Consider the sequence of numbers an defined by the iteration (1) for n ≥ k, where pr 1 ≤ r ≤ k) are non-negative numbers with and the starting values are given. So an is a weighted average of the previous k terms.

The basic inequalities

The inequalities in question are(1)

and(2)

for x > 1.

Recall that for a 2π-periodic function f , the Fourier coefficients on the interval (–Π, Π) are

and the Fourier series for f is

The celebrated theorem of Weierstrass, dating from 1885, states that continuous functions can be uniformly approximated by polynomials on any bounded, closed interval. But just how well can we approximate by polynomials of a certain degree? Let us introduce some notation to facilitate the discussion. For an interval I (which will usually be ), denote by C (I) the space of continuous functions on I, and write for (the notation is often used). Uniform convergence of fn to f (on I) equates to the statement that . Denote by the space of polynomials of degree not more than n. This is a linear subspace of C (I) of dimension n + 1. We write

The prolific output of G. H. Hardy included a number of inequalities, each known, in its own context, simply as ‘Hardy’s inequality’. Here we give an account of one of them, together with some applications and generalisations. It relates to averages.

Consider the problem of finding triples of numbers (x1, x2, x3) and (y1, y2, y3) satisfying (1)

$${x_1} + {x_2} + {x_3} = {y_1} + {y_2} + {y_3}$$

(2)

$$x_1^2 + x_2^2 + x_3^2 = y_1^2 + y_2^2 + y_3^2.$$

Given positive numbers xj, yj such that $\sum\limits_{i = 1}^n {{x_j}} = \sum\limits_{j = 1}^n {{y_j}} $ , it can happen that $\sum\limits_{j = 1}^n {x_j^2} = \sum\limits_{j = 1}^n {y_j^2} $ : for example, $({x_j}) = (7,\,3,\,2),\,({y_j}) = (6,\,5,\,1)$ . However, such cases are exceptional.

The ‘midpoint’ approximation to the integral $$\int_0^1 f $$ is

$${M_n}\left( f \right) = {1 \over n}\sum\limits_{r = 1}^n f \left( {{{2r - 1} \over {2n}}} \right)$$

Here we give an account of two appealingly simple techniques to generate monotonic sequences that were developed by the author and Grahame Bennett. Sadly, Bennett died in 2016 after a lifetime of distinguished contributions in the field of inequalities. This article is dedicated to him.

A famous theorem of Weierstrass, dating from 1885, states that any continuous function can be uniformly approximated by polynomials on a bounded, closed real interval.

D'Alembert's ratio test, a very basic plank in the theory of infinite series, can be stated as follows:

Suppose that an > 0 for all n ≥ 1. Then:

1. (i) if for some n0 and some ρ < 1, we have for all n ≥ n0, then is convergent;

2. (ii) if for some n0, we have for all n ≥ n0, then is divergent.

The well-known Fibonacci numbers Fn are defined by the recurrence relation

Fn = Fn – 1 + Fn – 2. (1)

together with the starting values F0 = 0, F1 = 1, or equivalently F1 = F2 = 1.

We record the first few:

The recurrence relation can also be applied backwards in the form Fn = Fn + 2 – Fn + 1 to define Fn for n < 0. An easy induction verifies that F−n = (−1)n – 1Fn for n > 0.

As usual, write

In [1], it is shown, by an elegant method, that for all n ⩾ 2, Hn is not an integer. The method can be traced back at least to [2, Exercise 251, p. 159]. Actually, rather more is shown: if Hn is expressed as a fraction an/bn in its lowest terms, then for all n ⩾ 2, bn is even. One way of looking at this is to say that once the term has entered the sum, the factor 2 persists in the denominator from then on. This suggests the following question:

(Q1) for primes p ⩾ 3, is bn a multiple of p for all n ⩾ p?

and more generally:

(Q2) for any prime p, is bn a multiple of pm for all n ⩾ pm?

In other words, once 1/pm has entered the sum, does the factor pm persist in the denominator?

Let dn denote the lowest common multiple of 1, 2, … , n. This number can be described as follows: for each prime p ⩽ n, let mp be the largest integer m such that pm ⩽ n. Then . Hence a positive answer to (Q2) would imply that bn simply equals dn.

Consider the integral

where b > a > 0. First, let us clarify why it even exists. Of course, convergence at infinity is ensured by the exponential terms, but the integrals of and e–ax/x and e–bx/x, taken separately, are divergent at 0, since these integrands equate asymptotically to 1/x as x → 0. However,

so (e–ax – e–bx)/x tends to the finite limit b – a as x → 0 and there is no problem integrating it on intervals of the form [0, r].

A neat way to evaluate I1 starts by expressing the integrand itself as an integral:

(1)

Inserting this into I1 converts it into a double integral.

Recall the integral definition of the gamma function: for a > 0. By splitting this integral at a point x ⩾ 0, we obtain the two incomplete gamma functions:

(1)

(2)

Γ(a, x)is sometimes called the complementary incomplete gamma function. These functions were first investigated by Prym in 1877, and Γ(a, x) has also been called Prym's function. Not many books give these functions much space. Massive compilations of results about them can be seen stated without proof in [1, chapter 9] and [2, chapter 8]. Here we offer a small selection of these results, with proofs and some discussion of context. We hope to convince some readers that the functions are interesting enough to merit attention in their own right.