Research Article On New Modifications Governed by Quantum Hahn’s Integral Operator Pertaining to Fractional Calculus Saima Rashid , 1 Aasma Khalid, 2 Gauhar Rahman , 3 Kottakkaran Sooppy Nisar, 4 and Yu-Ming Chu 5,6 1 Department of Mathematics, Government College University, Faisalabad 38000, Pakistan 2 Department of Mathematics, Government College Women University Faisalabad, Madina Town, Faisalabad 38000, Pakistan 3 Department of Mathematics, Shaheed Benazir Bhutto University, Sheringal, Pakistan 4 Department of Mathematics, College of Arts and Sciences, Prince Sattam Bin Abdulaziz University, Wadi Al Dawasir 11991, Saudi Arabia 5 Department of Mathematics, Huzhou University, Huzhou 313000, China 6 Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science & Technology, Changsha 410114, China Correspondence should be addressed to Yu-Ming Chu; chuyuming@zjhu.edu.cn Received 17 April 2020; Accepted 23 June 2020; Published 14 July 2020 Academic Editor: Gestur Ólafsson Copyright © 2020 Saima Rashid 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. In the article, we present several generalizations for the generalized Čebyšev type inequality in the frame of quantum fractional Hahn’s integral operator by using the quantum shift operator σ Ψ q ðςÞ = qς + ð1 − qÞσðς ∈ ½l 1 , l 2 , σ = l 1 + ω/ð1 − qÞ,0< q < 1, ω ≥ 0Þ. As applications, we provide some associated variants to illustrate the efficiency of quantum Hahn’s integral operator and compare our obtained results and proposed technique with the previously known results and existing technique. Our ideas and approaches may lead to new directions in fractional quantum calculus theory. 1. Introduction In 1882, Čebyšev discovered a fascinating and significantly valuable integral inequality as follows: 1 l 1 − l 2 ð l 2 l 1 Q x ðÞU x ðÞdx ≤ 1 l 1 − l 2 ð l 2 l 1 Q x ðÞdx  ! 1 l 1 − l 2 ð l 2 l 1 U x ðÞdx  ! , ð1Þ if Q and U are two integrable and synchronous functions on ½l 1 , l 2 , where the functions Q and U are said to be synchro- nous on ½l 1 , l 2  if Q x ðÞ − Q y ðÞ ð Þ U x ðÞ − U y ðÞ ð Þ ≥ 0 ð2Þ for all x, y ∈ ½l 1 , l 2 . It is well known that the Čebyšev inequality (1) has wild applications in the fields of pure and applied mathematics [1–10]. Recently, the generalizations and variants for the Čebyšev inequality (1) have attracted the attention of many researchers [11–20]. Quantum di fference operators are receiving an increase of interest due to their applications [21, 22]. Roughly speaking, quantum calculus can substitute the classical derivative by a di ffer- ence operator, which allows to deal nondi fferentiation functions. Let q ∈ ð0, 1Þ, ω ≥ 0, I ⊆ ℝ be an interval such that ω 0 = ω/ð1 − qÞ ∈ I , and H 1 : I ⟶ ℝ be a real-valued function. Then, the Hahn difference operator D q,ω [23] is defined by D q,ω H 1 ς ðÞ = H 1 qς + ω ð Þ − H 1 ς ðÞ ς q − 1 ð Þ + ω , ς ≠ ω 0 , H 1 ′ ω 0 ð Þ, ς = ω 0 , 8 > < > : ð3Þ if H 1 is differentiable at ω 0 . Hindawi Journal of Function Spaces Volume 2020, Article ID 8262860, 12 pages https://doi.org/10.1155/2020/8262860