Geometric progression in an inequality

Question

Problem: Show that if $a>0$ and $n>3$ is an integer then $$\frac{1+a+a^2 \cdots +a^n}{a^2+a^3+ \cdots a^{n-2}} \geq \frac{n+1}{n-3}$$

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I am unable to prove the above the inequality.

I used the geometric progression summation formula to reduce it to proving $\frac{a^{n+1}-1}{a^2(a^{n-3}-1)} \geq \frac{n+1}{n-3}$.

Also writing it as $$\frac{1+a+a^2 \cdots +a^n}{n+1} \geq \frac{a^2+a^3+ \cdots a^{n-2}}{n-3}$$ seems to suggest that some results on mean-inequalities can be used but I can't figure out what that is.

Answer 1

In fact, that identity is useful. In details, by AM-GM inequality, $$ \left(n-3\right)\left(1+a^n\right) \ge 2\left(a^2+\cdots+a^{n-2}\right) $$ and $$ \left(n-3\right)\left(a+a^{n-1}\right) \ge 2\left(a^2+\cdots+a^{n-2}\right) $$ Thus, $$ LHS = \frac{1+a+a^{n-1}+a^{n}}{a^2+\cdots+a^{n-2}}+1 \ge \frac{4}{n-3}+1 =RHS $$

Answer 2

What you may also consider, but this is more clunky than other suggestions, is to write $\lambda(n,a) = 1 + a + ... + a^n$ and $\mu(n,a) = a^2 + a^3 + ... + a^{n-2}$ and notice that $\lambda(n,a=1) = n + 1$, $\mu(n,a=1) = n - 3$, so that the equality holds for $a = 1$. Then you would have to show that the function $g(n,a) = \frac{\lambda(n,a)}{\mu(n,a)}$ has a maximum at $a = 1$, so would need to calculate $\frac{d g(n,a)}{d a}$ and $\frac{d^2 g(n,a)}{d^2 a}$.

Answer 3

The inequality holds trivially for $a = 1$, and it is invariant under the substitution $a \to 1/a$. Therefore it suffices to prove the inequality for $\mathbf{a > 1}$.

The hyperbolic sine is a convex function on $[0, \infty)$, so for $a > 1$ the function $$ f(x) = 2 \sinh \bigl(\log a \cdot \frac x2 \bigr) = a^{x/2} - a^{-x/2} = \frac{a^x - 1}{a^{x/2}} \quad (x \ge 0) $$ is also convex, and therefore $$ \frac{f(x) - f(0)}{x-0} = \frac{a^x - 1}{x \, a^{x/2}} $$ is increasing. It follows that for $0 \le x \le y$ $$ \frac{a^y - 1}{y \, a^{y/2}} \ge \frac{a^x - 1}{x \, a^{x/2}} $$ which is equivalent to $$ \frac{a^y-1}{a^x-1} \ge a^{(y-x)/2} \frac yx \, . $$

The desired inequality follows as a special case for $x = n-3$ and $y = n+1$, but we have shown that the inequality can be generalized to arbitrary positive real numbers as exponents.