Holder inequality

The supremum is defined to be the smallest number M such that l/ (r)l M almost everywhere. The term almost everywhere has a precise mathematical meaning for which we refer to the literature [4]. We almost never use it in the remainder of this paper. We therefore conclude that if v L°° then the expectation value of the external potential is finite. To show this we used that dGL1. But we also know that n L3 and if we make use of the Holder inequality... 

By making use of the strong monotonicity of A, Holder s inequality, and Lemma 1.2, we obtain the estimate... 

Multiply this equation by Then Holder s inequality implies the... 

Applying Holder s inequality and using the strong monotonicity of A, the monotonicity of / , Lemma 1.2 and the estimates for the norms, we obtain... 

For p = q = 2. Holder s inequality is known as the Cauchy-Schwartz inequality. 

It is easy to show that the rate of reaction as described by the lumped dispersed phase analysis underestimates the actual rate by using the Holder s inequality. Let g c) = and h(c) Thus,... 

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