New quantum boundaries for quantum Simpson’s and quantum Newton’s type inequalities for preinvex functions
Muhammad Aamir Ali, Mujahid Abbas, Hüseyin Budak, Praveen Agarwal, Ghulam Murtaza, Yu‐Ming Chu
Abstract
Abstract In this research, we derive two generalized integral identities involving the $q^{\varkappa _{2}}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>q</mml:mi><mml:msub><mml:mi>ϰ</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:msup></mml:math> -quantum integrals and quantum numbers, the results are then used to establish some new quantum boundaries for quantum Simpson’s and quantum Newton’s inequalities for q -differentiable preinvex functions. Moreover, we obtain some new and known Simpson’s and Newton’s type inequalities by considering the limit $q\rightarrow 1^{-}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>q</mml:mi><mml:mo>→</mml:mo><mml:msup><mml:mn>1</mml:mn><mml:mo>−</mml:mo></mml:msup></mml:math> in the key results of this paper.