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

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, ۵۹ (۲۰۱۰) ۱۸۵۲--۱۸۶۴), 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