Sentence examples for the required inequality from inspiring English sources
which is the required inequality.
almost surely, which is the required inequality.
By (8), we see that the required inequality is proved.
Thus, we get the required inequality in (3.9).
Let be another additive mapping, which satisfies the required inequality.
Using (2.41) in, we have the required inequality in (2.27).
To prove the uniqueness of T, let S : X→ Y be another additive mapping satisfying the required inequalities.
Applying inequality (2.23) on for as well as the inequalities (3.6) and (3.2) by choosing,,,,, we easily get the required inequalities.
Remark 2.5 The required inequalities in WHCF-Theorem and WPCF-Theorem turn into equalities for x 1 = x 2 = ⋯ = x n = s.
Letting M = A, G, H, P r, I, L, L p in (3.1), (3.2) and (3.3), we can get the required inequalities for a different weight function w ( x ), and the details are left to the interested reader.
Moreover, while variance in payoff can violate the required inequalities of payoffs for a valid PD matrix, a PD game between two individuals is invalid if just a single individual has an invalid PD matrix.
