The Hölder and Minkowski Inequalities Utilizing a Fractional Operator Involvement of Pseudo-Operator

Abstract

A generalized integral operator of order $\alpha$ of a real function $f$ including a parameter set $P$, namely $K_P^\alpha f(t)$ has been introduced by O. P. Agrawal (Computers and Mathematics with Applications, 59 (2010) 1852--1864), which is a generalization of some important fractional integrals such as the Riemann-Liouville fractional integral. Using pseudo-analysis, this paper introduces a pseudo-operator integral of order $\alpha$ including a parameter set $P$ in a semiring $([a, b], \oplus, \odot)$, which is a generalization of $K_P^\alpha f(t)$. We also discuss some particular cases and we obtain the well-known H\"{o}lder's and Minkowski's inequalities for this kind of pseudo-operator integral. The results given in this paper provide a generalization of several inequalities obtained in earlier studies.

Keywords

• riemann-liouville fractional integral
• pseudo-analysis
• semiring
• h\"{o}lder's inequality
• minkowski's inequality