On new generalized quantum integrals and related Hermite–Hadamard inequalities
Hasan Kara, Hüseyin Budak, Necmettin Alp, Humaira Kalsoom, Mehmet Zeki Sarıkaya
Abstract
Abstract In this article, we introduce a new concept of quantum integrals which is called ${}^{\kappa _{2}}T_{q}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mi>T</mml:mi> <mml:mi>q</mml:mi> <mml:none /> <mml:mprescripts /> <mml:none /> <mml:msub> <mml:mi>κ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mmultiscripts> </mml:math> -integral. Then we prove several properties of this concept of quantum integrals. Moreover, we present several Hermite–Hadamard type inequalities for ${}^{\kappa _{2}}T_{q}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mi>T</mml:mi> <mml:mi>q</mml:mi> <mml:none /> <mml:mprescripts /> <mml:none /> <mml:msub> <mml:mi>κ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mmultiscripts> </mml:math> -integral by utilizing differentiable convex functions. The results presented in this article are unification and generalization of the comparable results in the literature.