Geometric Inequalities via a Symmetric Differential Operator Defined by Quantum Calculus in the Open Unit Disk

The present investigation covenants with the concept of quantum calculus besides the convolution operation to impose a comprehensive symmetric q-differential operator defining new classes of analytic functions. We study the geometric representations with applications. The applications deliberated to indicate the certainty of resolutions of a category of symmetric differential equations type Briot-Bouquet.


Introduction
In this effort, we deal with the structure of q-calculus, which develops an interesting technique for calculations and organizes different classes of operators and specific transformations. The significance of q-calculus appeared in a huge number of applications including physical problems. The symmetric q-activation normally achieves q-difference equations (may involve derivative). A close connection between these operators and symmetries of q -symmetric operator is accordingly to be estimated (see [1][2][3][4][5][6][7][8][9]). In recent investigation, we deliver a process for deriving and interpreting from a symmetry possessions and infirm analogy with the traditional cases. By combining the q-calculus and the symmetric Salagean differential operator, we introduce a novel modified symmetric Salagean q-differential operator. Via employing this operator, we deliver new classes of analytic functions.

Preliminaries
This section gives out the mathematical processing to deliver the suggested SDOs and complex conformable operator for some classes of analytic functions in the open unit disk ∪ = fξ ∈ ℂ : |ξ|<1g. Let ∧ be the category of smooth function elicited as pursue A function γ ∈ ∧ is known as a starlike with respect to ξ = 0 if the straight line segment combining the origin to all else point of γ embedding completely in γðξ : |ξ|<1Þ. The aim is that each point of γðξ : |ξ|<1Þ must be manifested via (0,0). A univalent function (γ ∈ Ⓢ;) is indicated to be convex in ∪ if the linear slice combining two ends of γðξ : |ξ|<1Þ stays completely in γðξ : |ξ|<1Þ. We denote these classes by S * and C for starlike and convex, respectively. In addition, suppose that the category P involves all functions γ analytic in ∪ with a positive real part in ∪ achieving γð0Þ = 1. Mathematically, γ ∈ S * if and only if ξγ ′ ðξÞ/γðξÞ ∈ P , and γ ∈ C if and only if 1 + ξγ ′ ′ ðξÞ/γ ′ ðξÞ ∈ P ; equivalently, for starlikeness and for convexity. For two functions, γ 1 and γ 2 belong to the category ∧ and are said to be subordinate, noting by γ 1 ≺ γ 2 , if we can find a Schwarz function ⊺ with ⊺ð0Þ = 0 and |⊺ðξÞ | <1 achieving γ 1 ðξÞ = γ 2 ð⊺ðξÞÞ, ξ ∈ ∪ (the detail can be located in [10]). Obviously, γ 1 ðξÞ ≺ γ 2 ðξÞ implies that γ 1 ð0Þ = γ 2 ð0Þ and γ 1 ð∪Þ ⊂ γ 2 ð∪Þ: We employ next facts, one can find it in [11].
For every nonnegative integer n, the q-integer number n symbolized by ½n, q is structured by where ½0, q = 0, ½1, q = 1, and lim q→1 − ½n, q = n. Consequently, the analytic function γ is written by the formula Clearly, we have Δ q ξ n = ½n, qξ n−1 : Consequently, for γ ∈ ∧, we attain For γ ∈ ∧, it realized that the Sàlàgean q-differential operator [12] has the formula such that k represents as a positive integer. A computation based on the definition of Δ q implies that S k q γðξÞ = γðξÞ * Π k q ðξÞ, where * is the convolution product. and Clearly, the well-known Salagean differential operator [6]. Consider the role γðξÞ and a constant 0 ≤ λ ≤ 1, we introduce the q-symmetric Salagean differential operator using the definition of Δ q as follows: Obviously, we indicate that when λ = 1ðS 1,k q = S k q Þ, we get the Salagean q-differential operator. We can call (11) as the symmetric Sàlàgean q-differential operator in ∪. Also, we have the following two limits.
which are represented to the well-known Salagean differential operator [6] and the symmetric differential operator [8], respectively. Depending on the definition of (11), we impose the recognizing classes: Journal of Function Spaces We obtain the following special cases: In our investigation, we focus on the geometric presentation of the special classes S * q ðλ, k, ℏÞ and J λ,♭ q ðA, B, kÞ via utilizing the basis information given in [11].

Main Results
Here, we focus in the geometric representations of the classes S * q ðλ, k, ℏÞ and J λ,♭ q ðA, B, kÞ and the outcomes of these classes.
Proof. Sort out a function ρ in the following construction: Via the main information, S λ,k q γðξÞ is of constrained limit turning; it infers that Rðξρ ′ ðξÞ + ρðξÞÞ > 0: Accordingly, via Lemma 3 (i), we acquire RðρðξÞÞ > 0 which incomes the first requested statement of the theorem. In opinion of the additional information, we obligate the subsequent subordination relative Now, based on Lemma 3 (i), there exists a fixed positive number ℓ > 0 satisfying ♭ = ♭ðℓÞ and the subordination inequality This leads to the conclusion Lastly, we assume the third fact, a direct reckoning reaches to According to the virtue of Lemma 3 (ii), there occurs a fixed positive number, say ℓ > 0 achieving RðρðξÞÞ > ℓ: Consequently, we obtain Hence, via Equation (29), it indicates that So by Noshiro-Warschawski and Kaplan Propositions, S λ,k q γðξÞ ∈ Ⓢ and of bounded turning in ∪. Via the derivative (25) and operating the real, one can attain the real relation Hence, according to Lemma 3 (ii), one get RðS λ,k q γðξÞ/ξÞ > 0: Via considering the logarithmic derivative on (25) and acting as areal part, we obtain the consequence conversation: Thus, according to Lemma 3 (iii), where θðξÞ = 1, we attain that RðS λ,k q γðξÞ/ξÞ > 0: Theorem 6. Suppose that γ ∈ S * q ðλ, k, ℏÞ, with ℏðξÞ ∈ C. Then Journal of Function Spaces where ð ∈ ∪ (analytic) satisfies that ðð0Þ = 0 and |ððξÞ | <1. In addition, for |ξ | = χ, S λ,k q reckoning fulfills the formula statement Proof. We note that γ ∈ S * q ðλ, k, ℏÞ, then one can gain this confirms that there occurs analytic function type Schwarz achieving the relations ðð0Þ = 0,|ððξÞ | <1 and A calculation gives us Via integrating left and right parts, one can achieve Thus, we have Via the definition and the properties of subordination, one can have Furthermore, we deliver that ℏðzÞ maps the disk 0 < |ξ | <χ < 1 onto a convex symmetric domain corresponding to the real axis, that is which implies that Via applying Equation (40), one can indicate that which leads to Hence, we have Corollary 7 [8]. If q ⟶ 1 in Theorem 6, then Theorem 8. Suppose that γ ∈ J λ,♭ q ðA, B, kÞ, then the odd construction formula fulfills the consequently subordination

Journal of Function Spaces
Proof. Let γ ∈ J λ,♭ q ðA, B, kÞ: Subsequently, we get that there occurs a function L ∈ JðA, BÞ with the formula This yields In addition, since L achieves the inequality taking account that the fractional functional express ð1 + AξÞ/ð1 + BξÞ is univalent and hence, consequently, we attain the relation Furthermore, the expression MðξÞ ∈ S * , which leads to the inequality that is, there occurs a Schwarz function κ ∈ ∪, | κðξÞ| ≤ |ξ| < 1, κð0Þ = 0 with the property which implies that there is ζ, |ζ | = r < 1 such that An operation, one can indicate that Thus, one can inform the recognizing inequality This implies the consequence result Next consequence outcomes of the above result can be located in [8,25,26] accordingly. Corollary 9. If λ = 1 in Theorem 8, then Corollary 10. If λ = 1, k = 1, and q ⟶ 1 in Theorem 8, then Corollary 11. If q ⟶ 1 in Theorem 8, then

Applications
We introduce an application of our outcomes based on finding the outcome of Briot-Bouquet equation (BBE) (see [11] for more information). This category of ODE is an association of ODE whose outcomes are formulas in the complex plane. Existence and uniqueness theorems include the utility of majors and minors (or subordination and superordination concepts) (see [28][29][30][31]). Investigation of rational first ODEs in the complex region implies the finding of new transcendental special functions which are now known as symmetric BBE By employing the q-differential operator (11), we have the q-formula of BBE A simple result of (64) can be recognized at β = 1: Thus, we investigate the status, γ ∈ ∧ and β = 0: The initial condition will be γð0Þ = ℏð0Þ = 0: Journal of Function Spaces