``AM-GM Ineq.'' and ``Young's Ineq. for Increasing Functions.''

The following proof is a special case of Young's Ineq. for Increasing Functions where $f(x)=x$ .

A visual proof that for positive real numbers the Arithmetic Mean ≥ Geometric Mean pic.twitter.com/4J3Vh0phCN
— Fermat's Library (@fermatslibrary) 2020年8月8日

$\sqrt{ab}\leq \displaystyle \int_0^{\sqrt{a}} f(u) du +\int_0^{\sqrt{b}} f^{-1}(v)dv$ $\displaystyle \int_0^{\sqrt{a}} u du +\int_0^{\sqrt{b}} v dv$ $=\dfrac{a}{2}+\dfrac{b}{2}$ .

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``AM-GM Ineq.'' and ``Young's Ineq. for Increasing Functions.''