another proof of Young inequality

Let

$$F(x)={\int }_0^x\varphi (t)𝑑t,\text{ and }G(x)={\int }_0^x{\varphi }^{-1}(t)𝑑t.$$

Since ${\varphi }^{-1}$ is strictly increasing, $G$ is strictly convex, hence lies above its supporting line, i.e. for every $c$ and $x\ne c$

$$G(b)>G(c)+G^{\prime }(c)(b-c)=G(c)+{\varphi }^{-1}(c)(b-c).$$

In particular, for $c=\varphi (a)$ we have

$$F(a)+G(b)>F(a)+G(\varphi (a))+a(b-\varphi (a))=ab,$$

because $F(a)+G(\varphi (a))=a\varphi (a)$.