Montgomery identity and Ostrowski-type inequalities via quantum calculus

In this paper, we prove a quantum version of Montgomery identity and prove some new Ostrowski-type inequalities for convex functions in the setting of quantum calculus. Moreover, we discuss several special cases of newly established inequalities and obtain different new and existing inequalities in the field of integral inequalities. © 2021 Thanin Sitthiwirattham et al., published by De Gruyter.National Natural Science Foundation of China, NSFC: 11971241; King Mongkut's University of Technology North Bangkok, KMUTNB: KMUTNB-62-KNOW-20Funding information : This research was funded by King Mongkut’s University of Technology North Bangkok (Contract no. KMUTNB-62-KNOW-20). This work was partially supported by National Natural Science Foundation of China (Grant no. 11971241).2-s2.0-8511715956

Survey of the qualitative properties of fractional difference operators: monotonicity, convexity, and asymptotic behavior of solutions

In this article we discuss some of the qualitative properties of fractional difference operators. We especially focus on the connections between the fractional difference operator and the monotonicity and convexity of functions. In the integer-order setting, these connections are elementary and well known. However, in the fractional-order setting the connections are very complicated and muddled. We survey some of the known results and suggest avenues for future research. In addition, we discuss the asymptotic behavior of solutions to fractional difference equations and how the nonlocal structure of the fractional difference can be used to deduce these asymptotic properties