Sentence examples for mean value inequality from inspiring English sources

Theorem 6.1 (Mean value inequality).

In the end, a mean value inequality is also derived.

For sufficiency suppose that the mean value inequality (9) is true.

In the end, a mean value inequality is extended to a Cartan-Hadamard manifold.

It follows from (5.74), the mean value inequality and (5.75) that (5.76).

In this section, we introduce a mean value inequality for Cartan-Hadamard manifold which is an extension of the result proved by Antczak [17] and Barani and Pouryayevali [12].

Next, using Lemma 5, the mean value theorem, the Schwarz inequality and Lemma 4, we have ∥ ∫ 0 ∞ N n, α, β ( s ) ( x, t ) [ f ( s ) − f ( s ) ( x ) s !

Applying the mean value theorem, the Hölder inequality, Lemma 3.1, and (2.1 4, it is clear that the two terms on the right-hand side of (3.16) can be estimated as follows: begin{aligned} bigl|langle v,F_{1} ranglebigr|+bigl|langle u,F_{2} ranglebigr|&leq Cint_{Omega}|mathbf{v}unabla u|,dx &leq C|mathbf{v}|_{L^{3}}|u|_{L^{6}}|nabla u| &leq C|mathbf{v}|_{1}|nabla u|^{2} &leq Cdelta|nabla u|^{2}.

Using mean value theorem and Young's inequality, only one learning parameter needs to be tuned online at each step of recursion.

where ∥ u ∥ ∞ = max t ∈ [ 0, 1 ] | u ( t ) |. Proof The proof follows easily from the mean value theorem and the Hölder inequality, so we omit it here.

by the mean value theorem and for the second inequality in (3.4), we have used the bound.