On the generalized proximate order of functions analytic on the unit disc

The concept of Lindel $\ddot{o}$f proximate order has been used extensively to study the functions of completely regular growth. The main drawback of this approach is that it completely ignores the value of lower order. To overcome this problem, Chyzykov et al. introduced the concept of generalized proximate order for irregular growth. In this paper we studied the existence of generalized proximate order for every functions analytic on the unit disc with some new results for functions having irregular growth.