Weighted arithmetic–geometric mean inequality

For a,b≥0 and θ∈(0,1): a^θ b^(1−θ) ≤ θa+(1−θ)b

Weighted arithmetic–geometric mean inequality

Proposition (Weighted AM–GM). For all $a,b\ge 0$ and all $\theta\in(0,1)$ ,

a^\theta b^{1-\theta}\le \theta a+(1-\theta)b.

Context. This inequality can be derived from convexity of the function $x\mapsto -\ln x$ on $(0,\infty)$ and is used to prove Hölder-type inequalities.

Proof sketch. Assume $a,b>0$ . Convexity of $-\ln$ implies

-\ln(\theta a+(1-\theta)b)\le -\theta\ln(a)-(1-\theta)\ln(b).

Exponentiating yields $\theta a+(1-\theta)b\ge a^\theta b^{1-\theta}$ . The cases $a=0$ or $b=0$ follow by continuity or direct inspection.

Example. With $\theta=\tfrac12$ , this becomes $\sqrt{ab}\le \tfrac{a+b}{2}$ .