Let \(\{m_r\}_{r=1}^n\) be defined by \(m_r= F_r F_{n+1-r}\), where \(n\) is a fixed natural number and \(F_1, F_2,\dots\) are the ordinary Fibonacci numbers. For every natural number \(k\) and \(n=4k\) the following inequalities are valid: \[ m_1> m_3>\cdots> m_{2k-1}> m_{2k}> m_{2k-2}>\cdots> m_2. \] There are similar inequalities for \(n=4k+1\), \(4k+2\), and \(4k+3\).