A weighted differential hardy inequality on $$\mathop {AC}\limits^ \circ (I)$$

One-dimensional differential Hardy inequality

“…However, our upper estimate in (7) is worse than the known one (see [2–4], and Remark 3.2 further). In case (6), our criterion is given in terms different from those in [1, 2], and [4]. The terms in [4] are close to ours, but the comparison analysis shows that our results and an estimate of type (7) are better than in [4] (see Remark 3.3).…”

“…If (3) or (4) is correct, then by Theorem 3.1 there follows the correctness of the corresponding integral Hardy inequality (see [1]). For example, if (3) holds, then where the condition coincides with the Muckenhoupt condition (see [3]), and inequality (1) is equivalent to the integral Hardy inequality (5).…”

“…In [1] it is shown that if for some , then inequality (1) does not hold. In the case or then inequality (1) is satisfied for all functions such that or , respectively.…”

“…In the case or then inequality (1) is satisfied for all functions such that or , respectively. For example, in case (3), it is equivalent to the weighted integral Hardy inequality (see [1]) which has been studied for all values of the parameters (see [2, 3], and [4]).…”

“…In [1] it is also shown that in the last case inequality (1) is satisfied for all functions such that and , that is, it is an overdetermined case. This case is studied in [1, 2], and [4].…”