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I recently ran across the inequality $$(a+b)^p\le 2^{p-1}(a^p + b^p),$$ valid for positive real $a,b$ and $p\ge1$.

It has an amusing 2 line proof; it is occasionally very useful. Does it have a name? You know, like "Bernstein's Other Inequality" or "Minkowski's Lesser Inequality", or something?

I've looked in Beckenbach and Bellmans' Inequalities book, without luck.

kimchi lover
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    That is the midpoint-convexity for the function $t \mapsto t^p$, see for example https://math.stackexchange.com/a/978687/42969 or https://math.stackexchange.com/a/700853/42969 – Martin R Aug 18 '24 at 07:16
  • @MartinR Thanks. I used the log convexity of $t\mapsto\cosh t$. – kimchi lover Aug 18 '24 at 11:18

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I just realized this is Hölder's inequality appied to the vectors $(a,b)$ and $(1,1)$ in $\mathbb R^2$: if $q=p/(p-1)$, then $$ |(a,b)\cdot(1,1)|\le \|(a,b)\|_p \|(1,1)\|_q,$$ and so $$ |(a,b)\cdot(1,1)|^p\le \|(a,b)\|_p^p \|(1,1)\|_q^p,$$ where the first factor is $a^p+b^p)$ and the last factor is $2^{p/q}=2^{p-1}$.

kimchi lover
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