Let \(g\) be a function analytic and locally univalent on the unit disk \(\mathbb{D}\) and assume that \[ A_g(z)= {1-|z|^2\over 2} {g''(z)\over g'(z)}-\overline z. \] The linear-invariance order of \(g\) defined by \(\alpha(g)=\sup\{|A_g(z)|: z\in\mathbb{D}\}\) is a quantity that plays an important role in the theory of analytic or meromophic functions. In this paper the author is concerned with the quantity \(\mu(g)= \text{inf}\{|A_g(z)|: z\in\mathbb{D}\}\) called the lower linear-invariance order. He gives several examples of functions, univalent or multivalent, for which this order is positive. The autor establishes several theorems about properties of \(g\), if \(\mu(g)> 0\). From one of them it follows that such functions are unbounded.