In [20] the authors have applied this inequality for the error bounds of general Riemman's quadrature formulae in terms of ∥ f ′ ∥ 1.
Take an arbitrary vector (x in I^{n}) and, for (k in{1,dots,n}), denote In what follows, we are going to prove that, for all (j in{2, dots,n}), (3.2) Then, applying this inequality for all (j in{2,dots,n}), summing up side by side, after simple reduction, we get Then, after dividing both sides of this inequality by (Lambda_{n}), we arrive at (3.1).
Finally, we apply these inequalities for f-divergence measure in information theory in Section 5.
Applying the inequalities for and for and using (3.3), for we have (4.51).
Applying the inequalities for, for and using (3.3), for we have (4.31).
On the other hand, we apply another inequality for the source term: f ε ( t ) ≤ σ ( r ) μ | u ( t + ε ) | r − 2 + | u ( t ) | r − 2 | u ε ( t ) |, Open image in new window (26).
We then apply this to establish a geometric Poincaré inequality for stable solutions of the above system for general vector fields X and Y.
Recently, Moudafi [9] applied this algorithm for variational inequalities; Moudafi and Elisabeth [10] studied the algorithm by using enlargement of a maximal monotone operator; Moudafi and Oliny [11] considered convergence of a splitting inertial proximal method.
Applying Hölder's inequality, for we have (4.29).
Observe that applying Young's inequality for convolutions leads to ∥ C ˜ k, q ∥ 2 ≤ 1.
