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- I would appreciate some help on proofs involving the Lp norm below, please. I looked up Holder's inequality, which might help with this problem, but I'm not sure how to set up the proofs.
How would I prove part (1)? By inspection, the l1 norm will be greater that the lp form when p > 1, since l1 is just the sum of the magnitudes...just not sure how to show work that is proof-based.[tex](a_1+a_2+a_3)^\alpha \ge a_1^\alpha+(a_2+a_3)^\alpha \ge a_1^\alpha + a_2^\alpha + a_3^\alpha [/tex]
[tex]a_1+a_2+a_3 \ge [a_1^\alpha + a_2^\alpha + a_3^\alpha]^{\frac{1}{\alpha} }[/tex]
and so on.
Bonus
[tex]f(x,y):=(x+y)^a-x^a-y^a[/tex]
[tex]x,y \ge 0 ,\ \ a \ge 1[/tex]
[tex]\frac{\partial f}{\partial x} \ge 0,\ \ \frac{\partial f}{\partial y} \ge 0, \ \ f(0,0)=0[/tex]
I see that, how does this relate to part 1? I’m slow at understanding it seems.I stopped at 3 but you should continue upto N
[tex](a_1+a_2+...+a_N)^\alpha \ge a_1^\alpha + a_2^\alpha + ... +a_N^\alpha[/tex]
[tex]a_1+a_2+...+a_N \ge (a_1^\alpha + a_2\alpha + ... +a_N^\alpha)^\frac{1}{\alpha}[/tex]
Let ##a_k=|x_k|##
Okay. this makes sense. Thanks. How would part 2 be proved? I'm thinking I would need to choose some a_i and α and by using the convexity of the function f = x^α. Any thoughts here?[tex](|x_1|+|x_2|+...+|x_N|)^p \ge |x_1|^p +|x_2|^p + ... +|x_N|^p[/tex]
[tex]l_1\ norm=|x_1|+|x_2|+...+|x_N| \ge (|x_1|^p +|x_2|^p + ... +|x_N|^p)^\frac{1}{p}=l_p \ norm[/tex]
Why don’t you do it ?I'm thinking I would need to choose some a_i and α and by using the convexity of the function f = x^α.
since you know the Lp norm is a norm, you should be able to recognize that (1) is implied by triangle inequality, i.e.How would I prove part (1)? By inspection, the l1 norm will be greater that the lp form when p > 1, since l1 is just the sum of the magnitudes...just not sure how to show work that is proof-based.
Are your hints regarding parts 2 & 3?