On the Value Distribution of $${\varvec{f}}^{{\varvec{n}}} {\varvec{f}}^{({\varvec{k}})}-\mathbf 1 $$ f n

On the Zeros of the Differential Polynomials φfl(f(k))n−a

“…Remark 1. Obviously, Theorem 2 improves the results of Xu et al [13], Karmakar and Sahoo [14], Xu and Ye [17], Chakraborty, Saha and Pal [18], and Chen and Huang [19]. The coefficient reduced to 5 when l = 2, or 1 l−2 when l ≥ 3.…”

“…Our results mainly consider the quantitative estimation for the differential polynomial using the reduced counting function. They improve and generalize the existing literature ( [14,19,28,[35][36][37][38][39][40]).…”

“…In 2011, J. F. Xu et al [13] proved the inequality T(r, f ) ≤ MN(r, 1; f 2 f (k) ) + S(r, f ), where M = 6 except for k = 2, M = 10. Later, Karmakar and Sahoo [14] found the coefficient is also 6 when k = 2 for the inequality. Moreover, they improved the result of Xu et al, and obtained a unified inequality for l(≥ 2), k(≥ 1).…”

On the Zeros of the Differential Polynomial φ(z)f2(z)f′(z)2 − 1

“…Note that the proof of Theorem 5 requires the conclusion of Theorem 2, this is, the proof only holds under the condition δ(∞; f ) ≤ 7/9. In this paper, we will use a new way to get a quantitative description of Theorem 5 (see [11][12][13]). In fact, we prove the following result.…”