Sentence examples for refinement of the inequality from inspiring English sources

Therefore, Theorem 2.1 is a refinement of the inequality (1.2).

We obtain a refinement of the inequality shown by Zhan.

The following proposition is a refinement of the inequality in Theorem 3.4(2).

In this article, we first present a refinement of the inequality (1.5).

In this section, we give a refinement of the inequality (1.5).

After that, we use it to establish a refinement of the inequality (1.7) for the Hilbert-Schmidt norm.

Then we use this inequality, the Erdös-Mordell inequality and another well-known inequality to deduce a refinement of the Oppenheim inequality.

The inequality (2.4) and the first inequality in (1.1) yield the following refinement of the first inequality in (1.1).

The second inequality of (1.2) is given in [8, page 354, 3.9.32] for For a refinement of the first inequality in (1.2) see Remark 1.3(1) and of the second inequality see Theorem 2.4.

If we apply inequality (4.10) for, we get the following refinement of the previous inequality.

We give some refinements of the inequalities of Aczél, Popoviciu, and Bellman.