Sentence examples for extended inequality from inspiring English sources

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

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.

Meanwhile, the extended inequality, operator expressions, reverse inequality, and equivalent forms are given.

end{aligned} (6) Moreover, we prove an extended inequality of (6) with multiparameters and a best possible constant factor.

Recently, Carneiro and Madrid [18] extended inequality (1.5) to the fractional setting (also see [20, 29 31] for the relevant results).

So, we have obtained a more accurate and extended inequality of (9) with multiparameters and a best possible constant factor (B lambda _{1},lambda_{2})).

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

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.