The Cauchy-Schwarz inequality is an inequality that relates the inner product of two vectors to the product of their norms. It is named after Augustin-Louis Cauchy and Hermann Schwarz, who independently discovered it in the 19th century.

Jensen's inequality states that if (f) is a convex function and (X) is a random variable, then the expected value of (f(X)) is greater than or equal to the value of (f) at the expected value of (X).

Chebyshev's inequality states that for any random variable (X) with mean (\mu) and variance (\sigma^2), the probability that the absolute value of the difference between (X) and (\mu) is greater than or equal to (k\sigma) is at most (1/k^2).

Markov's inequality states that for any random variable (X) with mean (\mu) and variance (\sigma^2), the probability that the absolute value of the difference between (X) and (\mu) is greater than or equal to (k\sigma) is at least (1/k^2).

Jensen's inequality is used in risk minimization in machine learning to bound the expected loss of a model. The expected loss is the average loss over all possible inputs, and Jensen's inequality can be used to show that the expected loss of a convex function is greater than or equal to the loss of the expected input.

Chebyshev's inequality is used in data science for outlier detection, hypothesis testing, and confidence intervals. In outlier detection, Chebyshev's inequality can be used to identify data points that are significantly different from the rest of the data. In hypothesis testing, Chebyshev's inequality can be used to test the hypothesis that the mean of a population is equal to a specified value. In confidence intervals, Chebyshev's inequality can be used to construct confidence intervals for the mean of a population.

Markov's inequality is used in data science for tail bounds, concentration inequalities, and large deviations theory. In tail bounds, Markov's inequality can be used to bound the probability that a random variable deviates from its mean by more than a certain amount. In concentration inequalities, Markov's inequality can be used to show that the sum of a large number of independent random variables is concentrated around its mean. In large deviations theory, Markov's inequality can be used to study the behavior of rare events.

Srinivasa Ramanujan, Harish-Chandra, and C. R. Rao are all Indian mathematicians who made significant contributions to the field of mathematical inequalities. Ramanujan discovered a number of new and remarkable inequalities, including the Rogers-Ramanujan identities and the Ramanujan-Sato series. Harish-Chandra developed a theory of harmonic analysis on semisimple Lie groups, which has been used to prove a number of important inequalities. C. R. Rao is a statistician who has made significant contributions to the theory of estimation and hypothesis testing, including the development of the Rao-Blackwell theorem and the Cramér-Rao inequality.

The Rogers-Ramanujan identities are a set of 17 identities that relate the values of the Rogers-Ramanujan continued fraction at various arguments. They were discovered by Srinivasa Ramanujan in 1917 and published in the Proceedings of the Cambridge Philosophical Society.

The Ramanujan-Sato series is a series that expresses the Rogers-Ramanujan continued fraction as a sum of hypergeometric series. It was discovered by Srinivasa Ramanujan and Daihachiro Sato in 1918 and published in the Proceedings of the Cambridge Philosophical Society.

The Harish-Chandra theory of harmonic analysis on semisimple Lie groups is a theory that studies the structure of semisimple Lie groups and their representations. It was developed by Harish-Chandra in the 1950s and 1960s and has been used to prove a number of important inequalities, including the Plancherel theorem and the Kazhdan-Lusztig conjecture.

The Rao-Blackwell theorem is a theorem that states that the minimum variance unbiased estimator of a parameter is the conditional expectation of the parameter given the sufficient statistic. It was proved by C. R. Rao in 1945 and has been used to develop a number of important statistical methods, including the Kalman filter and the Wiener filter.

The Cramér-Rao inequality is an inequality that provides a lower bound on the variance of any unbiased estimator of a parameter. It was proved by Harald Cramér in 1946 and has been used to develop a number of important statistical methods, including the maximum likelihood estimator and the method of moments.

All of the Indian mathematical inequalities listed have been used in data science. The Cauchy-Schwarz inequality is used in the analysis of variance and regression. Jensen's inequality is used in risk minimization and dimensionality reduction. Chebyshev's inequality is used in outlier detection and hypothesis testing. Markov's inequality is used in tail bounds and concentration inequalities.