Young's inequality

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In mathematics, Young's inequality states that if a, b, p and q are positive real numbers with 1/p + 1/q = 1 then we have

ab \le \frac{a^p}{p} + \frac{b^q}{q}.

Equality holds for ap = bq since ab = a(b^q)^{1 \over q} = aa^{p \over q} = a^p = {a^p \over p} + {b^q \over q}.

Young's inequality is a special case of the inequality of weighted arithmetic and geometric means.

[edit] Usage

Young's inequality is used in the proof of the Hölder inequality.

[edit] Proof

We know that the function f(x) = ex is convex, since its second derivative is positive for any value. Thus, it follows:

ab = e^{\ln(a)}e^{\ln(b)} = e^{{1 \over p}\ln(a^p) + {1 \over q}\ln(b^q)} \le {1 \over p}e^{\ln(a^p)}+{1 \over q}e^{\ln(b^q)} = {a^p \over p} + {b^q \over q}.

Here we used the defining property of convex functions: for any t between 0 and 1 inclusively,

f(tx+(1-t)y)\leq t f(x)+(1-t)f(y).
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