[For the entire collection see Zbl 0746.00079.]
(HELP stands for Hardy, Everitt, Littlewood and Pólya.) This is a survey paper too rich in contents, with intricate results and conditions, to allow details in such a review (of a review). Suffice to say that the HELP integral inequalities are of the form \[ \begin{split} \left(\int^ b_ a(q(x)f(x)^ 2+p(x)f'(x)^ 2)dx\right)^ 2\\ \leq K\int^ b_ aw(x)f(x)^ 2dx\int^ b_ aw(x)^{-1}(q(x)f(x)-p'(x)f'(x)- p(x)f''(x))^ 2dx.\end{split} \] (The integrals may be improper.) Great emphasis is put on determining the (best possible) constant \(K\). In most cases this proves to be too difficult analytically but a good numerical method is offered. Connections are established with Sturm-Liouville differential equations, the spectral theory of their differential operators and orthogonal polynomials.
The analogous inequalities for infinite series are given a similar treatment.
{There are several misprints, more amusing than troublesome, such as “in principal” in line 19 on page 293. Sections 3.3.1 and 3.3.2 are misnumbered to 3.2.1 and 3.2.2. In line 10 on page 290, (3.18) should stand rather than (3.17).}.