Growth analysis of analytic functions is very important part of research in
the field of complex analysis and many researchers are involved in this area
during past decades. Collecting ideas from Heittokangas et al. (Meromorphic
functions of finite $\varphi $-order and linear q-difference equations, J.
Difference Equ. Appl., 27 (9) (2021), 1280-1309) and Bela\"{\i}di et
al. (Study of complex oscillation of solutions of a second order linear
differential equation with entire coefficients of $(\alpha ,\beta ,\gamma )$%
-order, WSEAS Trans. Math., 21 (2022), 361-370), here in this paper,
we have defined the $(\alpha ,\beta ,\gamma )$-Nevanlinna order and\ $%
(\alpha ,\beta ,\gamma )$-Nevanlinna type of an analytic function $f$ in the
unit disc $U$. We have also established some growth properties of the
composition of two analytic functions in the unit disc on the basis of their
$(\alpha ,\beta ,\gamma )$-Nevanlinna order, $(\alpha ,\beta ,\gamma )$%
-Nevanlinna lower order, $(\alpha ,\beta ,\gamma )$-Nevanlinna type and $%
(\alpha ,\beta ,\gamma )$-Nevanlinna weak type as compared to the growth of
their corresponding left and right factors, where $\alpha ,\beta ,\gamma $
are continuous non-negative functions defined on $(-\infty ,+\infty )$.