Cramer-Rao type integral inequalities were known for obtaining lower bounds for Bayesian risk for squared error loss functions for estimators based on fixed sample size. A sequential Cramer-Rao type integral inequality was derived by Prakasa Rao (Proc. Andhra Pradesh Akademi of Sciences (2000) 5: 23–38). An improved version of this inequality is obtained using a recent result on improved Cauchy-Schwartz inequality due to Walker (Statist. Probab. Lett. (2017) 122: 86–90).
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Б. Л. С. Пракаса Рао (2018) studied this question.