Chernoff (1981) obtained an upper bound for the variance of a function of a standard normal random variable, using Hermite polynomials. Chen (1980) gave a different proof, using the Cauchy-Schwarz inequality, and extended the inequality to the case of a multivariate normal. Here it is shown how similar upper bounds can be obtained for other distributions, including discrete ones. Moreover, by using a variation of the Cramer-Rao inequality, analogous lower bounds are given for the variance of a function of a random variable which satisfies the usual regularity conditions. Matrix inequalities are also obtained. All these bounds involve the first two moments of derivatives or differences of the function.
Theophilos Cacoullos. "On Upper and Lower Bounds for the Variance of a Function of a Random Variable." Ann. Probab. 10 (3) 799 - 809, August, 1982. https://doi.org/10.1214/aop/1176993788