Generalized gauss-chebyshev inequalities for unimodal distributions

Letg be an even function on ℝ which is nondecreasing in |x|. Letk be a positive constant. Sharp inequalities relatingP(|X|≥k) toEg(X) are obtained for random variablesX which are unimodal with mode 0, and for random variablesX which are unimodal with unspecified mode. The bounds in the mode 0 case generalize an inequality due to Gauss (1823), whereg(x)=x ². The bounds in the second case generalize inequalities of Vysochanskiĭ and Petunin (1980, 1983) and Dharmadhikari and Joag-dev (1985).