Sentence examples for extends inequality from inspiring English sources

for t > 0, which extends inequality (14).

for s > 0 and real number x ≥ 0. Remark 31 It is easy to see that inequality (75) refines and extends inequalities (10) and (58).

In [3] (see also [4]), Elbert extended inequality (2) to the one-dimensional p-Laplacian equation.

Let extend inequality (35) to Z≥2 non-negative scalars x 1,…,x Z.

Using the Hermite-Hadamard inequality for the convex function, we can extend inequality (3.1) on the left and on the right hand side as follows: (34).

We can extend inequality (1.9) given in the previous section to matrices by using the Frobenius inner product as follows: Let.

Here, (lambdacirc Koplus_{p} mucirc L) denotes the (L_{p} -Blaschke combination of K and L_{p} -Blaschkeend inequality (1.2) to general (L_{p})-mixed-brightness integrals.

To prove that formula (2.4.2) holds for arbitrary functions f in (B_{infty,1}^m({mathbb R})), we have to extend inequality (2.4.3) to the class (B_{infty,1}^m({mathbb R})).

In this paper, we first extend inequality (1.10) to dual quermassintegrals forms, that is, the extremums of dual quermassintegrals for the polars of general (L_{p} -projection bodies are obtained.

Bhatia and Davis [5] extended inequality (1) to the matrix case, they showed that it holds for positive semi-definite matrices, in the following form: bigl|!bigl|!bigl|A^{frac{1}{2}}bigr|ac{1}{2}}bigr|lebigl|bigl|lebigl|H_{nu}!bigl|H_{nu}(A, B bigr|!bigr|!bigr|lebiggl|!biggl|!biggl|frac{A+B}{2}biggr|!biggr|!biggr|, (9) where (|!|!|cdot|!|!|) is any invariant unitary norm.

For several results which generalize, improve and extend inequalities (1.1), we refer the interested reader to [2 18].