Sentence examples for by using the inequality from inspiring English sources

By using the inequality techniques and a useful Lyapunov functional, some new testable algebraic criteria are obtained for ensuring the existence and global exponential stability of periodic solution of the system.

Similarly, by using the inequality, (2.18).

By using the inequality for nonnegative and and (2.3).

By using the inequality, we derive from (3.17) that (3.18).

Since,, By using the inequality,, we have (28).

Then we turn to consider the problems on (1.1) and (1.2) by using the inequality.

By using the inequalities (2.11).

By using the inequalities (2.6) and (2.11) of [9, Lemma ], we can conclude that (2.6).

The monotonicity, log-convexity, log-concavity of extended hypergeometric functions are deduced by using the inequalities on extended beta functions.

Note that all known trapezoid-Grüss, midpoint-Grüss, and Ostrowski-Grüss type inequalities, which are proved by applying the inequalities (1.3), (1.6), and (1.7), can be improved by using the inequalities of Theorem 4.1.

In this section, we give several inequalities involving covariance matrices, an alternative based on the spectral norm of the relative gain of the covariance adjusted estimator and its upper bound by using the inequalities in Section 2.