The Hölder and Minkowski Inequalities Utilizing a Fractional Operator Involvement of Pseudo-Operator - Hadiseh Fallah Andevari; Azizollah Babakhani; Daniela Oliveira (Sahand Communications in Mathematical Analysis)
A generalized integral operator of order $α$ of a real function $f$ including a parameter set $P$, namely $K_P^α f(t)$ 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 $α$ including a parameter set $P$ in a semiring $([a, b], \oplus, \odot)$, which is a generalization of $K_P^α f(t)$. We also discuss some particular cases and we obtain the well-known Hölder'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.