Hello all, I'm trying to find a good resource for a discussion on the relation between say, the p-norm of a vector (from a finite dimensional vector space) and its Euclidean norm. In my search on the internet and in various books, I only encounter basic, standard inequalities such as the Cauchy-Schwarz and Holder's inequality. 


Are there textbooks that go more in depth than these two? 


In particular, I'm interested in the following: if I have two unit vectors $\psi$ and $\phi$ (from $R^d$, say), that are $\epsilon$-close, meaning that $\|\psi - \phi\|\_2 \leq \epsilon$, then what can one say about $\|\psi\|\_p - \|\phi\|\_p$? Intuitively, they must be close as well, but does the closeness depend on $d$, the dimension of the vector-space?


Any references or links or pointers would be greatly appreciated!


Thanks,
Henry



Inequalities and bounds for relating p-norms (Reference request)
It is no better than what you get from Holder. Take the case where one of the vectors is zero. 


