Optimal lower and upper bounds for the $L_p$-mean deviation of functions of a random variable

Kurzfassung/Abstract

Lower and upper bounds for the variance (that is the mean deviation in $L_2$) of functions of a random variable are well-known in the literature where they are called Cacoullos, Chernoff, or Poincaré type inequality. A similar upper bound is known for the mean deviation of functions of a random variable in $L_1$. In the present paper we give a general approach to how one can obtain lower and upper bounds for the mean deviation of functions of a random variable in $L_p$, $1\le p<\infty$. Especially, we obtain each inequality of the above type stated in the literature. Moreover, we also state new inequalities. To be concrete, we demonstrate such new inequalities for the class of probability measures with Lebesgue density
$$f_p(x;\mu,\alpha)= c_p(\alpha)\cdot\exp(-|x-\mu|^p/\alpha),\quad p\in [1,\infty).$$
Further, we develop lower and upper bounds for non-centered $p$th absolute moments of functions of a random variable, $p\in [1,\infty)$. For both types of inequalities we show that they are optimal in a certain sense. Our approach using Hille-Tamarkin operators is also new for the known results.