Entire functions and some of their growth properties on the basis of generalized order (α,β)

For any two entire functions f, g defined on finite complex plane \mathbb{C}, the ratios \frac{M_{f\circ g}(r)}{M_{f}(r)} and \frac{M_{f\circ g}(r)}{M_{g}(r)} as r\rightarrow \infty are called the growth of composite entire function f\circ g with respect to f and g respectively in terms of their maximum moduli. Several authors have worked about growth properties of functions in different directions. In this paper, we have discussed about the comparative growth properties of f\circ g, f and g, and derived some results relating to the generalized order (\alpha ,\beta ) after revised the original definition introduced by Sheremeta, where \alpha , \beta are slowly increasing continuous functions defined on (-\infty,+\infty ). Under different conditions, we have found the limiting values of the ratios formed from the left and right factors on the basis of their generalized order (\alpha ,\beta ) and generalized lower order (\alpha,\beta ), and also established some inequalities in this regard.

Keywords:

Entire function , Growth , composition , generalized order (alpha , beta ) , generalized lower order (alpha , beta )