There are some useful inequalities for , where p is a number ranging from 0 to infinity. These are the top 3 useful inequalities (note some of them only work for p less than 1, or p greater than 1).

**1)**

For , , where .

Proof:

**2)**

If , for all .

Proof:

Hence .

**3)**

For , for .

Proof:

By convexity of for ,

Multiplying throughout by gives

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