Left-Right Quantum Derivatives and Definite Integrals

Abstract

In this work, the concepts of quantum derivative and quantum integral are renamed to be the left quantum derivative and the left de.. . nite quantum integral. Symmetrically to the left, a new quantum derivative (the right) and de.. . nite quantum integral (the right) are de.. . ned. Some properties of these new concepts are investigated and as well as according todo these new concepts some inaccuracies in quantum integral inequalities are corrected. Moreover, some new quantum Hermite-Hadamard type inequalities are established. Hosted file M_Kunt_, A_ W_ Baidar_and_Z_\selectlanguage{polish}Ş\selectlanguage{english}anl\selectlanguage{polish}ı.

Quantum analog of some trapezoid and midpoint type inequalities for convex functions

“…In [23], the authors have denoted ( 2) and ( 4) respectively as left quantum derivative and definite left quantum integral and it has been written in this wise:…”

“…In recent years Tariboon and Ntouyas in [31], generalized the classic quantum derivative and integral. Also, in [23], the authors by using the notions of left and right quantum derivative and integral have derived a similar generalization of classic quantum derivative and integral. In many research papers, the quantum analogue of Hermite-Hadamard type inequalities has been obtained via the generalized form of quantum integral, which were given in [31].…”

“…[23] Let a function f be defined on [a, b]. Then the right quantum integral of f on [a, b] is defined by b…”