Exercise 9, § 1

Functions of a Real Variable, III, 117

With the notation of exerc. 6 show that

\mu_1 G_1 + \mu_2 G_2 + \cdots + \mu_n G_n \leq \lambda_1 a_1 + \lambda_2 a_2 + \cdots + \lambda_n a_n

provided that $\lambda_k \geq 0$ for each $k$ and

\mu_k = k \left( (\lambda_k \beta_k)^{1/k} - \beta_{k+1}^{1/k} \right) \quad \text{for } 1 \leq k \leq n

with $\beta_k \geq 0$ for all $k$ , $\beta_1 \leq 1$ and $\beta_{n+1} = 0$ (method of exerc. 8).

In particular, for suitable choices of $\lambda_k$ and $\beta_k$ one has

G_1 + G_2 + \cdots + G_n + n G_n \leq 2a_1 + \left( \frac{3}{2} \right)^2 a_2 + \cdots + \left( \frac{n+1}{n} \right)^n a_n

and in particular (Carleman’s inequality)

G_1 + G_2 + \cdots + G_n < e(a_1 + a_2 + \cdots + a_n);

e A_n \geq \Gamma_n e^{G_n / \Gamma_n}

where $\Gamma_n = (G_1 + G_2 + \cdots + G_n)/n$ (strengthening of the Carleman inequality);

\frac{A_n}{G_n} \geq \frac{1}{2} \left( \frac{\Gamma_n}{G_n} + \frac{G_n}{\Gamma_n} \right)

if the sequence $(a_k)$ is increasing:

n G_n - m G_m \leq a_{m+1} + a_{m+2} + \cdots + a_n.

Solution

No solution has been written yet.