On Discrete Inequalities Involving Arithmetic, Geometric, and Harmonic Means

Abstract

Si dimostra: se A(n), G(n), e H(n) denotano la media aritmetica, geometrica ed armonica dei primi n interi positivi , allora si ha che per n $\geq$ 2: \[ \begin{array}{cc} \frac{H(n)}{H(n-1)}-\frac{H(n+1)}{H(n)}< & \frac{G(n)}{G(n-1)}-\frac{G(n+1)}{G(n)}\\ \qquad\qquad\qquad\qquad< & \frac{A(n)}{A(n-1)}-\frac{A(n+1)}{A(n)} \end{array} \]

We prove: if A(n), G(n), and H(n) denote the arithmetic, geometric, and harmonic means of the first n positive integers, then we have for n $\geq$ 2: \[ \begin{array}{cc} \frac{H(n)}{H(n-1)}-\frac{H(n+1)}{H(n)}< & \frac{G(n)}{G(n-1)}-\frac{G(n+1)}{G(n)}\\ \qquad\qquad\qquad\qquad< & \frac{A(n)}{A(n-1)}-\frac{A(n+1)}{A(n)} \end{array} \]