74. Some algebraical lemmas

It will be convenient to prove at this stage a number of elementary inequalities which will be useful to us later on.

(i) It is evident that if $\alpha >1$ and $r$ is a positive integer then $$r{\alpha }^r>{\alpha }^{r-1}+{\alpha }^{r-2}+\cdots +1.$$ Multiplying both sides of this inequality by $\alpha –1$, we obtain $$r{\alpha }^r(\alpha –1)>{\alpha }^r–1;$$ and adding $r({\alpha }^r–1)$ to each side, and dividing by $r(r+1)$, we obtain $$\begin{array}{}\text{(1)}& \frac{{\alpha }^{r+1}–1}{r+1}>\frac{{\alpha }^r–1}{r}(\alpha >1).\end{array}$$ Similarly we can prove that $$\begin{array}{}\text{(2)}& \frac{1–{\beta }^{r+1}}{r+1}<\frac{1–{\beta }^r}{r}(0<\beta <1).\end{array}$$

It follows that if $r$ and $s$ are positive integers, and $r>s$, then $$\begin{array}{}\text{(3)}& \frac{{\alpha }^r–1}{r}>\frac{a^s–1}{s},\frac{1–{\beta }^r}{r}<\frac{1–{\beta }^s}{s}.\end{array}$$ Here $0<\beta <1<\alpha$. In particular, when $s=1$, we have $$\begin{array}{}\text{(4)}& {\alpha }^r–1>r(\alpha –1),1–{\beta }^r<r(1–\beta ).\end{array}$$

(ii) The inequalities (3) and (4) have been proved on the supposition that $r$ and $s$ are positive integers. But it is easy to see that they hold under the more general hypothesis that $r$ and $s$ are any positive rational numbers. Let us consider, for example, the first of the inequalities (3). Let $r=a/b$, $s=c/d$, where $a$, $b$, $c$, $d$ are positive integers; so that $ad>bc$. If we put $\alpha ={\gamma }^{bd}$, the inequality takes the form $$({\gamma }^{ad}–1)/ad>({\gamma }^{bc}–1)/bc;$$ and this we have proved already. The same argument applies to the remaining inequalities; and it can evidently be proved in a similar manner that $$\begin{array}{}\text{(5)}& {\alpha }^s–1<s(\alpha –1),1–{\beta }^s>s(1–\beta ),\end{array}$$ if $s$ is a positive rational number less than $1$.

(iii) In what follows it is to be understood that all the letters denote positive numbers, that $r$ and $s$ are rational, and that $\alpha$ and $r$ are greater than $1$, $\beta$ and $s$ less than $1$. Writing $1/\beta$ for $\alpha$, and $1/\alpha$ for $\beta$, in (4), we obtain $$\begin{array}{}\text{(6)}& {\alpha }^r–1<r{\alpha }^{r-1}(\alpha –1),1–{\beta }^r>r{\beta }^{r-1}(1–\beta ).\end{array}$$ Similarly, from (5), we deduce $$\begin{array}{}\text{(7)}& {\alpha }^s–1>s{\alpha }^{s-1}(\alpha –1),1–{\beta }^s<s{\beta }^{s-1}(1–\beta ).\end{array}$$

Combining (4) and (6), we see that $$\begin{array}{}\text{(8)}& r{\alpha }^{r-1}(\alpha –1)>{\alpha }^r–1>r(\alpha –1).\end{array}$$ Writing $x/y$ for $\alpha$, we obtain $$\begin{array}{}\text{(9)}& r{x}^{r-1}(x–y)>x^r–y^r>r{y}^{r-1}(x–y)\end{array}$$ if $x>y>0$. And the same argument, applied to (5) and (7), leads to $$\begin{array}{}\text{(10)}& s{x}^{s-1}(x–y)<x^s–y^s<s{y}^{s-1}(x–y).\end{array}$$