Bound product of two numbers by sum of their squares

Question

How can I bound the product of two reals by the sum of their squares?

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Let $s = a^2 + b^2$ and $p = ab$. Can I find a constant $C$ and an exponent $\alpha$ such that this holds? $$ p \leq Cs^\alpha $$ If not, I am interested in any other bound involving the two variables.

Answer 1

$$ab\le \frac{a^2 + b^2}{2}$$ which follows from $(a-b)^2 \ge 0$. I dont think you can do better than this.