Value distribution for n^th difference operator of meromorphic functions with maximal deficiency sum

Abstract

In this paper, we obtain the relationship between the characteristic function of meromorphic function having maximal deficiency sum and its higher order exact difference. We improve and generalise several results of Zhaojun Wu (Journal of Inequalities and Applications 530, [3]) to a great extent. We obtain an analogue of Shah and Singh (Mathematische Zeitschrift 65:171–174, [4]) in an improved form for $\Delta _{c}f$. Also we establish a lower bound for $K(\Delta _{c}f)=\limsup \limits _{r\rightarrow \infty }\frac{N\left( r,\Delta _{c}f\right) +N\left( r,\frac{1}{\Delta _{c}f}\right) }{T(r,\Delta _{c}f)}$. Under certain conditions we also show that $K(\Delta _{c}^{n}f)=0$, for any integer $n\ge 1$.