Research Article Geometric Inequalities via a Symmetric Differential Operator Defined by Quantum Calculus in the Open Unit Disk Rabha W. Ibrahim, 1,2 Rafida M. Elobaid , 3 and Suzan J. Obaiys 4 1 Informetrics Research Group, Ton Duc Thang University, Ho Chi Minh City, Vietnam 2 Faculty of Mathematics & Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam 3 Deanship of Educational Services, Prince Sultan University, Saudi Arabia 4 School of Mathematical and Computer Sciences, Heriot-Watt University Malaysia, Malaysia Correspondence should be addressed to Rafida M. Elobaid; robaid@psu.edu.sa Received 11 February 2020; Revised 8 July 2020; Accepted 14 July 2020; Published 18 August 2020 Academic Editor: Adrian Petrusel Copyright © 2020 Rabha W. Ibrahim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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. 1. 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 [1–9]). In recent investigation, we deliver a process for deriving and interpreting from a symmetry possessions and infirm analogy with the traditional cases. By combin- ing the q-calculus and the symmetric Salagean differential operator, we introduce a novel modified symmetric Sala- gean q-differential operator. Via employing this operator, we deliver new classes of analytic functions. 2. 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 γξ ðÞ = ξ + 〠 ∞ n=2 γ n ξ n , ξ ∈∪: ð1Þ 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 mani- fested 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, respec- tively. 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, R ξγ ′ ξ ðÞ/γξ ðÞ   > 0, ð2Þ Hindawi Journal of Function Spaces Volume 2020, Article ID 6932739, 8 pages https://doi.org/10.1155/2020/6932739