Carleman-Knopp type inequalities via Hardy inequalities

Abstract

Abstract. Some new Carleman-Knopp type inequalities are proved as "end point" inequalities of modern forms of Hardy's inequalities. Both finite and infinite intervals are considered and both the cases p q and q < p are investigated. The obtained results are compared with similar results in the literature and the sharpness of the constants is discussed for the power weight case. Moreover, some reversed Carleman-Knopp inequalities are derived and applied.Mathematics subject classification (2000): 26D15, 26D07.

Weighted integral inequalities with the geometric mean operator

“…(13) Now, on the strength of (6) the upper bound from (7) and (12) imply the result of ([11], (1.3)): if 0 < p < q < , then sup,> + (s i)eU A'/P s, IIGII < infS/P 0 )As/P 0 ) (14) s>l with slightly better factors on both sides and the upper bound from (10) and (13) …”

On Carleman and Knopp's Inequalities

“…o olya. Also, this inequality has been generalized in a number of ways and here we just mention the fairly recent papers [13,14,17,19,20] and the references given in these papers.…”

Two-Weight Criteria for the Multidimensional Hardy-Type Operator in p-Convex Banach Function Spaces and Some Applications

“…Note that the simplest case of (2.4) with v = w = 1 and p = q = 1 was considered in [20] and in [25]. Later this inequality was generalized in various ways by many authors in [12,22,23,24,31,39,40,41,46] and etc. The sufficient conditions for general weights ensuring the validity of the two-weight strong type inequalities for some sublinear operator are given in the following theorem.…”