Exercise 10, § 3

a) If $h, u, v'$ are primitives of real regulated functions on an open interval ]$a, b[$ of $\mathbf{R}$, if $v$ is a primitive of $v'$, and if $v(x) > 0$ on this interval, then one has the Redheffer identity

at those points of ]$a, b[$ where the derivatives are defined.

*b) Let $v, w$ be two functions $> 0$ on ]$a, b[$, which are primitives of regulated functions $v' > 0, w' \leq 0$. Deduce from a) that for every function $u$ which is a primitive of a regulated function $u'$ on ]$a, b[$ and such that $\liminf_{x \to a, x > a} u(x) = 0$, the hypothesis that the integral $\int_a^b \frac{w(x)}{v'(x)} {u'}^2(x) dx$ is convergent implies that the integral $\int_a^b \frac{u'(x)}{v(x)} u^2(x) dx$ is convergent and that

is finite, and also the inequality (Putting $h = w/v'$, first observe that $\int_c^x dt/h(t) \leq v(x)/w(x)$ for $a < c < x < b$ and remark that by the Cauchy-Schwarz inequality

to deduce that

and then integrate the Redheffer identity.)

c) Deduce from (*) that if $u$ is the primitive of a regulated function on ]0, 1], if

and if the integral $\int_0^1 {u'}^2(t)\,dt$ converges, then so does $\int_0^1 (u(t)/t)^2\,dt$ and for all $\alpha > 0$ one has the inequality

d) Let $\alpha$ be a number $> 0$, let K be a function $> 0$, differentiable and decreasing on ]0, $+\infty[$ and such that $\lim_{x \to +\infty} K(x) = 0$. If $u$ is the primitive of a regulated function on ]0, $+\infty[$, such that $\liminf_{x \to 0,\ x > 0} u(x) = 0$, and if the integral $\int_0^{+\infty} x^{1-\alpha} K(x){u'}^2(x)\,dx$ converges, then the function $x^{-\alpha} K(x)u^2(x)$ tends to 0 as $x$ tends to 0 or to $+\infty$, the integral

converges, and one has

(take $v(x) = x^\alpha,\ h(x) = x^{1-\alpha} K(x)$ in the Redheffer identity).

In particular, for $\alpha = \frac{1}{2}$ and $K(x) = x^{-1/2}$, one has

(Hardy-Littlewood inequality).

e) Let $\alpha \geq -1$, let K be a function $\geq 0$, differentiable and increasing on [0, $+\infty[$. Suppose that the integrals

and

converge. Then $K(x)u^2(x)$ tends to 0 as $x$ tends to $+\infty$, and one has

(take $v(x) = \exp(-cx^\alpha)$ in the Redheffer identity, with a suitable constant $c$).

In particular, if $a$ and $b$ are constants such that $b + 1 \geq a$ and $a + b \geq 0$, one has

if the integrals on the right-hand side converge (generalized H. Weyl inequality).

f) If $0 < \alpha < 2$ and $u$ is a primitive of a regulated function on $[0, \alpha]$ and $u(0) = 0$, one has

(generalized Opial inequality). (Apply (*) suitably, replacing $u(x)$ by $e^{-x} u(x)$.)

g) If $u$ is a primitive of a regulated function on $[0, b]$ and $u(0) = 0$, then

(Hlawka's inequality) (same method as in f).

h) If $u$ is the primitive of a regulated function on $\mathbf{R}$ then, for all $t \in \mathbf{R}$,

if the two integrals on the right-hand side converge (consider the two intervals $[-\infty, t]$ and $[t, +\infty[$; take $v(x) = e^{\alpha x}$ on the two intervals, $h(x) = 1/\alpha$ on the first interval and $h(x) = -1/\alpha$ on the second, then choose $\alpha > 0$ suitably).