Modes and moments of unimodal distributions (Q1096238)

The paper considers the author's [(*) Nagoja Math. J. 104, 29-42 (1986; Zbl 0613.60065)] inequality relationships between the mode \(\alpha\) of a unimodal distribution and its absolute moment of order \(p,\beta_ p=\int | x|^ p\mu (dx)\) or its absolute central moment of order \(p,\gamma_ p=\int | x-m|^ p\mu (dx)\). (Here m is the mean of the distribution which is considered finite). It was shown by the author (*) that, for some constants \(A_ p\) and \(B_ p\), \[ | \alpha | \leq A_ p\beta_ p^{1/p}\quad and\quad | \alpha -m| \leq B_ p\gamma_ p^{1/p}. \] The present paper provides an alternative proof of these inequalities as well as explicit expressions for \(B_ p\), \(A_{1/2}\), \(A_ 1\), \(A_ 2\) and a lower bound for \(A_ p\), \(p>0.\) The paper also constructs inequalities referring to the mode \(\alpha\) of a unimodal distribution and what it defines to be the g-moment \((\int g(x)\mu (dx))\) of the distribution and the central g-moment \((\int g(x)\mu (dx))\) of the distribution.