New quantum estimates in the setting of fractional calculus theory
Saima Rashid, Zakia Hammouch, Rehana Ashraf, Dumitru Bǎleanu, Kottakkaran Sooppy Nisar
Abstract
Abstract In this article, the investigation is centered around the quantum estimates by utilizing quantum Hahn integral operator via the quantum shift operator ${}_{\eta}\psi_{\mathfrak{q}}(\zeta)=\mathfrak{q}\zeta+(1-\mathfrak{q})\eta$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mmultiscripts><mml:mi>ψ</mml:mi><mml:mi>q</mml:mi><mml:none/><mml:mprescripts/><mml:mi>η</mml:mi><mml:none/></mml:mmultiscripts><mml:mo>(</mml:mo><mml:mi>ζ</mml:mi><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:mi>q</mml:mi><mml:mi>ζ</mml:mi><mml:mo>+</mml:mo><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>q</mml:mi><mml:mo>)</mml:mo><mml:mi>η</mml:mi></mml:math> , $\zeta\in[\mu,\nu]$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ζ</mml:mi><mml:mo>∈</mml:mo><mml:mo>[</mml:mo><mml:mi>μ</mml:mi><mml:mo>,</mml:mo><mml:mi>ν</mml:mi><mml:mo>]</mml:mo></mml:math> , $\eta=\mu+\frac{\omega}{(1-\mathfrak{q})}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>η</mml:mi><mml:mo>=</mml:mo><mml:mi>μ</mml:mi><mml:mo>+</mml:mo><mml:mfrac><mml:mi>ω</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>q</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mfrac></mml:math> , $0<\mathfrak{q}<1$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>0</mml:mn><mml:mo><</mml:mo><mml:mi>q</mml:mi><mml:mo><</mml:mo><mml:mn>1</mml:mn></mml:math> , $\omega\geq0$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ω</mml:mi><mml:mo>≥</mml:mo><mml:mn>0</mml:mn></mml:math> . Our strategy includes fractional calculus, Jackson’s $\mathfrak{q}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>q</mml:mi></mml:math> -integral, the main ideas of quantum calculus, and a generalization used in the frame of convex functions. We presented, in general, three types of fractional quantum integral inequalities that can be utilized to explain orthogonal polynomials, and exploring some estimation problems with shifting estimations of fractional order $\varrho_{1}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>ϱ</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math> and the $\mathfrak{q}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>q</mml:mi></mml:math> -numbers have yielded fascinating outcomes. As an application viewpoint, an illustrative example shows the effectiveness of $\mathfrak{q}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>q</mml:mi></mml:math> , ω -derivative for boundary value problem.