On the uniqueness of a meromorphic function and its higher difference operator under the purview of two shared sets

Abstract

In the paper, we investigate the uniqueness problem of a meromorphic function and its difference operator to the most general setting via two shared set problems and thus improve a recent result of Chen–Chen (Bull Malays Math Sci Soc 35(3): 765-774, 2012) .

1 Introduction, definitions and results

At the outset, for standard definitions and notations of value distribution theory, we refer our readers to follow [14]. Let us denote by $\mathbb {\overline{C}}={\mathbb {C}}\cup \{\infty \}$, ${\mathbb {C}}^*={\mathbb {C}}-\{0\}$, where ${\mathbb {C}}$ denotes the set of all complex numbers. Furthermore, throughout the paper, by any meromorphic function, we shall always mean that it is meromorphic in ${\mathbb {C}}$. For any non-constant meromorphic function h, we define $S(r, h) = o(T(r, h))$, $(r\rightarrow \infty , r\not \in E)$, where E denotes any set of positive real numbers having finite linear measure.

For such a non-constant meromorphic function f and $S\subseteq \mathbb {\overline{C}}$, we define

$$\begin{aligned} E_{f}(S)=\bigcup _{a\in S}\{(z,p)\in {\mathbb {C}}\times {\mathbb {N}}: f(z)=a\; {\textit{with}}\ {\textit{multiplicity}}\; p\} \end{aligned}$$

and

$$\begin{aligned} \overline{E}_{f}(S)=\bigcup _{a\in S}\{(z,1)\in {\mathbb {C}}\times {\mathbb {N}}: f(z)=a\}. \end{aligned}$$

For two non-constant meromorphic functions f and g if $E_{f}(S)=E_{g}(S)\left( \overline{E}_{f}(S)=\overline{E}_{g}(S)\right)$, then we say f, g share the set S counting multiplicities (ignoring multiplicities) or CM(IM) in short. If S contains only one element, then we say that f, g share the value a CM(IM).

In 2001, this notion of set sharing was further refined by Lahiri [17, 18] with the introduction a scaling between the notion of CM and IM sharing, called weighted sharing. Below, we explain the notion.

Definition 1.1

([17, 18]) Let k be a non-negative integer or infinity. For $a\in \overline{{\mathbb {C}}}$, we denote by $E_f(a; k)$ the set of all a-points of f, where an a-point of multiplicity m is counted m times if $m\le k$ and $k + 1$ times if $m > k$. If $E_f(a; k) = E_g(a; k)$, we say that f, g share the value a with weight k.

We write f, g share (a, k) to mean that f, g share the value a with weight k. Clearly, if f, g share (a, k), then f, g share (a, p) for any integer $p, 0\le p < k$. Also, we note that f, g share a value a IM or CM if and only if f, g share (a, 0) or $(a,\infty )$, respectively.

Definition 1.2

[17] For $S\subset \mathbb {\overline{C}}$, we define $E_f(S,k)=\cup _{a\in S}E_f(a;k)$, where k is a non-negative integer $a\in S$ or infinity. Clearly, $E_f(S)=E_f(S,\infty )$ and $\overline{E}_f(S)=E_f(S,0)$. Further, if $E_f(S,k)=E_g(S,k)$ for two non-constant meromorphic functions f and g, then we say that f and g share the set S with weight k or f and g share (S, k) in short.

The notion of set sharing for the uniqueness of meromorphic functions was trail-blazed by Gross [10, 11] in 1976. After that, a large amount of research have been added in the literature during the last 4 decades. Recently, the analogous study for a meromorphic function and its shift or difference operator via shared set(s) have become a matter of prime interest to the researchers. A number of results in this direction have also been obtained by many researchers throughout the last decade [8, 22, 23, 26]. In this paper, we walk along this new direction and prove a theorem which improves a recent result by Chen-Chen in [8].

Let f(z) be a meromorphic function and $c\in {\mathbb {C}}^*$. We define the shift and difference operators of f(z) by $f(z+c)$ and $\Delta _c f(z):=f(z+c)-f(z)$, respectively. We further define

$$\begin{aligned} \Delta _c^pf:=\Delta _c^{p-1}(\Delta _c f),\;\; \forall p\in {\mathbb {N}}-\{1\}\;\; and\;\; \Delta _c^pf=\Delta _cf\;\; for\;\; p=1. \end{aligned}$$

(1.1)

In this new direction, in 2010, Zhang [26] proved the following result for the uniqueness of f(z) and $f(z+c)$.

Theorem 1.3

[26] Let $S=\{w: w^n+\alpha w^{n-m}+\beta =0\}$, where $\alpha , \beta$ be two non-zero constants, such that $w^n+\alpha w^{n-m}+\beta$ has only simple zeros and $m\ge 2$, $n\ge 2\,m+4 , \gcd (m,n)=1$. Further, suppose that f(z) be a non-constant meromorphic function of finite order. If $E_{f(z)}({\mathcal {S}},\infty )=E_{f(z+c)}({\mathcal {S}},\infty )$ and $E_{f(z)}(\{\infty \},\infty )=E_{f(z+c)}(\{\infty \},\infty )$, then $f(z)\equiv f(z+c)$.

In view of Theorem 1.3, one may naturally ask

Question 1.4

What can be said about the uniqueness of f and $\Delta ^{p}_{c}(f)$ while sharing two sets ?

With respect to Question 1.4, in 2012, Chen-Chen [8] proved the following result.

Theorem 1.5

[8] Let ${\mathcal {S}}$ be defined as in Theorem 1.3. Let f(z) be a non-constant meromorphic function of finite order satisfying $E_{f(z)}({\mathcal {S}},\infty )=E_{\Delta _cf}({\mathcal {S}},\infty )$ and $E_{f(z)}(\{\infty \},\infty )=E_{\Delta _cf}(\{\infty \},\infty )$. If

$$\begin{aligned} N\left( r,\frac{1}{\Delta _cf}\right) =T(r,f)+S(r,f), \end{aligned}$$

then $f(z)\equiv \Delta _cf$.

Note that, pertinent to Question 1.4, Chen-Chen became successful to obtain the uniqueness of f and $\Delta ^{p}_{c}(f)$ while sharing two sets, but in that case, they considered the extra condition

$$\begin{aligned} N\left( r,\frac{1}{\Delta _cf}\right) =T(r,f)+S(r,f), \end{aligned}$$

(1.2)

with compared to Theorem 1.3. Also, they did not show the sharpness of this condition. Therefore, natural query arises:

Question 1.6

Is it possible to obtain the uniqueness of f and $\Delta _{c}(f)$ while sharing two sets without the condition (1.2) ?

Another observation reads that the least possible cardinality of the main range set in Theorem 1.5 is 9. Therefore, in this respect, one would naturally be interested to investigate the following problem.

Question 1.7

Is it possible to obtain the uniqueness of f and $\Delta _{c}(f)$ while sharing two sets having the cardinality of the main range set less than 9 ?

Another point to be noted is that in Theorem 1.5, authors considered only finite ordered meromorphic functions and in this case also they did not show the sharpness. Therefore, one must raise the following question in this respect.

Question 1.8

Is it possible to obtain the uniqueness of f and $\Delta _{c}(f)$ while sharing two sets, where f may or may not be of finite order ?

Now apropos of Theorem 1.5, we pose the last question of the paper as follows.

Question 1.9

Is it possible to obtain the uniqueness of f and $\Delta _{c}(f)$ while sharing two sets under more relaxed nature of sharing than those obtained in Theorem 1.5 ?

In this paper, we answer all the questions placed above from Question 1.6–1.9 in affirmative. Moreover, we answer all these questions in a more general setting using $\Delta ^{p}_{c}(f)$ in place of $\Delta _{c}(f)$.

Consider the polynomial defined by

$$\begin{aligned} P(w)=aw^n+bw^{2m}+cw^m+d, \end{aligned}$$

(1.3)

where $n>2\,m, a,b,c,d\in \mathbb {C^*}$ with $\frac{c^2}{4bd}=\frac{n(n-2m)}{(n-m)^2}$ and $a\ne \gamma _i$ for every i, where $\gamma _i=-\frac{2bme_i^{2m}+cme_i^m}{ne_i^n}$ for $i\in \{1,2,\dots m\}$ with $e_i$ being the roots of $w^m=-\frac{2dn}{c(n-m)}$ and $m,n\in {\mathbb {N}}$ with $gcd(m,n)=1.$ Now

$$\begin{aligned} P'(w)=anw^{n-1}+2bmw^{2m-1}+cmw^{m-1}. \end{aligned}$$

Therefore, the zeroes of $P'(w)$ are the roots of

$$\begin{aligned} anw^{n-1}+2bmw^{2m-1}+cmw^{m-1}=0. \end{aligned}$$

For any zero ‘s’ of $P'(w)$, we have

$$\begin{aligned} ans^{n-1}+2bms^{2m-1}+cms^{m-1}=0, \end{aligned}$$

that is

$$\begin{aligned} as^n=-\frac{2bms^{2m}+cms^{m}}{n}. \end{aligned}$$

Now, for $s=0$

$$\begin{aligned} P(0)=d(\ne 0), \end{aligned}$$

and since $\frac{c^2}{4bd}=\frac{n(n-2m)}{(n-m)^2},$ for $s\ne 0$, we have

$$\begin{aligned} P(s)&=-\frac{2bms^{2m}+cms^m}{n}+bs^{2m}+cs^m+d\\&=\frac{-2bms^{2m}-cms^m+bns^{2m}+cns^m+dn}{n}\\&=\frac{(bn-2bm)s^{2m}+(cn-cm)s^m+dn}{n}\\&=\frac{c^{2}(n-m)^{2}s^{2m}+4ndc(n-m)s^m+4(nd)^{2}}{4n^{2}d}\\&=\frac{c^2(n-m)^2\bigl (s^m+\frac{2nd}{c(n-m)}\bigr )^2}{4n^2d}. \end{aligned}$$

Therefore, ‘s’ is a zero of P(w) if $s^m=-\frac{2dn}{c(n-m)}$, i.e., if, $s\in \{e_1, e_2, \dots e_m\}.$ But then, we would have

$$\begin{aligned} ae_i^n=-\frac{2bme_i^{2m}+cme_i^m}{n}, \end{aligned}$$

for $i\in \{1,2,\dots ,m\},$ which is a contradiction, since $a\ne \gamma _i=-\frac{2bme_i^{2m}+cme_i^m}{ne_i^n}$. Thus, $P(w)=0$ has only simple roots.

Consider the rational function

$$\begin{aligned} R(w)=\frac{-aw^n}{bw^{2m}+cw^m+d}=\frac{-aw^n}{b(w-\alpha _1)(w-\alpha _2)\dots (w-\alpha _{2m})}, \end{aligned}$$

(1.4)

where $\alpha _i$’s are the roots of $bw^{2m}+cw^m+d=0$. Since $\frac{c^2}{4bd}=\frac{n(n-2m)}{(n-m)^2}\ne 1$, all $\alpha _i$’s are distinct.

From (1.4), we get using $\frac{c^2}{4bd}=\frac{n(n-2m)}{(n-m)^2}$

$$\begin{aligned} R'(w)&=\frac{-anw^{n-1}(bw^{2m}+cw^m+d)+aw^{n-1}(2mbw^{2m}+cmw^m)}{(bw^{2m}+cw^m+d)^2}\\&=\frac{-aw^{n-1}\{b(n-2m)w^{2m}+c(n-m)w^m+dn\}}{(bw^{2m}+cw^m+d)^2}\\&=\frac{-ac^2(n-m)^2w^{n-1}\bigl (w^m+\frac{2nd}{c(n-m)}\bigr )^2}{4nd(bw^{2m}+cw^m+d)^2}. \end{aligned}$$

Thus

$$\begin{aligned} R'(w)=\frac{-ac^2(n-m)^2w^{n-1}(w-e_1)^2(w-e_2)^2\dots (w-e_m)^2}{4ndb^2(w-\alpha _1)^2(w-\alpha _2)^2\dots (w-\alpha _{2m})^2}. \end{aligned}$$

(1.5)

Therefore, from (1.5), we know that $w=0$ is a root of $R(w)=0$ with multiplicity n and $w=e_i$ is a root of $R(w)-c_i=0$ with multiplicity 3, where

$$\begin{aligned} c_i=\frac{-ae_i^n}{b(e_i-\alpha _1)(e_i-\alpha _2)\dots (e_i-\alpha _{2m})}, \end{aligned}$$

for $i=1,2,\dots ,m$. Therefore

$$\begin{aligned} R(w)-c_i=\frac{-a(w-e_i)^3Q_{i(n-3)}(w)}{b(w-\alpha _1)(w-\alpha _2)\dots (w-\alpha _{2m})}, \end{aligned}$$

(1.6)

where $Q_{i(n-3)}(w)$ is a polynomial of degree $(n-3)$ for each $i=1,2,\dots m$. Obviously $c_i\ne 1$.

Now, we state the main result of the paper as follows.

Theorem 1.10

Let $S=\{w| P(w)=0\}$, where P(w) is given by (1.3) and $n\ge (4m+4)$. Suppose that f and $\Delta _{c}^{p}(f)$ are two non-constant meromorphic functions satisfying $E_f(S,2)=E_{\Delta _{c}^{p}(f)}(S,2)$ and $E_f(\infty ,0)=E_{\Delta _{c}^{p}(f)}(\infty ,0)$. Then, $f\equiv \Delta _{c}^{p}(f)$.

Remark 1.11

Observe that, in Theorem 1.10, we have been able to prove the uniqueness of f and $\Delta ^{p}_{c}(f)$ without the condition (1.2).

Remark 1.12

The least cardinality of the main range set in Theorem  1.10 is 8, whereas it is 9 in Theorem 1.5.

Remark 1.13

Theorem 1.10 is true for meromorphic function of any order not restricted to finite order only.

Remark 1.14

In Theorem 1.10, the nature of sharing is significantly relaxed than those obtained in Theorem 1.5.

Note that Remarks 1.11, 1.12, 1.13, 1.14 answer Questions 1.6, 1.7, 1.8, 1.9 respectively, in affirmative and in a more general way than was asked.

Next, we construct a set S containing 8 elements and a function f that satisfies the conditions of Theorem  1.10.

Example 1.15

Consider the polynomial

$$\begin{aligned} P(w)= & {} w^8+3w^2+24w+49. \end{aligned}$$

Clearly, P(w) satisfies all the conditions in 1.3. Let ${S=\{w|P(w)=0\}}$. Consider the function $f(z)= \sin z \exp \left( \dfrac{z}{2\pi }\log 2\right)$. Now, it is easy to verify that ${E_f(S,2)=E_{\Delta _{2\pi }(f)}(S,2)}$ and ${E_f(\infty ,0)=E_{\Delta _{2\pi }{(f)}}(\infty ,0)}$ and ${f=\Delta _{2\pi }(f)}$.

Now, we invoke the following definitions which will be needed in the sequel.

Definition 1.16

[16] For $a\in \mathbb {\overline{C}}$, we denote by $N(r,a;f|=1)$ the counting function of simple a-points of f. By $N(r,a;f|\le m)$ ($N(r,a;f|\ge m)$), we mean the counting function of those a-points of f, where the multiplicity of each a-point is not greater (not less) than m, m is a positive integer and each a-point is counted according to it’s multiplicity.

$\overline{N}(r,a;f|\le m)$($\overline{N}(r,a;f|\ge m)$) are defined in a similar way, where we ignore the multiplicity while counting the a-points of f.

Definition 1.17

[4] We denote by $\overline{N}(r, a; f |= k)$ the reduced counting function of those a-points of f whose multiplicities are exactly k where $k \ge 2$ is an integer.

Definition 1.18

Let fand g be two non-constant meromorphic functions, such that f and g share (a, 0), where $a\in \mathbb {\overline{C}}$. Let $z_0$ be an a-point of f with multiplicity p, an a-point of g with multiplicity q. We denote by $\overline{N}_L(r, a; f)$ ($\overline{N}_L(r, a; g)$) the counting function of those a-points of f and g where $p> q (q > p)$, each a-point is counted only once.

Definition 1.19

Let f and g be two non-constant meromorphic functions, such that f and g share (a, 0). Let $z_0$ be an a-point of f with multiplicity p and an a-point of g with multiplicity q. We denote by $N_{\textrm{E}}^{1)}(r,a;f)$ the counting function of those a-point of f and g, where $p=q=1$.

Clearly $N_{\textrm{E}}^{1)}(r,a;f)=N(r,a;f|=1)$, when f, g share (a, k), where $k\ge 1$.

Definition 1.20

[4] Let f and g be two non-constant meromorphic functions, such that f and g share (1, k) where $1 \le k < \infty$. Let $z_0$ be a 1-point of f with multiplicity p, a 1-point of g with multiplicity q. We denote by $\overline{N}_{\textrm{E}}^{(k+1} (r, 1; f)$the counting function of those 1-points of f and g where $p = q \ge k + 1$, each point in this counting function is counted only once.

We can define $\overline{N}_{\textrm{E}}^{(k+1} (r, 1; g)$ in a similar way.

Definition 1.21

[10, 11, 14] Let f, g share (a, 0). We denote by $\overline{N_*}(r, a; f, g)$ the reduced counting function of those a-points of f whose multiplicities differ from the multiplicities of the corresponding a-points of g.

We note that, $\overline{N}_*(r, a; f, g) = \overline{N}_*(r, a; g, f)$ and $\overline{N}_*(r, a; f, g) = \overline{N}_L(r, a; f)+\overline{N}_L(r, a; g)$.

Definition 1.22

[19] Let $a, b_1,b_2, \cdots , b_q$ $\in \mathbb {\overline{C}}$. We denote by $N(r,a;f | g\ne b_1,b_2,b_3,\cdots ,b_q)$ the counting function of those a-points of f, counted according to multiplicity, which are not the $b_i$-points of g for $i\in 1,2,\cdots ,q$.

2 Lemmas

Let F and G be two non-constant meromorphic functions defined in ${\mathbb {C}}$.

Let

$$\begin{aligned} H=\Big (\frac{F''}{F'}-\frac{2F'}{F-1}\Big )-\Big (\frac{G''}{G'}-\frac{2G'}{G-1}\Big ) \end{aligned}$$

and

$$\begin{aligned} V=\Big (\frac{F'}{F-1}-\frac{F'}{F}\Big )-\Big (\frac{G'}{G-1}-\frac{G'}{G}\Big )=\frac{F'}{F(F-1)}-\frac{G'}{G(G-1)}. \end{aligned}$$

Lemma 2.1

[24, 25] If F and G be two non-constant meromorphic functions, such that they share (1, 0) and $H\not \equiv 0$, then

$$\begin{aligned} N_\textrm{E}^{1)}(r,1;F)\le N(r,H)+S(r,F)+S(r,G). \end{aligned}$$

Lemma 2.2

[20] If $N(r,0;f^{(k)}|f\ne 0)$ denotes the counting function of those zeros of $f^{(k)}$ which are not the zeros of f, where a zero of $f^{(k)}$ is counted according to it’s multiplicity, then

$$\begin{aligned} N(r,0;f^{(k)}|f\ne 0)\le k\overline{N}(r,\infty ;f)+N(r,0;f|<k)+k\overline{N}(r,0;f|\ge k)+S(r,f). \end{aligned}$$

Lemma 2.3

[3] Let F and G be two non-constant meromorphic functions sharing (1, 2), then

$$\begin{aligned} 2\overline{N}_L(r,1;F)+3\overline{N}_L(r,1;G)+2\overline{N}_\textrm{E}^{(3}(r,1;F)+\overline{N}(r,1;F|=2)\le N(r,1;G)-\overline{N} (r,1;G). \end{aligned}$$

Let f be a non-constant meromorphic function and

$$\begin{aligned} F=R(f), \,\,\, G=R(\Delta _{c}^{p}(f)), \end{aligned}$$

(2.1)

where R(w) is given by (1.4).

Lemma 2.4

Let F and G be given by (2.1). If F and G share $(1,\lambda )$, where $0\le \lambda <\infty$, then

$$\begin{aligned} \overline{N}_L(r,1;F)\le \frac{1}{\lambda +1}\Big [\overline{N}(r,0;f)+\overline{N}(r,\infty ;f)-N_{\bigotimes }(r,0;f')\Big ]+S(r,f) \end{aligned}$$

and

$$\begin{aligned} \overline{N}_L(r,1;G)\le \frac{1}{\lambda +1}\Big [\overline{N}(r,0;\Delta _{c}^{p}(f))+\overline{N}(r,\infty ;\Delta _{c}^{p}(f))-N_{\bigotimes }(r,0;\Delta _{c}^{p}(f)')\Big ]+S(r,\Delta _{c}^{p}(f)), \end{aligned}$$

where $N_{\bigotimes }(r,0;f')=N\big (r,0;f'|f\ne 0,w_1,\dots w_n\big )$ and $w_i$ being the roots of $P(w)=0,$ which is given by (1.3) and $N_{\bigotimes }(r,0;\Delta _{c}^{p}(f)')$ is defined similarly like $N_{\bigotimes }(r,0;f').$

Proof

Since F and G share $(1,\lambda )$, we have using Lemma  2.2

$$\begin{aligned} \overline{N}_L(r,1;F)&\le \overline{N}(r,1;F|\ge \lambda +2)\\&\le \frac{1}{\lambda +1}\big [N(r,1;F)-\overline{N}(r,1;F)\big ]\\&\le \frac{1}{\lambda +1}\bigg [\sum _{j=1}^{n}\Big (N(r,w_j;f)-\overline{N}(r,w_j;f)\Big )\bigg ]\\&\le \dfrac{1}{\lambda +1}\left[ N(r,0;f'|f\ne 0)-N_{\bigotimes }(r,0;f')\right] \\&\le \dfrac{1}{\lambda +1}\left[ N(r,0;\dfrac{f'}{f})-N_{\bigotimes }(r,0;f')\right] \\&\le \dfrac{1}{\lambda +1}\left[ T(r,\dfrac{f'}{f})-N_{\bigotimes }(r,0;f')+O(1)\right] \\&\le \dfrac{1}{\lambda +1}\left[ N(r,\dfrac{f'}{f})-N_{\bigotimes }(r,0;f')\right] +S(r,f)\\&\le \frac{1}{\lambda +1}\Big [\overline{N}(r,0;f)+\overline{N}(r,\infty ;f)-N_{\bigotimes }(r,0;f')\Big ]+S(r,f). \end{aligned}$$

Second inequality can be proved in a similar way. $\square$

Lemma 2.5

Let F and G be given by (2.1). Suppose that $n\ge (4m+4)$. Then, $V\equiv 0$ implies $F\equiv G$.

Proof

Since $V\equiv 0$, we have $\frac{F'}{F-1}-\frac{F'}{F}=\frac{G'}{G-1}-\frac{G'}{G}$. Integrating, we get

$$\begin{aligned} \frac{1}{F}-\frac{B}{G}={1-B}, \end{aligned}$$

(2.2)

where $B\ne 0$ is a constant. As $F=R(f),$ $G=R(\Delta _{c}^{p}(f)),$ we know that

$$\begin{aligned} T(r,F)= & {} nT(r,f)+O(1);\\ T(r,G)= & {} nT(r,\Delta _{c}^{p}(f))+O(1). \end{aligned}$$

From (2.2), we get

$$\begin{aligned} T(r,f)=T(r,\Delta _{c}^{p}(f))+O(1). \end{aligned}$$

Define $h_1=\displaystyle \frac{1}{F}$ and $h_2=\displaystyle -\frac{B}{G}.$ Therefore

$$\begin{aligned} h_1+h_2=1-B. \end{aligned}$$

If possible, let $B\ne 1.$ Then, using second fundamental theorem, we have

$$\begin{aligned} T(r,h_1)&\le \overline{N}(r,0;h_1)+\overline{N}(r,1-B;h_1)+\overline{N}(r,\infty ;h_1)+S(r,h_1)\\&\le \overline{N}(r,\infty ;F)+\overline{N}(r,\infty ;G)+\overline{N}(r,0;F)+S(r,f)\\&\le \overline{N}(r,\infty ;f)+\sum _{j=1}^{2m}\overline{N}(r,\alpha _j;f)+\overline{N}(r,\infty ;\Delta _{c}^{p}(f))\\&\quad +\sum _{j=1}^{2m}\overline{N}(r,\alpha _j;\Delta _{c}^{p}(f))+\overline{N}(r,0;f)+S(r,f), \end{aligned}$$

which implies that

$$\begin{aligned} nT(r,f)\le (4m+3)T(r,f)+S(r,f), \end{aligned}$$

a contradiction to our assumption that $n\ge (4m+4)$. Hence, $B=1$, which implies $F\equiv G.$ $\square$

Lemma 2.6

Let F and G be given by (2.1) and suppose $H\not \equiv 0$. If F, G share $(1,\lambda )$ and $f, \Delta _{c}^{p}(f)$ share $(\infty ,k)$, where $0\le \lambda<\infty , 0\le k<\infty$, then

$$\begin{aligned} \Big [(n-2m)(k+1)-1\Big ]\overline{N}(r,\infty ;f|\ge k+1)&\le \Big [(n-2m)(k+1)-1\Big ]\overline{N}(r,\infty ;\Delta _{c}^{p}(f)|\ge k+1)\\&\le \frac{\lambda +2}{\lambda +1}\Big [\overline{N}(r,0;f)+\overline{N}(r,0;\Delta _{c}^{p}(f))\Big ]\\ {}&+ \frac{2}{\lambda +1}\overline{N}(r,\infty ;f)+S(r,f)+S(r,\Delta _{c}^{p}(f)). \end{aligned}$$

Proof

Since $H\not \equiv 0$, from Lemma 2.5, we get $V\not \equiv 0$ for $n\ge (4m+4)$. Now, using the Milloux theorem (see p.55 [14]), we have

$$\begin{aligned} m(r,V)=S(r,f)+S(r,\Delta _{c}^{p}(f)). \end{aligned}$$

Let ${z_0}$ be a pole of f of multiplicity ${\ge k+1}$. Then, ${z_0}$ is a pole of F of multiplicity ${p(\ge (n-2m)(k+1))}$. Again, as ${f,\Delta _{c}^p(f)}$ share ${(\infty ,k)}$, so ${z_0}$ is also a pole of G of multiplicity ${q\ge (n-2m)(k+1)}$.

Now, from the definition of V, it follows that $z_0$ is a zero of V with multiplicity at least $(n-2m)(k+1)-1$. Therefore, using Lemma 2.4, we obtain

$$\begin{aligned}{} & {} \Big [(n-2m)(k+1)-1\Big ]\overline{N}\Big (r,\infty ;f|\ge k+1\Big )\\{} & {} \quad =\Big [(n-2m)(k+1)-1\Big ]\overline{N}\Big (r,\infty ;\Delta _{c}^{p}(f)|\ge k+1\Big )\\{} & {} \quad \le N(r,0;V)\\{} & {} \quad \le N(r,\infty ;V)+m(r,V)+O(1)\\{} & {} \quad \le N(r,\infty ;V)+S(r,f)+S(r,\Delta _{c}^{p}(f))\\{} & {} \quad \le \overline{N}(r,0; F)+\overline{N}(r,0;G)+\overline{N}_*(r,1;F,G)+S(r,f)+S(r,\Delta _{c}^{p}(f))\\{} & {} \quad \le \overline{N}(r,0;f)+\overline{N}(r,0;\Delta _{c}^{p}(f))+\overline{N}_L(r,1;F)+\overline{N}_L(r,1;G) +S(r,f)+S(r,\Delta _{c}^{p}(f))\\{} & {} \quad \le \overline{N}(r,0;f)+\overline{N}(r,0;\Delta _{c}^{p}(f))+\frac{1}{\lambda +1}\left[ \overline{N}(r,0;f)+\overline{N}(r,0;\Delta _{c}^{p}(f))+\overline{N}(r,\infty ;f)\right. \\{} & {} \qquad \left. +\overline{N}(r,\infty ;\Delta _{c}^{p}(f))\right] +S(r,f)+S(r,\Delta _{c}^{p}(f))\\{} & {} \quad \le \frac{\lambda +2}{\lambda +1}\Big [\overline{N}(r,0;f)+\overline{N}(r,0;\Delta _{c}^{p}(f))\Big ]+\frac{2}{\lambda +1}\overline{N}(r,\infty ;f)+S(r,f)+S(r,\Delta _{c}^{p}(f)). \end{aligned}$$

$\square$

Lemma 2.7

Let F and G be given by (2.1) and $H\not \equiv 0.$ If F, G share $(1,\lambda )$ and $f, \Delta _{c}^{p}(f)$ share $(\infty ,k),$ where $0\le \lambda<\infty , 0\le k<\infty$, then

$$\begin{aligned} N_{\textrm{E}}^{1)}(r,1;F)&\le \overline{N}_L(r,1;F)+\overline{N}_L(r,1;G)+\overline{N}(r,0;f)+\overline{N}(r,0;\Delta _{c}^{p}(f))\\ {}&\,\,\,\,\,+\sum _{j=1}^{m}\overline{N}(r,e_j;f)+\sum _{j=1}^{m}\overline{N}(r,e_j;\Delta _{c}^{p}(f))+\overline{N}_*(r,\infty ;f,\Delta _{c}^{p}(f))\\ {}&\,\,\,\,\,+\overline{N}_0(r,0;f')+\overline{N}_0(r,0;\Delta _{c}^{p}(f)')+S(r,f)+S(r,\Delta _{c}^{p}(f)), \end{aligned}$$

where $\overline{N}_0(r,0;f')$ denotes the reduced counting function of those zeros of $f'$, which are not the zeros of $f(f-e_1)(f-e_2)\dots (f-e_m)$ and $F-1$, $\overline{N}_0(r,0;\Delta _{c}^{p}(f)')$ is defined similarly.

Proof

We have

$$\begin{aligned} F= & {} \frac{-af^n}{b(f-\alpha _1)(f-\alpha _2)\dots (f-\alpha _{2m})},\\ G= & {} \frac{-a\Delta _{c}^{p}(f)^n}{b(\Delta _{c}^{p}(f)-\alpha _1)(\Delta _{c}^{p}(f)-\alpha _2)\dots (\Delta _{c}^{p}(f)-\alpha _{2m})} \end{aligned}$$

and

$$\begin{aligned} F'= & {} \frac{{-a(n-2m)f^{n-1}(f-e_1)^2}(f-e_2)^2\dots (f-e_m)^2f'}{b(f-\alpha _1)^2(f-\alpha _2)^2\dots (f-\alpha _{2m})^2},\\ G'= & {} \frac{{-a(n-2m)\Delta _{c}^{p}(f)^{n-1}(\Delta _{c}^{p}(f)-e_1)^2}(\Delta _{c}^{p}(f)-e_2)^2\dots (\Delta _{c}^{p}(f)-e_m)^2\Delta _{c}^{p}(f)'}{b(\Delta _{c}^{p}(f)-\alpha _1)^2(\Delta _{c}^{p}(f)-\alpha _2)^2\dots (\Delta _{c}^{p}(f)-\alpha _{2m})^2}. \end{aligned}$$

From Lemma 2.1

$$\begin{aligned} N_{\textrm{E}}^{1)}(r,1;F)\le N(r,H)+S(r,F)+S(r,G). \end{aligned}$$

Now, simple zeros of $(f-\alpha _1), (f-\alpha _2),\dots , (f-\alpha _{2\,m})$ are the simple poles of F and simple zeros of $(\Delta _{c}^{p}(f)-\alpha _1),(\Delta _{c}^{p}(f)-\alpha _2),\dots , (\Delta _{c}^{p}(f)-\alpha _{2\,m})$ are simple poles of G. Again multiple zeros of $(f-\alpha _1), (f-\alpha _2),\dots , (f-\alpha _{2\,m})$; $(\Delta _{c}^{p}(f)-\alpha _1), (\Delta _{c}^{p}(f)-\alpha _2),\dots , (\Delta _{c}^{p}(f)-\alpha _{2\,m})$ are the zeros of $f'$ and $\Delta _{c}^{p}(f)'$, respectively.

Now, we see that all possible poles of H may occur at (i) multiple poles of F and G having different multiplicities; (ii) multiple zeros of F and G; (iii) zeros of $F-1$ and $G-1$ having different multiplicities; (iv) zeros of $F'$ which are not the zeros of $F(F-1)$; (v) zeros of $G'$ which are not the zeros of $G(G-1)$.

Since all poles of H are simple, we get the conclusion of this lemma. $\square$

Lemma 2.8

Let F and G be given by (2.1) and $H\not \equiv 0$. If F, G share (1, 2) and $f, \Delta _{c}^{p}(f)$ share $(\infty ,0)$, then

$$\begin{aligned} (m+n)T(r,f)+mT(r,\Delta _{c}^{p}(f))\le & {} 2\overline{N}(r,0;f)+2\overline{N}(r,0;\Delta _{c}^{p}(f)) +2\sum _{j=1}^{m}\overline{N}(r,e_j;f)\\{} & {} +2\sum _{j=1}^{m}\overline{N}(r,e_j;\Delta _{c}^{p}(f)) + 3\overline{N}(r,\infty ;f)+S(r,f)+S(r,\Delta _{c}^{p}(f)). \end{aligned}$$

Proof

Since F and G share $(1,2), N_\textrm{E}^{1)}(r,1;F)=N(r,1;F|=1)$. By Lemmas 2.3 and 2.7, we see that

$$\begin{aligned}{} & {} \overline{N}(r,1;F)+\overline{N}(r,1;G)\nonumber \\{} & {} \quad =\overline{N}(r,1;F|=1)+\overline{N}(r,1;F|=2) +\overline{N}_\textrm{E}^{(3}(r,1;F)\nonumber \\{} & {} \qquad +\overline{N}_L(r,1;F)+\overline{N}_L(r,1;G)+\overline{N}(r,1;G)\nonumber \\{} & {} \quad \le \overline{N}(r,1;F|=1)+\overline{N}(r,1,F|=2)+\overline{N}_\textrm{E}^{(3}(r,1;F)\nonumber \\{} & {} \qquad +\overline{N}_L(r,1;F)+\overline{N}_L(r,1;G)+N(r,1;G)-2\overline{N}_L(r,1;F)\nonumber \\{} & {} \qquad -3\overline{N}(r.1;G)-2\overline{N}_{\textrm{E}}^{(3}(r,1;F)-\overline{N}(r,1;F|=2)\nonumber \\{} & {} \quad \le \overline{N}(r,1;F|=1)+N(r,1;G)-\overline{N}_\textrm{E}^{(3}(r,1;F)-\overline{N}_L(r,1;F)\nonumber \\{} & {} \qquad -2\overline{N}_L(r,1;G)\nonumber \\{} & {} \quad \le \overline{N}_L(r,1;F)+\overline{N}_L(r,1;G)+\overline{N}(r,0;f)+\overline{N}(r,0;\Delta _{c}^{p}(f))\nonumber \\{} & {} \qquad +\sum _{j=1}^{m}\overline{N}(r,e_j;f)+\sum _{j=1}^{m}\overline{N}(r,e_j;\Delta _{c}^{p}(f))+\overline{N}_*(r,\infty ;f,\Delta _{c}^{p}(f))\nonumber \\{} & {} \qquad +\overline{N}_0(r,0;f')+\overline{N}_0(r,0;\Delta _{c}^{p}(f)')\nonumber \\{} & {} \qquad +nT(r,\Delta _{c}^{p}(f))-\overline{N}_\textrm{E}^{(3}(r;1;F)-\overline{N}_L(r,1;F)-2\overline{N}_L(r,1;G)\nonumber \\{} & {} \qquad +S(r,f)+S(r,\Delta _{c}^{p}(f))\nonumber \\{} & {} \quad \le \overline{N}(r,0;f)+\overline{N}(r,0;\Delta _{c}^{p}(f))+\sum _{j=1}^{m}\overline{N}(r,e_j;f)+\sum _{j=1}^{m}\overline{N}(r,e_j;\Delta _{c}^{p}(f))\nonumber \\{} & {} \qquad +\overline{N}(r,\infty ;f)+nT(r,\Delta _{c}^{p}(f))+\overline{N}_0(r,0;f')+\overline{N}_0(r,0;\Delta _{c}^{p}(f)')\nonumber \\{} & {} \qquad +S(r,f)+S(r,\Delta _{c}^{p}(f)). \end{aligned}$$

(2.3)

Now, by second fundamental theorem, we have

$$\begin{aligned}{} & {} (n+m)T(r,f)+(n+m)T(r,\Delta _{c}^{p}(f))\nonumber \\{} & {} \quad \le \overline{N}(r,1;F)+\overline{N}(r,0;f)+\sum _{j=1}^{m}\overline{N}(r,e_j;f)\nonumber \\{} & {} \qquad +\overline{N}(r,\infty ;f)+\overline{N}(r,1;G)+\overline{N}(r,0;\Delta _{c}^{p}(f))\nonumber \\{} & {} \qquad +\sum _{j=1}^{m}\overline{N}(r,e_j;\Delta _{c}^{p}(f))+\overline{N}(r,\infty ;\Delta _{c}^{p}(f))\nonumber \\{} & {} \qquad -N_0(r,0;f')-N_0(r,0;\Delta _{c}^{p}(f)')+S(r,f)+S(r,\Delta _{c}^{p}(f)). \end{aligned}$$

(2.4)

Hence, using (2.3) in (2.4), we get

$$\begin{aligned}{} & {} (m+n)T(r,f)+mT(r,\Delta _{c}^{p}(f))\nonumber \\{} & {} \quad \le \overline{N}(r,0;f)+\sum _{j=1}^{m}\overline{N}(r,e_i;f)+\overline{N}(r,\infty ;f)\nonumber \\{} & {} \qquad +\overline{N}(r,0;\Delta _{c}^{p}(f))+\sum _{j=1}^{m}\overline{N}(r,e_j;\Delta _{c}^{p}(f))+\overline{N}(r,\infty ;\Delta _{c}^{p}(f))\nonumber \\{} & {} \qquad -N_0(r,0;f')-N_0(r,0;\Delta _{c}^{p}(f)')+\overline{N}(r,0;f)\nonumber \\{} & {} \qquad +\overline{N}(r,0;\Delta _{c}^{p}(f))+\sum _{j=1}^{m}\overline{N}(r,e_j;f)+\sum _{j=1}^{m}\overline{N}(r,e_j;\Delta _{c}^{p}(f))\nonumber \\{} & {} \qquad +\overline{N}(r,\infty ;f)+\overline{N}_0(r,0;f')+\overline{N}_0(r;0;\Delta _{c}^{p}(f)')\nonumber \\{} & {} \qquad +S(r,f)+S(r,\Delta _{c}^{p}(f))\nonumber \\{} & {} \quad \le 2\overline{N}(r,0;f)+2\overline{N}(r,0;\Delta _{c}^{p}(f))+2\sum _{j=1}^{m}\overline{N}(r,e_j;f)\nonumber \\{} & {} \qquad +2\sum _{j=1}^{m}\overline{N}(r,e_j;\Delta _{c}^{p}(f))+3\overline{N}(r,\infty ;f)+S(r,f)+S(r,\Delta _{c}^{p}(f)). \end{aligned}$$

(2.5)

This proves the lemma. $\square$

Lemma 2.9

[6, 7] Let $\phi (z)=c^2(z^{n-m}-E)^2-4bd(z^{n-2m}-E)(z^n-E)$, where $c,b,d,E\in \mathbb {C^*},\frac{c^2}{4bd}= \frac{n(n-2m)}{(n-m)^2}$, gcd$(m,n)=1, n>2\,m$. Let ${\omega = e^{2\pi i/m}}$. Then, for ${l=0,1,\cdots ,m-1}$

• (i) $\phi (z)$ has no multiple zero, when $E\ne \omega ^l$.

• (ii) $\phi (z)$ has exactly one multiple zero, when $E=\omega ^l$

and its multiplicity is 4. In particular, when $E=1$, then the multiple zero is 1.

3 Proof of the theorem

Proof of Theorem 1.10

Let F and G be given by (2.1). Since f and $\Delta _{c}^{p}(f)$ share (S, 2), so we have F, G share (1, 2). Now, we consider the following cases to complete the proof of the theorem.

Case-1 Suppose that $H\not \equiv 0$. Then, $F\not \equiv G$, and so, by Lemma 2.5, we have $V\not \equiv 0$ for $n\ge 4m+4$. Now, putting $\lambda =2$ and $k=0$ in Lemma 2.6, we obtain

$$\begin{aligned} (n-2m-1)\overline{N}(r,\infty ;f)&\le \frac{4}{3}\Big [\overline{N}(r,0;f)+\overline{N}(r,0;\Delta _{c}^{p}(f))\Big ]+\frac{2}{3}\overline{N}(r,\infty ;f)\\&\quad +S(r,f)+S(r,\Delta _{c}^{p}(f)). \end{aligned}$$

That is

$$\begin{aligned} \overline{N}(r,\infty ;f)&\le \frac{4}{3n-6m-5}\Big [\overline{N}(r,0;f)+\overline{N}(r,0;\Delta _{c}^{p}(f))\Big ]\nonumber \\&\quad +S(r,f)+S(r,\Delta _{c}^{p}(f)). \end{aligned}$$

(3.1)

From (3.1) and Lemma 2.8, we get

$$\begin{aligned} (m+n)T(r,f)+mT(r,\Delta _{c}^{p}(f))&\le 2\overline{N}(r,0;f)+2\overline{N}(r,0;\Delta _{c}^{p}(f))\nonumber \\&\quad +2\sum _{j=1}^{m}\overline{N}(r,e_j;f)+2\sum _{j=1}^{m}\overline{N}(r,e_j;\Delta _{c}^{p}(f))\nonumber \\&\quad +\frac{12}{3n-6m-5}\Big [\overline{N}(r,0;f)+\overline{N}(r,0;\Delta _{c}^{p}(f))\Big ]\nonumber \\&\quad +S(r,f)+S(r,\Delta _{c}^{p}(f))\nonumber \\&\le \Big [2+2m+\frac{12}{3n-6m-5}\Big ]T(r,f)\nonumber \\&\quad +\Big [2+2m+\frac{12}{3n-6m-5}\Big ]T(r,\Delta _{c}^{p}(f))\nonumber \\&\quad +S(r,f)+S(r,\Delta _{c}^{p}(f)). \end{aligned}$$

(3.2)

Similarly, we obtain

$$\begin{aligned} (m+n)T(r,\Delta _{c}^{p}(f))+mT(r,f)&\le \Big [2+2m+\frac{12}{3n-6m-5}\Big ]T(r,f)\nonumber \\&\quad +\Big [2+2m+\frac{12}{3n-6m-5}\Big ]T(r,\Delta _{c}^{p}(f))+S(r,f)+S(r,\Delta _{c}^{p}(f)). \end{aligned}$$

(3.3)

Adding (3.2) and (3.3), we get

$$\begin{aligned} (n+2m)\Big [T(r,f)+T(r,\Delta _{c}^{p}(f))\Big ]&\le \Big [4+4m+\frac{24}{3n-6m-5}\Big ]\Big [T(r,f)+T(r,\Delta _{c}^{p}(f))\Big ]\\&\quad +S(r,f)+S(r,\Delta _{c}^{p}(f)). \end{aligned}$$

This implies that

$$\begin{aligned} n-4-2m-\frac{24}{3n-6m-5}&\le 0, \end{aligned}$$

which is a contradiction for $n\ge (4m+4)$.

Case-2 Suppose that $H\equiv 0$. Then

$$\begin{aligned} \frac{F''}{F'}-\frac{2F'}{F-1}=\frac{G''}{G'}-\frac{2G'}{G-1}. \end{aligned}$$

On integration, we have,

$$\begin{aligned} \frac{1}{G-1}=\frac{A}{F-1}+B, \end{aligned}$$

where $A(\ne 0), B$ are constants. Hence

$$\begin{aligned} G=\frac{(B+1)F+(A-B-1)}{BF+(A-B)} \end{aligned}$$

(3.4)

and

$$\begin{aligned} T(r,G)=T(r,F)+O(1). \end{aligned}$$

(3.5)

We now consider the following subcases.

Subcase-2.1 Let $B\ne 0,-1$. Then, (3.4) and the fact that $f, \Delta _{c}^{p}(f)$ share $(\infty , 0)$ implies $\infty$ is a Picard exceptional value of $f, \Delta _{c}^{p}(f)$. It follows that

$$\begin{aligned} \overline{N}(r,\infty ;F)&=\sum _{j=1}^{2m}\overline{N}(r,\alpha _j;f), \end{aligned}$$

(3.6)

$$\begin{aligned} \overline{N}(r,\infty ;G)&=\sum _{j=1}^{2m}\overline{N}(r,\alpha _j;\Delta _{c}^{p}(f)). \end{aligned}$$

(3.7)

Subcase-2.1.1 Suppose that $A-B-1\ne 0$. Then, from (3.4), we get

$$\begin{aligned} \overline{N}\left( r,-\frac{A-B-1}{B+1};F\right) =\overline{N}(r.0;G). \end{aligned}$$

Because $\frac{A-B-1}{B+1}\ne \frac{A-B}{B}$. For if, $\frac{A-B-1}{B+1}=\frac{A-B}{B}$, then $1=0$, which is absurd. Now, in view of (3.5), by the second fundamental theorem, we have

$$\begin{aligned} nT(r,f)&\le \overline{N}(r,\infty ;F)+\overline{N}(r,0;F)+ \overline{N}\left( r,-\frac{A-B-1}{B+1};F\right) +S(r,F)\\&\le \sum _{j=1}^{2m}\overline{N}(r,\alpha _j;f)+\overline{N}(r,0;f)+\overline{N}(r,0;\Delta _{c}^{p}(f))+S(r,f)\\&\le (2m+2)T(r,f)+S(r,f), \end{aligned}$$

which contradicts our assumption $n\ge (4m+4)$.

Subcase-2.1.2 Suppose that $A-B-1=0$. Now, from (3.4), we have

$$\begin{aligned} G=\frac{(B+1)F}{BF+1}. \end{aligned}$$

(3.8)

From (3.8), we find that

$$\begin{aligned} \overline{N}(r,-\frac{1}{B};F)=\overline{N}(r,\infty ; G)=\sum _{j=1}^{2m}\overline{N}(r,\alpha _j;\Delta _{c}^{p}(f)). \end{aligned}$$

Again using the second fundamental theorem and in view of (3.5), we get

$$\begin{aligned} nT(r,f)&\le \overline{N}(r,0;F)+\overline{N}(r,\infty ; F)+\overline{N}\left( r,-\frac{1}{B};F\right) +S(r,F)\\&\le \overline{N}(r,0;f)+ \sum _{j=1}^{2m}\overline{N}(r,\alpha _j;f)+\sum _{j=1}^{2m}\overline{N}(r,\alpha _j;\Delta _{c}^{p}(f))+S(r,f)\\&\le (4m+1)T(r,f)+S(r,f), \end{aligned}$$

which is a contradiction to our assumption that $n\ge (4m+4).$

Subcase-2.2 Let $B=0$. Then, (3.4) takes the following form:

$$\begin{aligned} G=\frac{F+A-1}{A}. \end{aligned}$$

(3.9)

Subcase-2.2.1 Suppose that $A-1\ne 0$. From (3.9), we find $\overline{N}(r,-(A-1);F)=\overline{N}(r,0;G)$. Hence, by the second fundamental theorem and (3.5), we have

$$\begin{aligned} nT(r,f)&\le \overline{N}(r,0;F)+\overline{N}(r,\infty ;F)+\overline{N}(r,-(A-1);F)+S(r,F)\\&\le \overline{N}(r,0;f)+\overline{N}(r,\infty ;f)+\sum _{j=1}^{2m}\overline{N}(r,\alpha _j;f)+\overline{N}(r,0;\Delta _{c}^{p}(f))+S(r,f)\\&\le (2m+3)T(r,f)+S(r.f), \end{aligned}$$

which is a contradiction to our assumption that $n\ge (4m+4).$

Subcase-2.2.2 Suppose that $A=1$, i.e., $F\equiv G$. Therefore, $R(f)=R(\Delta _{c}^{p}(f))$, where R is given by (1.4). Hence, we obtain

$$\begin{aligned} bf^{2m}\Delta _{c}^{p}(f)^{2m}(f^{n-2m}-\Delta _{c}^{p}(f)^{n-2m})+cf^m\Delta _{c}^{p}(f)^m(f^{n-m}-\Delta _{c}^{p}(f)^{n-m})+d(f^n-\Delta _{c}^{p}(f)^n)=0. \end{aligned}$$

(3.10)

Set $h=\frac{f}{\Delta _{c}^{p}(f)}$. Substituting $f=h\Delta _{c}^{p}(f)$ in (3.10), we obtain

$$\begin{aligned} bh^{2m}\Delta _{c}^{p}(f)^{2m}(h^{n-2m}-1)+ch^m\Delta _{c}^{p}(f)^m(h^{n-m}-1)+d(h^n-1) =0. \end{aligned}$$

(3.11)

If h is not constant, from (3.11), we conclude that

$$\begin{aligned} \Bigg (2bh^{m}(h^{n-2m}-1)\Delta _{c}^{p}(f)^m+c(h^{n-m}-1)\Bigg )^2=\phi (h), \end{aligned}$$

(3.12)

where $\phi (h)=c^2(h^{n-m}-1)^2-4bd(h^n-1)(h^{n-2m}-1)$. From Lemma  2.9, we know that 1 is a multiple zero of $\phi (z)$ of multiplicity 4 and all other zeros of $\phi (z)$ are simple. Let all the zeros of $\phi (z)$ be $1,\beta _1, \beta _2,\dots ,\beta _{2n-2\,m-4}$. Therefore, (3.12) implies that every zero of $(h-\beta _j),(j=1, 2, \dots , (2n-2m-4 ))$ is of multiplicity at least 2. Then, by the second fundamental theorem, we have

$$\begin{aligned} (2n-2m-6)T(r,h)&\le \sum _{j=1}^{2n-2m-4}\overline{N}(r.\beta _j;h)+S(r,h)\\&\le \frac{1}{2}\sum _{j=1}^{2n-2m-4}N(r,\beta _j;h)+S(r,h)\\&\le (n-m-2)T(r,h)+S(r,h), \end{aligned}$$

which is a contradiction to our assumption that $n\ge (4m+4)$. Therefore, h is constant. From (3.11), we have $h^{n-2\,m}-1=0, h^{n-m}-1=0, h^n-1=0$, which imply $h=1$, and hence, $f\equiv \Delta _{c}^{p}(f)$.

Subcase-2.3 Let $B=-1$. From (3.4), we have

$$\begin{aligned} G=\frac{A}{-F+(A+1)} \end{aligned}$$

(3.13)

Subcase-2.3.1 Let $A+1\ne 0$. Then, from (3.13), we find $\overline{N}(r,A+1;F)=\overline{N}(r,\infty ;G)$. The fact that $f, \Delta _{c}^{p}(f)$ share $(\infty ,0)$ implies f omits $\infty$. Now, by the second fundamental theorem, we have

$$\begin{aligned} nT(r,f)&\le \overline{N}(r,0;F)+\overline{N}(r,\infty ;F)+\overline{N}(r,A+1;F)+S(r,F)\\&\le \overline{N}(r,0;f)+\sum _{j=1}^{2m}\overline{N}(r,\alpha _j;f)+\sum _{j=1}^{2m}\overline{N}(r,\alpha _j;\Delta _{c}^{p}(f))+S(r,f)\\&\le (4m+1)T(r,f)+S(r,f), \end{aligned}$$

which is a contradiction to our assumption that $n\ge (4m+4)$.

Subcase-2.3.2 Suppose that $A+1 =0$, and hence, $FG\equiv 1$, which implies that

$$\begin{aligned} \frac{a^2f^n\Delta _{c}^{p}(f)^n}{b^2(f-\alpha _1)(f-\alpha _2)\dots (f-\alpha _{2m})(\Delta _{c}^{p}(f)-\alpha _1)(\Delta _{c}^{p}(f)-\alpha _2)\dots (\Delta _{c}^{p}(f)-\alpha _{2m})}=1. \end{aligned}$$

(3.14)

Since $f, \Delta _{c}^{p}(f)$ share $(\infty ,0)$, from (3.14), we get that f omits $\infty$. Also, the zeros of $(f-\alpha _j),(j=1,2,\dots , 2m)$ are of order at least n. Hence, using the second fundamental theorem, we get

$$\begin{aligned} (2m-1)T(r,f)\le & {} \sum \limits _{j=1}^{2m}\overline{N}(r,\alpha _{j};f)+\overline{N}(r,\infty ;f)+S(r,f)\\\le & {} \frac{1}{n}\sum \limits _{j=1}^{2m} N(r,\alpha _{j};f)+S(r,f)\\\le & {} \frac{2m}{n} T(r,f)+S(r,f), \end{aligned}$$

which is a contradiction for $n\ge 4m+4$. $\square$