We study Hardy-type integral inequalities with remainder terms for smooth compactly-supported functions in convex domains of finite inner radius. New \( L_{1} \)- and \( L_{p} \)-inequalities are obtained with constants depending on the Lamb constant which is the first positive solution to the special equation for the Bessel function. In some particular cases the constants are sharp. We obtain one-dimensional inequalities and their multidimensional analogs. The weight functions in the spatial inequalities contain powers of the distance to the boundary of the domain. We also prove that some function depending on the Bessel function is monotone decreasing. This property is essentially used in the proof of the one-dimensional inequalities. The new inequalities extend those by Avkhadiev and Wirths for \( p=2 \) to the case of every \( p\geq 1 \).

我们研究在有限内半径的凸域中具有光滑紧支撑函数的余项的Hardy型积分不等式。新\（L_ {1} \） -和\（L_ {p} \）通过Lamb常数获得常数不等式，该常数是Bessel函数特殊方程式的第一个正解。在某些特定情况下，常数很尖锐。我们获得一维不等式及其多维类似物。空间不等式中的权函数包含到域边界的距离的幂。我们还证明，取决于贝塞尔函数的某些函数是单调递减的。此属性本质上用于证明一维不等式。新的不等式将Avkhadiev和Wirths对于\（p = 2 \）的不等式扩展 到每个\（p \ geq 1 \）的情况。