{\left( -1\right) ^{n}q^{\frac{1}{2} n(n+u)}}{\left[ q,q\right] _{n}\left[ q^{u+1},q\right] _{n}}z^{2n+u}, \end{equation*}$

where $z\in \mathbb{C},$ $u > -1,$ $q\in \left(0, 1\right).$ The functions $J_{u}^{1}(z, q)$ and $J_{u}^{2}(z, q)$ are the $q$ -extensions of the classical Bessel functions of the first kind. For more study about $q$ -extensions of Bessel functions (see ^[6,20,21]). Since neither $J_{u}^{1}(z, q)$ nor $J_{u}^{2}(z, q)$ belongs to $\mathcal{A}$ , first we perform normalizations of $J_{u}^{1}(z, q)$ and $J_{u}^{2}(z, q)$ as:

$\begin{eqnarray*} f_{u}^{1}(z,q) & = &2^{u}C_{u}(q)z^{1-n-u}J_{u}^{1}(z,q) \\ & = &\sum\limits_{n = 0}^{\infty }\frac{\left( -1\right) ^{n}q^{n(n+u)}}{4^{n}\left[ q,q \right] _{n}\left[ q^{u+1},q\right] _{n}}z^{n+1} \\ & = &z+\sum\limits_{n = 2}^{\infty }\frac{\left( -1\right) ^{n-1}q^{(n-1)(n-1+u)}}{ 4^{n-1}\left[ q,q\right] _{n-1}\left[ q^{u+1},q\right] _{n-1}}z^{n}. \end{eqnarray*}$

Similarly

$\begin{eqnarray*} f_{u}^{2}(z,q) & = &C_{u}(q)z^{1-n-u}J_{u}^{2}(z,q) \\ & = &\sum\limits_{n = 0}^{\infty }\frac{\left( -1\right) ^{n}q^{\frac{1}{2}n(n+u)}}{ \left[ q,q\right] _{n}\left[ q^{u+1},q\right] _{n}}z^{n+1}, \\ & = &z+\sum\limits_{n = 2}^{\infty }\frac{\left( -1\right) ^{n-1}q^{\frac{1}{2} (n-1)(n-1+u)}}{\left[ q,q\right] _{n-1}\left[ q^{u+1},q\right] _{n-1}}z^{n}, \end{eqnarray*}$

where

$\begin{equation*} C_{u}(q) = \frac{\left[ q,q\right] _{\infty }}{\left[ q^{u+1},q\right] _{\infty }}, \ z\in \mathbb{C} ,\text{ }u \gt -1,\text{ }q\in \left( 0,1\right) . \end{equation*}$

Now clearly, the functions $f_{u}^{1}(z, q)$ and $f_{u}^{2}(z, q)\in$ $\mathcal{A}.$

Now, by using the above idea of convolution and normalized Jackson and Hahn-Exton $q$ -Bessel functions, we introduce a new operators $B_{u}^{q}$ and $B_{u, 1}^{q}$ as follows:

$\begin{equation} B_{u}^{q}f(z) = f_{u}^{1}(z,q)\ast f(z) = z+\sum\limits_{n = 2}^{\infty }\varphi _{1}a_{n}z^{n} \end{equation}$

(1.11)

and

$\begin{equation} B_{u,1}^{q}f(z) = f_{u}^{2}(z,q)\ast f(z) = z+\sum\limits_{n = 2}^{\infty }\varphi _{2}a_{n}z^{n}, \end{equation}$

(1.12)

where

$\begin{equation*} \varphi _{1} = \frac{\left( -1\right) ^{n-1}q^{(n-1)(n-1+u)}}{4^{n-1}\left[ q,q \right] _{n-1}\left[ q^{u+1},q\right] _{n-1}} \end{equation*}$

and

$\begin{equation*} \varphi _{2} = \frac{\left( -1\right) ^{n-1}q^{\frac{1}{2}(n-1)(n-1+u)}}{\left[ q,q\right] _{n-1}\left[ q^{u+1},q\right] _{n-1}}. \end{equation*}$

From the definition (1.11) and (1.12), it can easy to verify that

$\begin{equation} zD_{q}\left( B_{u+1}^{q}f(z)\right) = \left( \frac{\left[ u\right] _{q}}{q^{u} }+1\right) B_{u}^{q}f(z)-\frac{\left[ u\right] _{q}}{q^{u}}B_{u+1}^{q}f(z) \end{equation}$

(1.13)

and

$\begin{equation*} zD_{q}\left( B_{u+1,1}^{q}f(z)\right) = \left( \frac{\left[ u\right] _{q}}{ q^{u}}+1\right) B_{u,1}^{q}f(z)-\frac{\left[ u\right] _{q}}{q^{u}} B_{u+1,1}^{q}f(z). \end{equation*}$

Finally Noor et al. introduced $q$ -Bernardi integral operator ^[28], which is defined by

$\begin{equation*} L_{\lambda }^{q} = L_{\lambda }^{q}f(z) = \frac{\left[ \lambda +1\right] _{q}}{ z^{\lambda }}\int \limits_{0}^{z}t^{\lambda -1}f(t)d_{q}t, \ \lambda \gt -1. \end{equation*}$

Remark 4. For $q\rightarrow 1-$ , then $L_{\lambda }^{q} = L$ , introduced by Bernardi in ^[2].

2. Preliminary results

Here we gave the generalization of two lemmas which was introduced in ^[3,27].

Lemma 1. Let $h(z)$ be an analytic and convex univalent in $E$ with

$\begin{equation*} \Re\left( vh(z)+\alpha \right) \gt 0\ \ \ \ (v,\alpha \in \mathbb{C} )\ \ \ \ {{and}} \ \ \ \ h(0) = 1. \end{equation*}$

If $p(z)$ is analytic in $E$ and $p(0) = 1,$ then

$\begin{equation} p(z)+\frac{zD_{q}p(z)}{vp(z)+\alpha }\prec h(z), \ z\in E, \end{equation}$

(2.1)

then

$\begin{equation*} p(z)\prec h(z). \end{equation*}$

Proof. Suppose that $h(z)$ is analytic and convex univalent in $E$ and $p(z)$ is analytic in $E$ . Letting $q\rightarrow 1-,$ in (2.1), we have

$\begin{equation*} p(z)+\frac{zp^{\prime }(z)}{vp(z)+\alpha }\prec h(z), \ z\in E, \end{equation*}$

then by Lemma in ^[26], we have

$\begin{equation*} p(z)\prec h(z). \end{equation*}$

Lemma 2. Let an analytic functions $p(z)$ and $g(z)$ in open unit disk $E$ with

$\begin{equation*} \Re p(z) \gt 0\ \ \ \ {{and}}\ \ \ \ h(0) = g(0). \end{equation*}$

suppose that $h(z)$ be convex functions in $E$ and let $U\geq 0,$ then

$\begin{equation} Uz^{2}D_{q}^{2}g(z)+p(z)g(z)\prec h(z) \end{equation}$

(2.2)

then

$\begin{equation*} g(z)\prec h(z), \ z\in E. \end{equation*}$

Proof. Suppose that $h(z)$ is convex in the open unit disk $E$ . Let $p(z)$ and $g(z)$ is analytic in $E$ with $\Re p(z) > 0$ and $h(0) = g(0)$ . Letting $q\rightarrow 1-,$ in (2.2), we have

$\begin{equation*} Uz^{2}g^{\prime \prime }(z)+p(z)g(z)\prec h(z), \ z\in E, \end{equation*}$

then by Lemma in ^[27], we have

$\begin{equation*} g(z)\prec h(z). \end{equation*}$

3. Main results

Theorem 1. Let $h(z)$ be convex univalent in $E$ with $\Re(h(z)) > 0$ and $h(0) = 1$ . If a function $f\in \mathcal{A}$ satisfies the condition

$\begin{equation*} \frac{zD_{q}\left( B_{u}^{q}f(z)\right) }{B_{u}^{q}f(z)}\prec h(z), \ z\in E, \end{equation*}$

then

$\begin{equation*} \frac{zD_{q}\left( B_{u+1}^{q}f(z)\right) }{B_{u+1}^{q}f(z)}\prec h(z), \ z\in E. \end{equation*}$

Proof. Let

$\begin{equation} p(z) = \frac{zD_{q}\left( B_{u+1}^{q}f(z)\right) }{B_{u+1}^{q}f(z)}. \end{equation}$

(3.1)

where $p$ is an analytic function in $E$ with $p(0) = 1.$ By using (1.13) into (3.1), we have

$\begin{equation*} p(z) = \left( \frac{\left[ u\right] _{q}}{q^{u}}+1\right) \frac{zB_{u}^{q}f(z) }{B_{u+1}^{q}f(z)}-\frac{\left[ u\right] _{q}}{q^{u}}. \end{equation*}$

Differentiating logarithmically with respect to $z$ , we have

$\begin{equation*} p(z)+\frac{zD_{q}p(z)}{p(z)+\frac{\left[ u\right] _{q}}{q^{u}}} = \frac{ zB_{u}^{q}f(z)}{B_{u+1}^{q}f(z)}. \end{equation*}$

By using Lemma $1,$ we get required result.

By taking $q\rightarrow 1-,$ in Theorem 1, then we have the following result.

Corollary 1. Let $h(z)$ be convex univalent in $E$ with $\Re(h(z) > 0$ and $h(0) = 1$ . If a function $f\in \mathcal{A}$ satisfies the condition

$\begin{equation*} \frac{z\left( B_{u}f(z)\right) ^{^{\prime }}}{B_{u}f(z)}\prec h(z), \ z\in E, \end{equation*}$

then

$\begin{equation*} \frac{z\left( B_{u+1}f(z)\right) ^{^{\prime }}}{B_{u+1}f(z)}\prec h(z), \ z\in E. \end{equation*}$

Theorem 2. Let $f\in \mathcal{A}.$ If $B_{u}^{q}f(z)\in k-\mathcal{ST}_{q},$ then $B_{u+1}^{q}f(z)\in k-\mathcal{ST}_{q}.$

Proof. Let

$\begin{equation*} p(z) = \frac{zD_{q}\left( B_{u+1}^{q}f(z)\right) }{B_{u+1}^{q}f(z)}. \end{equation*}$

From (1.13), we have

$\begin{equation*} \left( \frac{\left[ u\right] _{q}}{q^{u}}+1\right) \frac{zB_{u}^{q}f(z)}{ B_{u+1}^{q}f(z)} = p(z)+\frac{\left[ u\right] _{q}}{q^{u}}. \end{equation*}$

Differentiating logarithmically with respect to $z$ , we have

$\begin{equation*} \frac{zD_{q}B_{u}^{q}f(z)}{B_{u}^{q}f(z)} = p(z)+\frac{zD_{q}p(z)}{p(z)+\frac{ \left[ u\right] _{q}}{q^{u}}}\prec p_{k,q}\left( z\right) . \end{equation*}$

Since $p_{k, q}\left(z\right)$ is convex univalent in $E$ given by (1.8) and

$\begin{equation*} \Re\left( p_{k,q}\left( z\right) \right) \gt \frac{2k}{2k+1+q}. \end{equation*}$

The proof of the theorem 3.2 follows by Theorem 1 and condition (1.9).

For $q\rightarrow 1-,$ in Theorem 2, then we have the following result.

Corollary 2. Let $f\in \mathcal{A}.$ If $B_{u}f(z)\in k-\mathcal{ST},$ then $B_{u+1}f(z)\in k-\mathcal{ST}.$

Theorem 3. Let $f\in \mathcal{A}.$ If $B_{u}^{q}f(z)\in k-\mathcal{UCV}_{q},$ then $B_{u+1}^{q}f(z)\in k-\mathcal{UCV}_{q}.$

Proof. By virtue of (1.10), and Theorem 2, we get

$\begin{eqnarray*} B_{u}^{q}f(z) &\in &k-\mathcal{UCV}_{q}\Leftrightarrow zD_{q}\left( B_{u}^{q}f(z)\right) \in k-\mathcal{ST}_{q} \\ && \\ &\Leftrightarrow &B_{u}^{q}zD_{q}f(z)\in k-\mathcal{ST}_{q} \\ && \\ &\Rightarrow &B_{u+1}^{q}zD_{q}f(z)\in k-\mathcal{ST}_{q} \\ && \\ &\Leftrightarrow &B_{u+1}^{q}f(z)\in k-\mathcal{UCV}_{q}. \end{eqnarray*}$

Hence Theorem 3 is complete.

For $q\rightarrow 1-,$ in Theorem 3, then we have the following result.

Corollary 3. Let $f\in \mathcal{A}.$ If $B_{u}f(z)\in k-\mathcal{UCV},$ then $B_{u+1}f(z)\in k-\mathcal{UCV}.$

Theorem 4. Let $f\in \mathcal{A}.$ If $B_{u}^{q}f(z)\in k-\mathcal{UCC}_{q},$ then $B_{u+1}^{q}f(z)\in k-\mathcal{UCC}_{q}.$

Proof. Since

$\begin{equation*} B_{u}^{q}f(z)\in k-\mathcal{UCC}_{q}, \end{equation*}$

then

$\begin{equation} \frac{zD_{q}B_{u}^{q}f(z)}{B_{u}^{q}g(z)}\prec p_{k,q}(z), \ \text{ for some } \ B_{u}^{q}g(z)\in k-\mathcal{ST}_{q}. \end{equation}$

(3.2)

Letting

$\begin{equation*} h(z) = \frac{zD_{q}B_{u+1}^{q}f(z)}{B_{u+1}^{q}g(z)} \end{equation*}$

and

$\begin{equation*} H(z) = \frac{zD_{q}B_{u+1}^{q}g(z)}{B_{u+1}^{q}g(z)}. \end{equation*}$

We see that $h(z), H(z)\in \mathcal{A}$ , in $E$ with $h(0) = H(0) = 1.$ By using Theorem 2, we have

$\begin{equation*} B_{u+1}^{q}g(z)\in k-\mathcal{ST}_{q} \end{equation*}$

and

$\begin{equation*} \Re\left( H(z)\right) \gt \frac{2k}{2k+1+q}. \end{equation*}$

Also note that

$\begin{equation} zD_{q}B_{u+1}^{q}f(z) = h(z)\left( B_{u+1}^{q}g(z)\right) . \end{equation}$

(3.3)

Differentiating both sides of (3.3), we obtain

$\begin{eqnarray} \frac{zD_{q}\left( zD_{q}B_{u+1}^{q}f(z)\right) }{B_{u+1}^{q}g(z)} & = &\frac{ zD_{q}B_{u+1}^{q}g(z)}{B_{u+1}^{q}g(z)}h(z)+zD_{q}h(z) \\ & = &H(z)h(z)+zD_{q}h(z). \end{eqnarray}$

(3.4)

By using the identity (1.13), we get

$\begin{eqnarray} \frac{zD_{q}B_{u}^{q}f(z)}{B_{u}^{q}g(z)} & = &\frac{B_{u}^{q}zD_{q}f(z)}{ B_{u}^{q}g(z)} \\ & = &\frac{zD_{q}\left( B_{u+1}^{q}zD_{q}f(z)\right) +\frac{\left[ u\right] _{q}}{q^{u}}\left( B_{u+1}^{q}zD_{q}f(z)\right) }{zD_{q}\left( B_{u+1}^{q}g(z)\right) +\frac{\left[ u\right] _{q}}{q^{u}}B_{u+1}^{q}g(z)} \\ && \\ & = &\frac{\frac{zD_{q}\left( B_{u+1}^{q}zD_{q}f(z)\right) }{B_{u+1}^{q}g(z)}+ \frac{\left[ u\right] _{q}}{q^{u}}\frac{\left( B_{u+1}^{q}zD_{q}f(z)\right) }{B_{u+1}^{q}g(z)}}{\frac{zD_{q}\left( B_{u+1}^{q}g(z)\right) }{ B_{u+1}^{q}g(z)}+\frac{\left[ u\right] _{q}}{q^{u}}} \\ && \\ & = &h(z)+\frac{zD_{q}h(z)}{H(z)+\frac{\left[ u\right] _{q}}{q^{u}}}. \end{eqnarray}$

(3.5)

From (3.2), (3.4), and (3.5), we conclude that

$\begin{equation*} h(z)+\frac{zD_{q}h(z)}{H(z)+\frac{\left[ u\right] _{q}}{q^{u}}}\prec p_{k,q}\left( z\right) . \end{equation*}$

On letting $U = 0$ and $B(z) = \frac{1}{H(z)+\frac{\left[ u\right] _{q}}{q^{u}}},$ we have

$\begin{equation*} \Re\left( B(z)\right) = \frac{\Re\left( H(z)+\frac{\left[ u\right] _{q}}{q^{u}}\right) }{\left \vert H(z)+\frac{\left[ u\right] _{q}}{q^{u}} \right \vert ^{2}} \gt 0. \end{equation*}$

Apply Lemma 2, we have

$\begin{equation*} h(z)\prec p_{k,q}\left( z\right) , \end{equation*}$

where $p_{k, q}\left(z\right)$ given by (1.8). Hence Theorem 4 is complete.

We can prove Theorem 5 by using a similar argument of Theorem 4

Theorem 5. Let $f\in \mathcal{A}.$ If $B_{u}^{q}f(z)\in k-\mathcal{UQC}_{q},$ then $B_{u+1}^{q}f(z)\in k-\mathcal{UQC}_{q}.$

Now in Theorem 6, we study the closure properties of the $q$ -Bernardi integral operator $L_{\lambda }^{q}.$

Theorem 6. Let $f\in \mathcal{A}$ and $\lambda > -\left(\frac{2k}{2k+1+q} \right).$ If $B_{u}^{q}f(z)\in k-\mathcal{ST}_{q},$ then $L_{\lambda }^{q}\left(B_{u}^{q}f(z)\right) \in k-\mathcal{ST}_{q}.$

Proof. From the definition of $L_{\lambda }^{q}f(z)$ and the linearity of the operator $B_{u}^{q},$ we have

$\begin{equation} zD_{q}\left( B_{u}^{q}L_{\lambda }^{q}f(z)\right) = (1+\lambda )B_{u}^{q}f(z)-\lambda B_{u}^{q}L_{\lambda }^{q}f(z). \end{equation}$

(3.6)

Substituting $p(z) = \frac{zD_{q}\left(B_{u}^{q}L_{\lambda }^{q}f(z)\right) }{ B_{u}^{q}L_{\lambda }^{q}f(z)}$ in $\left(3.6\right)$ , we have

$\begin{equation} p(z) = (1+\lambda )\frac{B_{u}^{q}f(z)}{B_{u}^{q}L_{\lambda }^{q}f(z)}-\lambda . \end{equation}$

(3.7)

Differentiating (3.7) with respect to $z$ , we have

$\begin{eqnarray*} \frac{zD_{q}\left( B_{u}^{q}f(z)\right) }{B_{u}^{q}f(z)} & = &\frac{ zD_{q}\left( B_{u}^{q}L_{\lambda }^{q}f(z)\right) }{B_{u}^{q}L_{\lambda }^{q}f(z)}+\frac{zD_{q}p(z)}{p(z)+\lambda } \\ & = &p(z)+\frac{zD_{q}p(z)}{p(z)+\lambda }. \end{eqnarray*}$

By Lemma 1, $p(z)\prec$ $p_{k, q}\left(z\right)$ , since $\Re \left(p_{k, q}\left(z\right) +\lambda \right) > 0.$ This completes the proof of Theorem 6.

By a similar argument we can prove Theorem 7 as below.

Theorem 7. Let $f\in \mathcal{A}$ and $\lambda > -\left(\frac{2k}{2k+1+q} \right).$ If $B_{u}^{q}f(z)\in k-\mathcal{U}\mathcal{CV}_{q},$ then $L_{\lambda }^{q}\left(B_{u}^{q}f(z)\right) \in k-\mathcal{U}\mathcal{CV} _{q}.$

Theorem 8. Let $f\in \mathcal{A}$ and $\lambda > -\left(\frac{2k}{2k+1+q} \right).$ If $B_{u}^{q}f(z)\in k-\mathcal{UCC}_{q},$ then $L_{\lambda }^{q}\left(B_{u}^{q}f(z)\right) \in k-\mathcal{UCC}_{q}.$

Proof. By definition, there exists a function

$\begin{equation*} B_{u}^{q}g(z)\in k-\mathcal{ST}_{q}, \end{equation*}$

so that

$\begin{equation} \frac{zD_{q}B_{u}^{q}f(z)}{B_{u}^{q}g(z)}\prec p_{k,q}(z). \end{equation}$

(3.8)

Now from (3.6), we have

$\begin{eqnarray} \frac{zD_{q}\left( B_{u}^{q}f(z)\right) }{B_{u}^{q}g(z)} & = &\frac{ zD_{q}\left( B_{u}^{q}L_{\lambda }^{q}\left( zD_{q}f(z)\right) \right) +\lambda \left( B_{u}^{q}L_{\lambda }^{q}\left( zD_{q}f(z)\right) \right) }{ zD_{q}\left( B_{u}^{q}L_{\lambda }^{q}g(z)\right) +\lambda B_{u}^{q}L_{\lambda }^{q}\left( g(z)\right) } \\ & = &\frac{\frac{zD_{q}\left( B_{u}^{q}L_{\lambda }^{q}\left( zD_{q}f(z)\right) \right) }{B_{u}^{q}L_{\lambda }^{q}\left( g(z)\right) } +\lambda \frac{\left( B_{u}^{q}L_{\lambda }^{q}\left( zD_{q}f(z)\right) \right) }{B_{u}^{q}L_{\lambda }^{q}\left( g(z)\right) }}{\frac{zD_{q}\left( B_{u}^{q}L_{\lambda }^{q}g(z)\right) }{B_{u}^{q}L_{\lambda }^{q}\left( g(z)\right) }+\lambda }. \end{eqnarray}$

(3.9)

Since $B_{u}^{q}g(z)\in k-\mathcal{ST}_{q},$ by Theorem 6, we have $L_{\lambda }^{q}\left(B_{u}^{q}g(z)\right) \in k-\mathcal{ST}_{q}.$ Taking

$\begin{equation*} H(z) = \frac{zD_{q}\left( B_{u}^{q}L_{\lambda }^{q}g(z)\right) }{ B_{u}^{q}\left( L_{\lambda }^{q}g(z)\right) }. \end{equation*}$

We see that $H(z)\in \mathcal{A}$ in $E$ with $H(0) = 1,$ and

$\begin{equation*} \Re\left( H(z)\right) \gt \frac{2k}{2k+1+q}. \end{equation*}$

Now for

$\begin{equation*} h(z) = \frac{zD_{q}\left( B_{u}^{q}L_{\lambda }^{q}f(z)\right) }{ B_{u}^{q}\left( L_{\lambda }^{q}g(z)\right) }. \end{equation*}$

Thus we obtain

$\begin{equation} zD_{q}\left( B_{u}^{q}L_{\lambda }^{q}f(z)\right) = h(z)B_{u}^{q}\left( L_{\lambda }^{q}g(z)\right) . \end{equation}$

(3.10)

Differentiating both sides of (3.10), we obtain

$\begin{eqnarray} \frac{zD_{q}\left( B_{u}^{q}D_{q}\left( zL_{\lambda }^{q}f(z)\right) \right) }{B_{u}^{q}\left( L_{\lambda }^{q}g(z)\right) } & = &\frac{zD_{q}\left( B_{u}^{q}\left( L_{\lambda }^{q}g(z)\right) \right) }{B_{u}^{q}\left( L_{\lambda }^{q}g(z)\right) }h(z)+zD_{q}h(z) \\ & = &H(z)h(z)+zD_{q}h(z). \end{eqnarray}$

(3.11)

Therefore from (3.9) and (3.11), we obtain

$\begin{equation*} \frac{zD_{q}\left( B_{u}^{q}f(z)\right) }{B_{u}^{q}g(z)} = \frac{ zD_{q}h(z)+H(z)h(z)+\lambda h(z)}{H(z)+\lambda }. \end{equation*}$

This in conjunction with (3.8) leads to

$\begin{equation} h(z)+\frac{zD_{q}h(z)}{H(z)+\lambda }\prec p_{k,q}\left( z\right) . \end{equation}$

(3.12)

On letting $U = 0$ and $B(z) = \frac{1}{H(z)+\lambda },$ we have

$\begin{equation*} \Re\left( B(z)\right) = \frac{\Re\left( H(z)+\lambda \right) }{ \left \vert H(z)+\lambda \right \vert ^{2}} \gt 0. \end{equation*}$

Apply Lemma 2, we have

$\begin{equation*} h(z)\prec p_{k,q}\left( z\right) . \end{equation*}$

where $p_{k, q}\left(z\right)$ given by (1.8). Hence Theorem 8 is complete.

We can prove Theorem 9 by using a similar argument of Theorem 8.

Theorem 9. Let $f\in \mathcal{A}$ and $\lambda > -\left(\frac{2k}{2k+1+q} \right).$ If $B_{u}^{q}f(z)\in k-\mathcal{UQC}_{q},$ then $L_{\lambda }^{q}\left(B_{u}^{q}f(z)\right) \in k-\mathcal{UQC}_{q}.$

4. Conclusions

Our present investigation is motivated by the well-established potential for the usages of the basic (or $q$ -) calculus and the fractional basic (or $q$ -) calculus in Geometric Function Theory as described in a recently-published survey-cum-expository review article by Srivastava ^[38]. We have studied new family of analytic functions involving the Jackson and Hahn-Exton $q$ -Bessel functions and investigate their inclusion relationships and certain integral preserving properties bounded by generalized conic domain $\Omega _{k, q}$ . Also we discussed some applications of our main results by using the $q$ -Bernardi integral operator. The convolution operator $B_{u, 1}^{q}$ , which are defined by (1.12) will indeed apply to any attempt to produce the rather straightforward results which we have presented in this paper.

Basic (or $q$ -) series and basic (or $q$ -) polynomials, especially the basic (or $q$ -) hypergeometric functions and basic (or $q$ -) hypergeometric polynomials, are applicable particularly in several diverse areas (see, for example, [38, p. 328]).

Moreover, in this recently-published survey-cum-expository review article by Srivastava ^[38], the so-called $(p, q)$ -calculus was exposed to be a rather trivial and inconsequential variation of the classical $q$ -calculus, the additional parameter $p$ being redundant (see, for details, [, p. 340]). This observation by Srivastava ^[38] will indeed apply also to any attempt to produce the rather straightforward $(p, q)$ -variations of the results which we have presented in this paper.

Acknowledgements

The third author is partially supported by Universiti Kebangsaan Malaysia grant (GUP-2019-032).

Conflicts of interest

The authors declare that they have no competing interests.

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