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

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