  The $\varphi$-order was introduced in 2009 for meromorphic functions in the
unit disc, and was used as a growth indicator for solutions of linear
differential equations. In this paper, the properties of meromorphic functions
in the complex plane are investigated in terms of the $\varphi$-order, which
measures the growth of functions between the classical order and the
logarithmic order. Several results on value distribution of meromorphic
functions are discussed by using the $\varphi$-order and the $\varphi$-exponent
of convergence. Instead of linear differential equations, the applications in
the complex plane lie in linear $q$-difference equations.
