New Generalizations of Hardy's Integral Inequality

Abstract

In this paper, we obtain some new Hardy type integral inequalities. These w inequalities generalize the results obtained by Y. Bicheng et al.

Some new generalizations of Hardy′s integral inequality

“…In this paper, we have the generalized Holder's inequality in L(p) space and the results of [1,3]. S. K. Sunanda et al 3…”

“…Let 0 < a < b < ∞, p > 1, 1/ p + 1/q = 1, f ≥ 0, and 0 < 1 − θ p (t) f p (t)dt, (1.4) where θ p (t) = 1/ p ∞ k=1 p k (−1) k−1 (a/t) k/q > 0 for t > a, and θ p (a) = 1/q. Oguntuase and Imoru [3] generalized (1.3) and (1.4) as follows. Let 0 < a < b < ∞, p > 1, 1/ p + 1/q = 1 − 1/r, f ≥ 0, r > 1, and 0 < (1.5)…”

The Minkowski inequality involving generalized k-fractional conformable integral

“…Thus, the new results involving integral inequalities have been possible; consequently, some applications have been made [16,17]. We mention a few of them, i.e., the inequalities of Minkowski, Holder, Hardy, Hermite-Hadamard, Jensen, among others [20][21][22][23][24][25][26]. Such applications of fractional integral operators motivate us to present the generalization of the existing fractional conformable operators and generalize the reverse Minkowski inequality [27][28][29][30][31] involving generalized k-fractional conformable integrals.…”

“…26) respectively, if integrals exist, where k > 0, s ∈ R \ {0}. For k > 0, s ∈ R \ {0}, α > 0 and p ≥ 1.…”

Generalized Hardy, Copson, Leindler and Bennett inequalities on time scales

“…Hardy inequalities (continuous or discrete) has been extensively studied in literature and many papers which deal with new proofs, generalizations and extensions have appeared in the literature; we refer the reader to the books [18], [19], [24] and the papers [4], [7], [10], [15]- [17], [21]- [23], [26]. There are a few papers [25], [27]- [29] which study dynamic inequalities of Hardy's type on time scales, i.e., where the domain of the unknown function is a so-called time scale T (which may be an arbitrary closed subset of the real numbers R).…”