Variance bounds by a generalization of the Cauchy-Schwarz inequality
General upper and lower bounds for Var[g(X)] are obtained, where X is a continuous random variable and g(x), \(x\in (-\infty,\infty)\), is a real- valued function with continuous derivatives of order \(2m+3\). The upper bound depends on \(2m+2\) derivatives of g. The result is a generalization of results of \textit{H. Chernoff} [Ann. Probab. 9, 533-535 (1981; Zbl 0457.60014)] and \textit{T. Cacoullos} [ibid. 10, 799-809 (1982; Zbl 0492.60021)].
scientific article; zbMATH DE number 4086640