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It might sound silly, but I am always curious whether Hölder's inequality $$\sum_{k=1}^n |x_k\,y_k| \le \biggl( \sum_{k=1}^n |x_k|^p \biggr)^{\!1/p\;} \biggl( \sum_{k=1}^n |y_k|^q \biggr)^{\!1/q} \text{ for all }(x_1,\ldots,x_n),(y_1,\ldots,y_n)\in\mathbb{R}^n\text{ or }\mathbb{C}^n.$$ can be derived from the Cauchy-Schwarz inequality.

Here $\frac{1}{p}+\frac{1}{q}=1$, $p>1$.

t.b.
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Sunni
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2 Answers2

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Yes it can, assuming nothing more substantial than the fact that midpoint convexity implies convexity. Here are some indications of the proof in the wider context of the integration of functions.

Consider positive $p$ and $q$ such that $1/p+1/q=1$ and positive functions $f$ and $g$ sufficiently integrable with respect to a given measure for all the quantities used below to be finite. Introduce the function $F$ defined on $[0,1]$ by $$ F(t)=\int f^{pt}g^{q(1-t)}. $$ One sees that $$ F(0)=\int g^q=\|g\|_q^q,\quad F(1)=\int f^p=\|f\|_p^p,\quad F(1/p)=\int fg=\|fg\|_1. $$ Furthermore, for every $t$ and $s$ in $[0,1]$, $$ F({\textstyle{\frac12}}(t+s))=\int h_th_s,\qquad h_t=f^{pt/2}g^{q(1-t)/2},\ h_s=f^{ps/2}g^{q(1-s)/2}, $$ hence Cauchy-Schwarz inequality yields $$ F({\textstyle{\frac12}}(t+s))^2\le\int h_t^2\cdot\int h_s^2=F(t)F(s). $$ Thus, the function $(\log F)$ is midpoint convex hence convex. In particular, $1/p=(1/p)1+(1/q)0$ with $1/p+1/q=1$ hence $$ F(1/p)\le F(1)^{1/p}F(0)^{1/q}, $$ which is Hölder's inequality $\|fg\|_1\le\|f\|_p\|g\|_p$.

Did
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    Wow, that is amazing. It is rarely seen in literatures mention Holder's inequality is implied by CS. Is this consideration known before? – Sunni Jul 02 '11 at 22:32
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    That midpoint convex implies convex does require some regularity on a function such as measurability. There's very little chance it won't hold here of course but this is probably worth mentioning. – Zarrax Jul 02 '11 at 23:27
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    @Zarrax: Right. In the present case $F$ is also convex on $[0,1]$ hence continuous on $(0,1)$. This is enough to prove that $\log F$ is convex (and continuous) on $(0,1)$ hence, rather than measurability issues which are not a problem here, the important missing step is that convexity of $\log F$ should be used on $1/p=(1/p)t+(1/q)(1-t)$, then the limits of $F(t)$ and $F(1-t)$ when $t\to0$ should be compared to $F(0)$ and $F(1)$. I deliberately omitted these technicalities in my post, which is the reason why the first paragraph announces some indications of the proof, rather than a proof. – Did Jul 02 '11 at 23:43
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    @Sunni, I think this can interest you: YUAN-CHUAN LI AND SEN-YEN SHAW, A Proof of Hölder's Inequality using the C-S Inequality, Journal of Inequalities in Pure and Applied Mathematics. – leo Jul 03 '11 at 04:13
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    In the book: "THE CAUCHY–SCHWARZ MASTER CLASS" of J. MICHAEL STEELE, in the case of finite sums, as you post, is the exercise 9.3. – leo Jul 03 '11 at 04:23
  • @Sunni: if you want to search the article this can be useful:volume 7, issue 2, article 62, 2006. Received 14 October, 2005; accepted 15 November, 2005. – leo Jul 03 '11 at 04:28
  • The paper by Li and Shaw mentioned by leo is available here: http://www.emis.de/journals/JIPAM/article679.html?sid=679 – Did Jul 03 '11 at 09:35
  • Thank you all. That is interesting. – Sunni Jul 04 '11 at 12:22
  • Is it correct to say that when Hölder is proven using Cauchy-Schwarz it is valid on a smaller class of functions than the real Hölder ? In the sense that it is not proven (at least not here) that Cauchy-Schwarz applies to all $h_s$ and $h_t$ given by any $f\in L^p \quad g\in L^q$ - in order for which we need $h_s,h_t \in L^2$ – James Well Mar 14 '17 at 07:23
  • @JamesWell Cauchy-Schwarz inequality $$\left(\int h_th_s\right)^2\leqslant\int h_t^2\cdot\int h_s^2$$ holds even when $h_t$ or $h_s$ is not square integrable since then the RHS is infinite. Thus, no, no "smaller class of functions than the real Hölder" here. – Did Mar 14 '17 at 07:36
  • Fair point ! Then since $h_sh_t$ is positive as I just realised, the integral is well-defined. My mistake – James Well Mar 15 '17 at 17:56
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I suggest the following consideration. We will prove the above inequality for rational $p,q\in(1,\infty)$ with $\frac1p+\frac1q= 1$, and the irrational cases follow by continuity.

If $p$ and $q$ are rational, let $p=\frac ab$ and $q=\frac ac$ with $b+c=a$ and $2^m\ge a$. Now by induction $$ \sum |x^{(1)}\dots x^{(2^m)}| \le \left( \sum |x^{(1)}\cdots x^{(2^{m-1})}|^2 \right)^{\frac12}\cdot\left( \sum |x^{(2^{m-1}+1)}\cdots x^{(2^{m})}|^2 \right)^{\frac12}$$ $$\le \left( \sum |x^{(1)}|^{2^m} \right)^{\frac1{2^m}}\cdots \left( \sum |x^{(2^m)}|^{2^m} \right)^{\frac1{2^m}}$$ where $x^{(i)}$ are sequences of length $n$, whose indices are omitted. By plugging in $$x^{(1)}= \dots= x^{(b)}= x^{\frac a{b2^m}},\quad x^{(b+1)}= \dots= x^{(b+c)}= y^{\frac a{c2^m}},\quad x^{(b+c+1)}= \dots = x^{(2^m)}=(xy)^{\frac1{2^m}}$$ we get $$\sum |xy|= \sum |x^{\frac a{2^m}}y^{\frac a{2^m}}(xy)^{\frac {2^m-a}{2^m}}| \le \left( \sum |x|^{p} \right)^{\frac{b}{2^m}}\cdot \left( \sum |y|^{q} \right)^{\frac c{2^m}}\cdot \left( \sum |xy| \right)^{\frac {2^m-a}{2^m}}$$ implying the inequality we aimed for by dividing the third term of the RHS, and putting the inequality to the $\frac{2^m}{a}$-th power.

Peter Patzt
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