Advancement on the study of growth analysis of differential polynomial and differential monomial in the light of slowly increasing functions

Study of the growth analysis of entire or meromorphic functions has generally been done through their Nevanlinna's characteristic function in comparison with those of exponential function. But if one is interested to compare the growth rates of any entire or meromorphic function with respect to another, the concepts of relative growth indicators will come. The field of study in this area may be more significant through the intensive applications of the theories of slowly increasing functions which actually means that $L(ar)\sim L(r)$ as $ r\rightarrow \infty $ for every positive constant $a$, i.e. $\underset{ r\rightarrow \infty }{\lim }\frac{L\left( ar\right) }{L\left( r\right) }=1$, where $L\equiv L\left( r\right) $ is a positive continuous function increasing slowly. Actually in the present paper, we establish some results depending on the comparative growth properties of composite entire and meromorphic functions using the idea of relative $_{p}L^{\ast }$-order, relative $_{p}L^{\ast }$- type, relative $_{p}L^{\ast }$-weak type and differential monomials, differential polynomials generated by one of the factors which extend some earlier results where $_{p}L^{\ast }$ is nothing but a weaker assumption of $L.$