Abstract
Quantum calculus (the calculus without limit) appeared for the first time in fluid mechanics, noncommutative geometry and combinatorics studies. Recently, it has been included into the field of geometric function theory to extend differential operators, integral operators, and classes of analytic functions, especially the classes that are generated by convolution product (Hadamard product). In this effort, we aim to introduce a quantum symmetric conformable differential operator (Q-SCDO). This operator generalized some well-know differential operators such as Sàlàgean differential operator. By employing the Q-SCDO, we present subclasses of analytic functions to study some of its geometric solutions of q-Painlevé differential equation (type III).
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1 Introduction
The conception of q-calculus model is a creative method for designs of the q-special functions. The procedure of q-calculus improves various kinds of orthogonal polynomials, operators, and special functions, which realize the form of their typical complements. The idea of q-calculus was principally realized by Carmichael [1], Jackson [2], Mason [3], and Trjitzinsky [4]. An analysis of this calculus for the early mechanism was offered by Ismail et al. [5]. Numerous integral and derivative features were formulated by using the convolution concept; for example, the Sàlàgean derivative [6], Al-Oboudi derivative (generalization of the Sàlàgean derivative) [7], and the symmetric Sàlàgean derivative [8]. It is significant to notify that the procedure of convolution finds its uses in different research, analysis, and study of the geometric properties of regular functions (see [9–11]). Here, we aim to study some geometric properties of a new quantum symmetric conformable differential operator (Q-SCDO). The classes of analytic functions are suggested by using the convolution product. The consequences are generalized classes in the open unit disk.
2 Methodology
This section provides the mathematical information that is used in this paper. Let ⋀ be the category of smooth functions given as follows:
where \(\cup=\{\xi\in\mathbb{C}: |\xi| < 1\}\).
Definition 1
Two functions ⋎1 and ⋎2 in ⋀ are said to be subordinate, denoted by \(\curlyvee_{1} \prec\curlyvee_{2}\), if we can find a Schwarz function ⊺ with \(\intercal(0)=0\) and \(|\intercal(\xi)|<1\) such that \(\curlyvee_{1}(\xi) = \curlyvee _{2}(\intercal(\xi))\), \(\xi\in\cup\) (the details can be found in [12]). Obviously, \(\curlyvee_{1}(\xi) \prec\curlyvee_{2}(\xi) \) implies \(\curlyvee_{1}(0) = \curlyvee_{2}(0) \) and \(\curlyvee_{1} (\cup) \subset \curlyvee_{2}(\cup)\). In addition, the subordinate \(\curlyvee_{1}(\xi) \prec_{r} \curlyvee _{1}(\xi)\), \(\xi\in\cup(r)\) is written by
Definition 2
For two functions ⋎1 and ⋎2 in ⋀, the Hadamard or convolution product is defined as
Definition 3
For each nonnegative integer n, the value of q-integer number, denoted by \([n]_{q} \), is defined by \([n]_{q} = \frac{1-q^{n}}{1-q}\), where \([0]_{q} =0\), \([1]_{q} =1\) and \(\lim_{q\rightarrow1^{-}} [n]_{q} =n\).
Example 2.1
\([1]_{0.5} =1\), \([2]_{0.5}=1.5\), \([3]_{0.5}=1.75\), \([2]_{0.75}=1.75\), \([3]_{0.5}=2.312\), \([2]_{0.99}= 1.99\), \([3]_{0.99}= 2.97\), \([3]_{1}= 3\).
Definition 4
The q-difference operator of ⋎ is written by the formula
Clearly, we have \(\Delta_{q} \xi^{n}= [n]_{q} \xi^{n-1}\). Consequently, for \(\curlyvee\in\bigwedge\), we have
For \(\curlyvee\in\bigwedge\), the Sàlàgean q-derivative factor [13] is formulated as follows:
where k is a positive integer.
A computation based on the definition of \(\Delta_{q}\) implies that
Obviously,
the Sàlàgean derivative factor [6].
Definition 5
Let \(\curlyvee(\xi)\in\bigwedge\), and let \(\nu\in[0,1]\) be a constant. Then Q-SCDO has the following operations:
so that \(\kappa_{1}(\nu,\xi)\neq - \kappa_{0}(\nu,\xi)\),
and
The value \(\nu=0\) indicates the Sàlàgean derivative
Moreover, the following operator can be located in [14], where
3 Convolution classes
Based on the definition (2.7), we introduce the following classes. Denote the following functions:
Thus, in terms of the convolution product, the factor (2.7) is formulated as follows:
Let ⋎ be a function from ⋀ and \(\sigma(\xi) \) be a convex univalent function in ∪ such that \(\sigma(0)=1\). The class \(\varXi^{k}_{q_{1},q_{2}}(\sigma) \) is defined by
Also, we define a special class involving the above functions when \(\nu \rightarrow0\), as follows:
When \(k=0\), we have Dziok subclass [15].
We denote by \(\mathcal{S}^{*}(\sigma)\) the class of all functions given by
and by \(\mathcal{C}^{*}(\sigma)\) the class of all functions
The following preliminary result can be found in [16, 17].
Lemma 3.1
IfKis smooth (analytic) in ∪, \(\curlyvee\in\mathcal{C}(\frac{1+\xi}{1-\xi})\)is convex and\(g \in S^{*}(\frac{1+\xi}{1-\xi})\)is starlike then
where\(\overline{\operatorname{co}}(K(\cup))\)is the closed convex hull of\(K(\cup)\).
Lemma 3.2
For analytic functions\(h, \hbar \in\cup\), the subordination\(h\prec\hbar\)implies that
where\(\xi=re^{i\theta}\), \(0< r<1\), andpis a positive number.
Some of the few studies in q-calculus are realized by comparison between two different values of calculus. Class \(\varXi^{\nu ,k}_{q_{1},q_{2}}(\sigma)\) shows the relation between the \(q_{1}\)- and \(q_{2}\)-calculus depending on the operator (2.7).
4 Inclusions
This section deals with the geometric representations of the class \(\varXi^{\nu,k}_{q_{1},q_{2}}(\sigma)\), \(q_{1}\neq q_{2}\) and their consequences.
Theorem 4.1
Let\(\curlyvee\in\bigwedge\)and let the function\(g:=\varPsi ^{k}_{q_{2}}*\curlyvee\in S^{*}(\frac{1+\xi}{1-\xi})\), \(\xi\in\cup\). If\(\curlyvee\in\varXi^{0,k}_{q_{1},q_{2}}(\sigma)\), \(q_{1}\neq q_{2}\)and the function\(\varPhi^{k}(\xi) \in\mathcal{C}(\frac{1+\xi}{1-\xi})\)then\(\curlyvee\in\varXi^{\nu,k}_{q_{1},q_{2}}(\sigma)\), \(\sigma(0)=1\).
Proof
Suppose that \(\curlyvee\in\varXi^{0,k}_{q_{1},q_{2}}(\sigma)\). This implies that there is a Schwarz function υ with \(\upsilon(0)=0\) and \(|\upsilon (\xi)|<1\) satisfying the following relation:
This leads to
By employing the convolution’s properties, we arrive at
Accordingly, by virtue of Lemma 3.1, we obtain
Since \(\sigma(\xi) \) is a convex univalent function in ∪ with \(\sigma(0)=1\), by the concept of subordination, we conclude that
which means that \(\curlyvee\in\varXi^{\nu,k}_{q_{1},q_{2}}(\sigma)\). This completes the proof. □
In this place, we note that the conclusion of Theorem 4.1 yields the following consequence:
Corollary 4.2
Let ⋎ be a function from ⋀ and\(\sigma(\xi) \)be a convex univalent function in ∪ such that\(\sigma(0)=1\). Then
In general, we have the following result:
Theorem 4.3
Let\(\curlyvee\in\bigwedge\)and let the function\(G:=\varPsi ^{k}_{q_{2}}*\varPhi^{k}_{\nu}*\curlyvee\in S^{*}(\frac{1+\xi}{1-\xi})\), \(\xi \in\cup\). If\(\rho_{1}:=\varPsi_{q_{1}}*\varPhi_{\nu}\prec_{r} \rho _{2}:=\varPsi_{q_{2}}*\varPhi_{\nu}\)for some\(r<1\)and the function\(\rho _{2} \in\mathcal{C}(\frac{1+\xi}{1-\xi})\)then
Proof
Suppose that \(\curlyvee\in\varXi^{\nu,k}_{q_{1},q_{2}}(\sigma)\). Then there is a Schwarz transform ω with \(\omega(0)=0\) and \(|\omega (\xi)|<1\) such that
This yields the following equality:
By considering the convolution’s properties, we obtain
Since \(\rho_{1}\prec_{r} \rho_{2}\), by letting \(r\rightarrow1\), we obtain \(\rho_{1}(\xi)= \rho_{2}(\xi)\). As a result, by Lemma 3.1, we deduce that
Since \(\sigma(\xi) \) is a convex univalent function in ∪ with \(\sigma(0)=1\), then by the definition of subordination, we obtain
which completes the proof. □
We note that if we replace the condition of Theorem 4.3 by \(\rho_{2} \prec_{r} \rho_{1}\) such that \(\rho_{1} \in\mathcal{C}(\frac {1+\xi}{1-\xi})\) then we obtain the same conclusion.
Theorem 4.4
Let\(\curlyvee\in\bigwedge\)and let the function\(H:=\varPsi^{k}_{q_{2}}*\varPhi^{k}_{\nu_{1}}*\curlyvee\in S^{*}(\frac{1+\xi }{1-\xi})\), \(\xi\in\cup\). If\(\varPhi^{k}_{\nu_{1}}\prec_{r} \varPhi^{k}_{\nu_{2}}\)for some\(r<1\)then
Proof
Suppose that \(\curlyvee\in\varXi^{\nu_{1},k}_{q_{1},q_{2}}(\sigma)\). Consequently, a Schwarz function ϑ exists with \(\vartheta (0)=0\) and \(|\vartheta(z)|<1\) such that
This yields
But the condition \(\varPhi^{k}_{\nu_{1}}\prec_{r} \varPhi^{k}_{\nu_{2}}\) implies that \(\varPhi^{k}_{\nu_{1}}(r \xi)= \varPhi^{k}_{\nu_{2}}(r \xi)\) (for some r). It is clear that \(\eta(\xi)=\xi\in\mathcal{C}(\frac{1+\xi}{1-\xi })\); therefore, by the convolution’s properties, we attain
Thus, in view of Lemma 3.1, we get
Since \(\sigma(\xi) \) is a convex univalent function in ∪ with \(\sigma(0)=1\), then by the definition of subordination, we obtain
which completes the proof. □
We record that if we change the condition of Theorem 4.4 by \(\varPhi^{k}_{\nu_{2}}\prec_{r} \varPhi^{k}_{\nu_{1}}\), we have
5 Integral inequalities
The following section deals with some inequalities containing the operator (2.7). For two functions \(h(\xi)=\sum a_{n}\xi^{n}\) and \(\hbar(\xi)=\sum b_{n}\xi^{n}\), we have \(h \ll\hbar\) if and only if \(|a_{n}| \leq|b_{n}|\), ∀n. This inequality is known as the majorization of two analytic functions.
Theorem 5.1
Consider the operator\([ \mathcal{S}_{\nu}^{k} ]_{q}\curlyvee(\xi)\), \(\curlyvee\in\bigwedge\). If the coefficients of ⋎ satisfy the inequality\(|\curlyvee_{n}| \leq(\frac{1}{n \nu})^{k}\), \(\nu\in (0,1)\)then
Proof
Let
Then, a straightforward computation implies that
Comparing Eq. (5.3) and the coefficients of \([ \mathcal{S}_{\nu}^{k} ]_{q}\curlyvee(\xi)\), which are satisfying
we conclude that \([ \mathcal{S}_{\nu}^{k} ]_{q}\curlyvee(\xi)\) is majorized by the function \(\sigma(\xi,\delta)\) for all \(\delta\geq1\). By the properties of majorization [18], we have
Thus, according to Lemma 3.2, we conclude that
□
In the same manner as in the proof of Theorem 5.1, one can get the next result:
Theorem 5.2
Consider the operator\([ \mathcal{S}_{\nu}^{k} ]_{q}\curlyvee(\xi)\), \(\curlyvee\in\bigwedge\). If the coefficients of ⋎ satisfy the inequality\(|\curlyvee_{n}| \leq(\frac{1}{n \nu})^{k}\), \(\nu\in (0,1)\)then
Moreover, the inequality in Theorem 5.1 can be studied in the following result:
Theorem 5.3
Consider the operator\(\mathcal{S}_{q}^{\kappa,k} \psi(z)\), \(\psi\in \varLambda\). If the coefficients ofψsatisfy the inequality\(|\vartheta_{n}| \leq(\frac{1}{n\kappa})^{k}\), \(\kappa\in(0,\infty)\)then there is a probability measureμon\((\partial U)^{2}\), for all\(\delta>1\).
Proof
Let \(\epsilon,\varepsilon\in\partial{\cup.}\) Then we have
By virtue of Theorem 1.11 in [19], the functional \(( \frac{1+\epsilon\xi}{1+\varepsilon\xi} )^{\delta}\) defines a probability measure μ in \((\partial{\cup})^{2}\) fulfilling
Then there is a constant c (diffusion constant) such that
This completes the proof. □
6 A class of differential equations
This section deals with an application of the operator (2.7) in a class of differential equations (for recent work see [20]). The class of quantum III-Painlevé differential equations has been studied recently in [21–23]. This class takes the formula
Rearranging Eq. (6.1), we have
subjected to the boundary conditions
where
Now by employing the operator (2.7), Eq. (6.2) becomes (called q-Painlevé differential equation of type III)
subjected to (6.3). Our aim is to study the geometric solution of (6.4) satisfying the boundary condition (6.3). For this purpose, we define the following analytic class:
Definition 6
For a function \(\curlyvee\in\bigwedge\) and a convex function \(\psi \in\cup\) with \(\psi(0)=0\), the function ⋎ is said to be in the class \(\mathbf{V}_{q} (\psi)\) if and only if
where \(\psi(\xi)\in\bigwedge\).
For the functions in the class \(\mathbf{V}_{q}(\psi)\), the following result holds.
Theorem 6.1
If the function\(\curlyvee\in\mathbf{V}_{q}(\psi)\)is given by (2.1), then
Proof
Let \(\curlyvee\in\mathbf{V}_{q} (\psi)\) have the expansion
Moreover, we let
Then by the definition of subordination, there is a Schwarz function ⊤ with \(\top(0)=0\) and \(|\top(\xi)|<1\) satisfying \(\mathsf {P}(\xi) = \psi(\top(\xi))\), \(\xi\in\cup\). Furthermore, if we assume that \(|\top(\xi)|=|\xi|<1\), then, in view of Schwarz lemma, there is a complex number τ with \(|\tau|=1\) satisfying \(\top(\xi)=\tau\xi\). Consequently, we obtain
It follows that
Since ψ is convex univalent in ∪, \(|\psi_{n}|\leq1\), ∀n; this implies that
Hence, the proof is complete. □
We need the following fact, which can be located in [12].
Lemma 6.2
Consider functions\(f_{1}, f_{2}, f_{3}: \cup\rightarrow\mathbb{C}\)such that\(\Re(f_{1}) \geq a\geq0\). If\(f\in\mathbb{H}[1,n]\) (the set of analytic functions having the expansion\(f(\xi)= 1+\varphi_{1}\xi+\cdots \)) and
then\(\Re(f(\xi))>0\).
Lemma 6.3
Let ♭ be convex in ∪ and suppose\(f_{1}, f_{2}, f_{3}: \cup \rightarrow\mathbb{C}\)are analytic functions such that\(\Re(f_{1}) \geq a\geq0\). If\(g \in\mathbb{H}[0,m]\) (the set of analytic functions with the expansion\(g(\xi)= g_{1}\xi^{m}+\cdots\)), \(m\geq1\)and
then\(g(\xi) \prec \flat(\xi)\).
Lemma 6.4
Let\(a,b,c \in\mathbb{R}\)be such that\(a\geq0\), \(b\geq-a\), \(c\geq-b\). If\(q \in\mathbb{H}[0,1]\), where\(q(\xi)= q_{1}\xi+\cdots\)and
then\(q(\xi) \prec \frac{\xi}{b+c}\), which is the best dominant.
Theorem 6.5
Let\(\curlyvee\in\mathbf{V}_{q}(\xi)\)and\(F(\xi)= \frac{\xi ( [ \mathcal{S}_{\nu}^{k} ]_{q}\curlyvee(\xi) )' }{ ( [ \mathcal{S}_{\nu}^{k} ]_{q}\curlyvee(\xi) )}\). If\(\Re(\xi F (\xi ))>-1\), \(\xi\in\cup\), then\([ \mathcal{S}_{\nu}^{k} ]_{q}\curlyvee\in\mathfrak{S}^{*}\) (starlike with respect to the origin).
Proof
Let \(F(\xi)=\frac{\xi ( [ \mathcal{S}_{\nu}^{k} ]_{q}\curlyvee(\xi ) )' }{ ( [ \mathcal{S}_{\nu}^{k} ]_{q}\curlyvee(\xi) )}\). Then a straightforward computation implies that
Hence, we obtain
It is clear that \(F \in\mathbb{H}[1,1]\) and
Then in view of Lemma 6.2, with \(a=0\), \(f_{1}=1\), \(f_{2}=0\) and \(f_{3}=1\), we have
that is, \([ \mathcal{S}_{\nu}^{k} ]_{q}\curlyvee\in\mathfrak{S}^{*}\) with respect to the origin. □
Theorem 6.6
Let\(\curlyvee\in\bigwedge\)and\(F(\xi)= \frac{\xi ( [ \mathcal{S}_{\nu}^{k} ]_{q}\curlyvee(\xi) )' }{[ \mathcal{S}_{\nu}^{k} ]_{q}\curlyvee(\xi)}\). If
whereψis convex in ∪, then\(\curlyvee\in\mathbf {V}_{q}(\psi)\).
Proof
Let \(\curlyvee\in\bigwedge\) and \(F(\xi)= \frac{\xi ( [ \mathcal{S}_{\nu}^{k} ]_{q}\curlyvee(\xi) )' }{[ \mathcal{S}_{\nu}^{k} ]_{q}\curlyvee(\xi)}\). As in the proof of Theorem 6.5, we have \(P(\xi)= \frac{\xi F'(\xi)}{F(\xi)}\). Then a calculation gives
Obviously, \(P\in\mathbb{H}[0,m]\), \(m=1\), and, by letting \(a=0\), \(f_{1}(\xi )=1\), and \(f_{2}(\xi)=1\), in view of Lemma 6.3, we have
Consequently, we get \(\curlyvee\in\mathbf{V}_{q}(\psi)\). □
Theorem 6.7
Let\(\curlyvee\in\bigwedge\)and\(F(\xi)= \frac{\xi ( [ \mathcal{S}_{\nu}^{k} ]_{q}\curlyvee(\xi) )' }{[ \mathcal{S}_{\nu}^{k} ]_{q}\curlyvee(\xi)}\). If
whereψis convex in ∪, then\(\curlyvee\in\mathbf {V}_{q}(\xi)\).
Proof
Let \(\curlyvee\in\bigwedge\) and \(F(\xi)= \frac{\xi ( [ \mathcal{S}_{\nu}^{k} ]_{q}\curlyvee(\xi) )' }{[ \mathcal{S}_{\nu}^{k} ]_{q}\curlyvee(\xi)}\). As in the proof of Theorem 6.5, we have \(P(\xi)= \frac{\xi F'(\xi)}{F(\xi)}\). Then a straightforward calculation gives
Obviously, \(P\in\mathbb{H}[0,1]\) and, by letting \(a=0\), \(b=1\), and \(c=0\), where \(c\geq-b\), in view of Lemma 6.4, we have
Consequently, we obtain \(\curlyvee\in\mathbf{V}_{q}(\xi)\). □
7 Conclusion
In this paper, we presented different types of integral inequalities based on q-calculus and conformable differential operator. These inequalities described the relations between the quantum conformable differential operators for different orders.
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Ibrahim, R.W., Elobaid, R.M. & Obaiys, S.J. On subclasses of analytic functions based on a quantum symmetric conformable differential operator with application. Adv Differ Equ 2020, 325 (2020). https://doi.org/10.1186/s13662-020-02788-6
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DOI: https://doi.org/10.1186/s13662-020-02788-6