Sentence examples for obtain a refinement of from inspiring English sources

We obtain a refinement of the inequality shown by Zhan.

Applying Theorem 7 we obtain a refinement of (15).

In this note, we obtain a refinement of (1.2) and a log-majorization inequality for eigenvalues.

Next, applying Theorem 5 we obtain a refinement of this inequality.

As a result, we obtain a refinement of the Hermite-Hadamard inequality for an r-convex function ( 0 ≤ r ≤ 1 ).

Also, we obtain a refinement of some known inequalities for a class of continuous concave or convex functions.

Finally, we obtain a refinement to a quasiconformal extension criterion of the main result.

Maligranda [2] obtained a refinement of the Dunkl-Williams inequality.

end{aligned} Moreover, Mićić, Pečarić, and Perić [7] obtained a refinement of (2).

As a direct consequence of the inequality (1.3), Kittaneh and Manasrah [1] obtained a refinement of the Heinz inequality as follows: H v ( a, b ) + r 0 a - b 2 ≤ a + b 2, (1.5).

Kittaneh and Manasrah [1] obtained a refinement of Young's inequality as follows: a v b 1 - v + r 0 a - b 2 ≤ v a + ( 1 - v ) b, (1.3).