A Markovian Gauss inequality for asymmetric deviations from the mode of symmetric unimodal distributions

For a random variable with a unimodal distribution and finite second moment Gaus (1823) proved a sharp bound on the probability of the random variable to be outside a symmetric interval around its mode. An alternative proof for it under the assumption of a finite r-th absolute moment is given (r ≥ 1), based on the Khintchine representation of unimodal random variables. A special instance of the resulting Narumi–Gaus inequality is the one with finite first absolute moment, which might be called a Markovian Gaus inequality.
For symmetric unimodal distributions with finite second moment Semenikh in(2019) generalized the Gaus inequality to arbitrary intervals. For the class of symmetricuni modal distributions with finite first absolute moment we construct a Markovian version of it. Related inequalities of Volkov (1969) and Sellke and Sellke (1997) will be discussed as well.