Inequalities Involving Functions and Their Integrals and Derivatives

On a variant of the Hardy inequality between weighted Orlicz spaces

“…This inequality was generalized to Inequalities in L p were studied by Muckenhoupt [46], Maz'ya and Rozin [44], Bradley [8], Kokilashvili [33], Sinnamon [53], Sawyer [51], Bloom and Kerman [6], Stepanov [54], and many others (we refer to the monographs [38,36,37,34,45] and references therein). As to the Orlicz-space result (1.2), several authors contributed to a complete characterization of admissible weights ω, r, , v and nondecreasing functions P, Q which allow for (1.2) with T f (x) = x 0 K(x, y)f (y) dy being the generalized Hardy operator with kernel K. To name just a few, we refer to the papers of Bloom and Kerman [5,6], Lai [40,41,42,43], Heinig and Maligranda [28], Heinig and Lai [27] and their references.…”

Popoviciu type inequalities for n-convex functions via extension of Montgomery identity

“…In case n = 1 the sum n−2 k=0 · · · is empty, so identity (9) reduces to the well-known Montgomery identity (see for instance [6])…”

Fractional Representation Formulae Under Initial Conditions and Fractional Ostrowski Type Inequalities

“…Let f : ra, bs Ñ R be differentiable on ra, bs, and let f 1 : ra, bs Ñ R be integrable on ra, bs, then the following Montgomery identity holds [2]: (1) f pxq " 1 b´a …”