Abstract
In this research paper, we improve some fractional integral inequalities of Minkowski-type. Precisely, we use a proportional fractional integral operator with respect to another strictly increasing continuous function ψ. The functions used in this work are bounded by two positive functions to get reverse Minkowski inequalities in a new sense. Moreover, we introduce new fractional integral inequalities which have a close relationship to the reverse Minkowski-type inequalities via ψ-proportional fractional integral, then with the help of this fractional integral operator, we discuss some new special cases of reverse Minkowski-type inequalities through this work. An open issue is covered in the conclusion section to extend the current findings to be more general.
Highlights
1 Introduction During their uncompromising effort in the expansion of mathematics, mathematicians have recently expanded the traditional calculus of derivatives and integrals for integer orders to the generalized form of conventional derivatives and integrals of noninteger order, these noninteger-order derivatives/integrals are referred to as fractional calculus, which during a few previous decades became one of very influential branches of mathematics, especially, when dealing with the differential/integral equations and inequalities
Rashid et al [27] presented a note on reverse Minkowski inequalities by using a generalized proportional fractional operator involving another function
Motivated by the results mentioned above, in particular, Theorems 1.5 and 1.6, we strive in our recent work to employ a generalized proportional fractional integral with respect to a strictly increasing continuous function ψ to establish the functional bounds in the reverse Minkowski type inequalities in terms of the fractional integral
Summary
During their uncompromising effort in the expansion of mathematics, mathematicians have recently expanded the traditional calculus of derivatives and integrals for integer orders to the generalized form of conventional derivatives and integrals of noninteger order, these noninteger-order derivatives/integrals are referred to as fractional calculus, which during a few previous decades became one of very influential branches of mathematics, especially, when dealing with the differential/integral equations and inequalities.The fractional calculus theory became important due to its significant applications in several areas such as physics, fluid dynamics, control theory, computer networking, signal processing, biology, image processing, and other areas. In (2006), Bougoffa presented the classical integral version of Minkowski inequality as follows: Theorem 1.1 ([18]) Consider positive functions η, ς in Lz[s, ω] with z ∈ [1, ∞). Rashid et al [27] presented a note on reverse Minkowski inequalities by using a generalized proportional fractional operator involving another function.
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