Some Sharp Chernoff type inequalities

Two sharp Chernoff type inequalities are obtained for star body in ${\mathbb{R}}^2$, one of which is an extension of the dual Chernoff-Ou-Pan inequality, and the other is the reverse Chernoff type inequality. Furthermore, we establish a generalized dual symmetric mixed Chernoff inequality for two planar star bodies. As a direct consequence, a new proof of the the dual symmetric mixed isoperimetric inequality is presented.