Sharp inequalities between skewness and kurtosis for unimodal distributions (Q1347189)

Let \(X\) be a random variable with zero mean and unit variance and define \(\gamma_ 1 = E(X^ 3)\), \(\gamma_ 2 = E(X^ 4) - 3\). The main result is an upper bound for \(\gamma_ 2\) in terms of \(\gamma_ 1\) in the case that \(X\) has a unimodal distribution with support \([a,b]\), where \(a + b \geq 0\). The bound is sharp.