Meromorphic functions of finite $\varphi$-order and linear $q$-difference equations

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.