Chapter I

Inequalities of convexity

1. The fundamental inequality of convexity

Let X be a set; in the vector space $\mathbf{R}^X$ of all finite numerical functions$^1$ defined on X, let P be the set of all positive real-valued functions on X. On the other hand, let $M$ be a numerical function,$^2$ finite or not, with values $\geqslant 0$, defined on P, such that:

$1^\circ$ $M(0) = 0$, and $M$ is positively homogeneous, that is, $M(\lambda f) = \lambda M(f)$ for every real number $\lambda > 0$.$^3$

$2^\circ$ $M$ is increasing in P, in other words the relation $f \leqslant g$ implies $M(f) \leqslant M(g)$.

$3^\circ$ $M$ is convex in P, in other words (TVS, II, §2, No. 8) satisfies the relation $M(f + g) \leqslant M(f) + M(g)$.

Example. — Suppose X is a finite set, for example the interval $[1, n]$ of $\mathbf{N}$; denoting by $x_i$ ($1 \leqslant i \leqslant n$) the coordinates of a vector $\mathbf{x} \in \mathbf{R}^n$, the functions

Remark. — Let S be a pointed convex cone contained in P (that is, a set such that $S + S \subset S$ and $\lambda S \subset S$ for $\lambda \geqslant 0$; cf. TVS, II, §2, No. 4); let $M$ be a numerical function (finite or not) with values $\geqslant 0$, defined on S and satisfying in S the above conditions $1^\circ, 2^\circ$ and $3^\circ$. Then $M$ can be extended to the entire set P, in such a way that the extended function (which we denote again by $M$) satisfies the same conditions: it suffices, for every function $f \in P$, to set $M(f) = +\infty$ if there does not exist any function $g \in S$ such that $f \leqslant g$, and $M(f) = \inf_{g \in S, f \leqslant g} M(g)$ in the contrary case. This procedure will be applied in Ch. IV, §1 to define the upper integral of a positive function.

(1) Fonction numérique finie—a function with values in $\mathbf{R}$—may also be translated as "real-valued function" (cf. GT, IV, §5, No. 1).

(2) Fonction numérique—"numerical function"—signifies a function with values in $\overline{\mathbf{R}}$ (TG, IV, §5, No. 1). The phrase "finite or not" is sometimes added as a reminder that the function may take on infinite values.

(3) Recall that in $\overline{\mathbf{R}}$, products such as $0 \cdot (+\infty)$ are not defined (GT, IV, §4, No. 3).

PROPOSITION 1. — Let $\varphi(t_1, t_2, \ldots, t_n)$ be a finite numerical function, defined and continuous for $t_i \geq 0$ ($1 \leq i \leq n$), such that: $1^\circ$ the relations $t_i > 0$ ($1 \leq i \leq n$) imply $\varphi(t_1, t_2, \ldots, t_n) > 0$; $2^\circ$ the function $\varphi$ is positively homogeneous; $3^\circ$ the set $K \subset \mathbf{R}^n$ defined by the relations $t_i \geq 0$ ($1 \leq i \leq n$), $\varphi(t_1, t_2, \ldots, t_n) \geq 1$ is convex. Under these conditions, if $f_1, f_2, \ldots, f_n$ are $n$ finite functions $\geq 0$ defined on $X$, such that $M(f_i) < +\infty$ for $1 \leq i \leq n$, then

One knows, by the Hahn-Banach theorem (TVS, II, §5), that $K$ is the intersection of the $n$ half-spaces $t_i \geq 0$ ($1 \leq i \leq n$) and a family of closed half-spaces $(U_\iota)_{\iota \in I}$, $U_\iota$ being defined by a relation of the form

where the $\alpha_{\iota k}$ are not all zero. By hypothesis, if $t = (t_i)$ is such that $t_i > 0$ for $1 \leq i \leq n$, then $\varphi(t_1, \ldots, t_n) > 0$, therefore there exists a $\lambda_0 > 0$ such that the relation $\lambda \geq \lambda_0$ implies $\lambda t \in K$; this shows that, for each $\iota \in I$, the relations $t_i \geq 0$ ($1 \leq i \leq n$) imply $\alpha_{\iota 1} t_1 + \cdots + \alpha_{\iota n} t_n \geq 0$ and therefore $\alpha_{\iota k} \geq 0$ for $1 \leq k \leq n$; it is then clear that $K$ is also the intersection of the half-spaces $t_i \geq 0$ ($1 \leq i \leq n$) and the $U_\iota$ such that $\beta_\iota \geq 0$; moreover, since the origin does not belong to $K$, there exists at least one index $\iota$ such that $\beta_\iota > 0$.

Now let $C$ be the convex cone in $\mathbf{R}^{n+1}$ defined by the relations $t_i \geq 0$ ($1 \leq i \leq n+1$), $t_{n+1} \leq \varphi(t_1, t_2, \ldots, t_n)$ (the closure of the convex cone generated in $\mathbf{R}^{n+1}$ by the convex set $K \times \{1\}$); it is immediate that $C$ is also defined by the relations $t_i \geq 0$ ($1 \leq i \leq n+1$) and

For every $x \in X$, we therefore have

for all $\iota \in I$. For every index $\iota$ such that $\beta_\iota > 0$, it follows from (4) and the hypotheses on $M$ that $M(\varphi(f_1, f_2, \ldots, f_n))$ is finite and

and this relation is also verified in an obvious manner if $\beta_\iota = 0$. We thus see that the point with coordinates

belongs to $C$, which proves the proposition.

2. The inequalities of Hölder and Minkowski

In this No. and in the following one, $X$ and $P$ have the same meaning as in No. 1, and $M$ denotes a function defined on $P$ that satisfies the conditions listed in No. 1.

PROPOSITION 2. — Let $\alpha$ and $\beta$ be two numbers such that $0 < \alpha < 1$, $0 < \beta < 1$, $\alpha + \beta = 1$. If $f$ and $g$ are two finite functions $\geq 0$ defined on $X$, and if $M(f)$ and $M(g)$ are finite, then

(Hölder’s inequality).

By Prop. 1, it all comes down to proving that, in $\mathbf{R}^2$, the set defined by the relations $t_1 \geq 0$, $t_2 \geq 0$, $t_1^\alpha t_2^\beta \geq 1$ is convex, or again (FRV, I, §4, No. 1, Def. 1) that the function $u(t) = t^{-\alpha/\beta}$ is convex for $0 < t < +\infty$. Now, setting $r = \alpha/\beta$, we have $D^2 u(t) = r(r+1)t^{-r-2}$ and, since $r > 0$, $D^2 u(t) > 0$ on $]0, +\infty[$, which proves the proposition (FRV, I, §4, No. 4, Cor. of Prop. 8).

COROLLARY. — Let $\alpha_i$ ($1 \leq i \leq n$) be $n$ numbers $\geq 0$ such that

We can restrict attention to the case that $\alpha_i > 0$ for every $i$. It suffices to argue by induction on $n$, by applying the inequality (5) to the numbers $\alpha = \alpha_1$ and $\beta = \sum_{i=2}^n \alpha_i$, and to the functions $f = f_1$, $g = (f_2^{\alpha_2} f_3^{\alpha_3} \cdots f_n^{\alpha_n})^{1/\beta}$.

PROPOSITION 3. — Let $p$ be a real number $\geq 1$. If $f$ and $g$ are two finite functions $\geq 0$ defined on $X$, then

(Minkowski’s inequality).

We can restrict attention to the case that $M(f^p)$ and $M(g^p)$ are finite. By Prop. 1, we are reduced to proving that the set in $\mathbf{R}^2$ defined by the relations $t_1 \geq 0,\ t_2 \geq 0,\ t_1^{1/p} + t_2^{1/p} \geq 1$ is convex, or again that the function $u(t) = (1 - t^{1/p})^p$ is convex for $0 \leq t \leq 1$. Now,

for $0 < t \leq 1$, whence the proposition.

3. The semi-norms $N_p$

Let $p$ be a real number $\geq 1$ and let $\mathscr{F}^p(X, M)$ be the set of finite numerical functions $f$ defined on $X$ such that $M(|f|^p)$ is finite. It is obvious that if $g$ is a function belonging to $\mathscr{F}^p(X, M)$ and if $|f| \leq |g|$, then $f$ also belongs to $\mathscr{F}^p(X, M)$; this remark and Minkowski’s inequality show that the sum of two functions in $\mathscr{F}^p(X, M)$ also belongs to this set; taking into account the fact that $M$ is positively homogeneous, we thus see that $\mathscr{F}^p(X, M)$ is a linear subspace of the space $\mathbf{R}^X$ of all finite numerical functions defined on $X$.

For every number $p > 0$ and every finite numerical function $f$ defined on $X$, set

then $N_p(\lambda f) = |\lambda| N_p(f)$ for every scalar $\lambda$; moreover, if $p \geq 1$ then, by (7),

which proves that $N_p$ is a semi-norm on the vector space $\mathscr{F}^p(X, M)$ (TVS, II, §1).

Proposition 4. — Let $p$ and $q$ be two real numbers $> 0$ and set $1/r = 1/p + 1/q$. For any finite numerical functions $f, g$ defined on $X$,

provided that $N_p(f)$ and $N_q(g)$ are finite.

For, the relation (9) may be written

which is none other than Hölder’s inequality (5) applied to the numbers $\alpha = r/p$ and $\beta = r/q$ and to the functions $|f|^p$ and $|g|^q$.

COROLLARY. — Suppose that $M(1) = 1$; then, for every finite numerical function $f$ defined on $X$, the mapping $p \mapsto N_p(f)$ is increasing in $]0, +\infty[$.

Applying the inequality (9) to the case that $g = 1$, we see that $N_r(f) \leq N_p(f)$ for all $q > 0$; since the number $r$ defined by $1/r = 1/p + 1/q$ runs over the set of numbers such that $0 < r < p$ when $q$ runs over the set of numbers $> 0$, the corollary is proved.

PROPOSITION 5. — For every finite numerical function $f$ defined on $X$, the set $I$ of values of $1/p$ ($p > 0$) such that $N_p(f)$ is finite is either empty or is an interval; if $I$ is not reduced to a point, then the mapping $\alpha \mapsto \log N_{1/\alpha}(f)$ is either convex on $I$ or is equal to $-\infty$ on the interior of $I$.

Let $r$ and $s$ be two distinct numbers $> 0$ such that $1/r$ and $1/s$ belong to $I$; it all comes down to proving that if

with $0 < t < 1$, then

or, what comes to the same,

a relation that may, by the definition of $N_p$, be written

Setting $\alpha = tp/r$, we have $1-\alpha = (1-t)p/s$ by the relation that defines $p$ as a function of $t, r, s$; whence $p = \alpha r + (1-\alpha)s$. Hölder’s inequality now yields

which is none other than the inequality (12).

Exercises

1) With the hypotheses of No. 1, show that the set of bounded functions on X such that $M(|f|)$ is finite is a subalgebra A of $\mathbf{R}^X$, and that the set of bounded functions on X such that $M(|f|) = 0$ is an ideal in A. If, moreover, $M(1)$ is finite, show that the mapping $f \mapsto M(f)$ is continuous when A is equipped with the topology of uniform convergence in X.

2) Let X be the interval $[0, +\infty[$ of $\mathbf{R}$, S the convex cone formed by the functions defined on X such that $0 \leq f(x) \leq kx$ on X (for a finite number $k > 0$ depending on $f$). Set $M(f) = 0$ for $f \in S$, and $M(f) = +\infty$ for every positive function $f$ defined on X and not belonging to S. Show that $M$ satisfies the conditions of No. 1, and that $M(x) = 0$, and $M(x^r) = +\infty$ for every number $r > 0$ not equal to 1.

3) Give an example where X is a set with two elements, $N_p(\mathbf{x})$ is finite for every $p > 0$ and every $\mathbf{x} \in \mathbf{R}^2$, but where there exist values of $p$ such that the mapping $p \mapsto N_p(\mathbf{x})$ is not differentiable at these points.

4) Deduce the inequality (6) from the inequality of the geometric mean

(FRV, III, §1, No. 1, Prop. 2). (Reduce to the case that $M(f_i) = 1$ for $1 \leq i \leq n$.)

5) Let $\alpha$ be a real number $> 1$ and let $\beta = 1 - \alpha < 0$. Let $g$ be a finite function, defined on X, such that $g(x) > 0$ for all $x \in X$ and such that $M(g) > 0$; show that for every finite function $f \geq 0$ defined on X such that $M(f)$ is finite,

(apply Hölder’s inequality suitably).

6) Deduce Minkowski’s inequality from Hölder’s inequality (find an upper bound for $M((f+g)^{p-1})$ with the help of Hölder’s inequality). If one assumes that $M(f+g) = M(f) + M(g)$ for every pair of functions $f, g$ defined and $\geq 0$ on $X$, deduce similarly from Exer. 5 the inequality

in the following cases:

a) $0 < p < 1$, $f$ and $g$ finite functions $\geq 0$ defined on $X$, such that $f(x) + g(x) > 0$ for all $x \in X$ and such that $M(f^p)$ and $M(g^p)$ are finite;

b) $p < 0$, $f$ and $g$ finite functions defined on $X$, such that $f(x) > 0$ and $g(x) > 0$ for all $x \in X$, $M(f^p)$ and $M(g^p)$ are finite, and $M((f+g)^p) > 0$.