The fundamental theorems and the standard notations of the Nevanlinna value distribution theory of meromorphic functions will be used (see Hayman [1], Yang [2] and Yi-Yang [3]). In addition, for a meromorphic function *f*(*z*), we use δ (*a, f*) to denote the Nevanlinna deficiency of *a* ∊ ℂ̃ = ℂ⋃{*∞*} where
$$\delta (a,\text{\hspace{0.17em}}f)=\underset{r\to +\infty}{\mathrm{lim}\text{\hspace{0.17em}}\mathrm{inf}}\frac{m(r,\text{\hspace{0.17em}}\frac{1}{f-a})}{T(r,\text{\hspace{0.17em}}f)}\text{=1}-\underset{r\to +\infty}{\mathrm{lim}\text{\hspace{0.17em}}sup}\frac{m(r,\text{\hspace{0.17em}}\frac{1}{f-a})}{T(r,\text{\hspace{0.17em}}f)}.$$
We also use *S*(*r, f*) to denote any quantity satisfying *S*(*r, f*) = *o*(*T*(*r, f*)) for all *r* outside a possible exceptional set *E* of finite logarithmic measure ${lim}_{r\to \infty}{\int}_{[1,r)\cap E}{\scriptstyle \frac{dt}{t}}<\infty $

Throughout this paper, we assume *m; n; k; t* are positive integers.

Many people were interested in the value distribution of different expressions of meromorphic functions and obtained lots of important theorems (see [1, 4–6]).

Recently, the topic of complex differences has attracted the interest of many mathematicians, and a number of papers have focused on the value distribution of complex differences and difference analogues of Nevanlinna theory (including [7–11]). By combining complex differentiates and complex differences, we proceed in this way in this paper.

Firstly, we study the Nevanlinna deficiencies related to a meromorphic function *f*(*z*), its derivative *f*′(*z*) and its differential-difference monomials
$${F}_{1}(z)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}f{(z)}^{m}\text{\hspace{0.17em}}f(z\text{\hspace{0.17em}}+\text{\hspace{0.17em}}c){f}^{\prime}(z)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}and\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{F}_{2}(z)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}f{(z+c)}^{n}f){f}^{\prime}(z),$$
and obtain the following theorems.

*Let f*(*z*) *be a meromorphic function of finite order satisfying*
$$\underset{r\to +\infty}{\mathrm{lim}\text{\hspace{0.17em}}\mathrm{sup}}\frac{T(r,\text{\hspace{0.17em}}f)}{T(r,\text{\hspace{0.17em}}{f}^{\prime})}\text{<}+\infty ,$$(1)
*and c be a non-zero complex constant. Then*
$$\delta (\infty ,\text{\hspace{0.17em}}{F}_{1})\ge \delta (\infty ,\text{\hspace{0.17em}}{f}^{\prime}).$$

*Let f*(*z*) *be a meromorphic function of finite order satisfying* (1)*, and c be a non-zero complex constant. Then*
$$\delta (\infty ,\text{\hspace{0.17em}}{F}_{2})\ge \delta (\infty ,\text{\hspace{0.17em}}{f}^{\prime}).$$

*It is easy to find meromorphic functions to make the inequalities in Theorems* *1.1* *and* *1.2* *hold. For example, let f*_{1}(*z*) = *e*^{z}, *then*
$$\delta (\infty ,\text{\hspace{0.17em}}{f}_{1}^{\text{'}})=\delta (\infty ,\text{\hspace{0.17em}}{F}_{1})=\delta (\infty ,\text{\hspace{0.17em}}{F}_{2})=\delta (\infty ,\text{\hspace{0.17em}}{f}_{1})=1,$$
*showing the equalities in Theorems 1.1* *and* *1.2* *may hold.*

*Let* ${f}_{2}\left(z\right)=\genfrac{}{}{0.1ex}{}{{z}^{3}+2}{z}$ *and c* = 1 = *m* = *n, then*
$${f}_{2}^{\text{'}}(z)=\frac{2{z}^{3}-2}{{z}^{2}}\text{\hspace{0.17em}}and\text{\hspace{0.17em}}{f}_{2}(z+1)=\frac{{z}^{3}+3{z}^{2}+3z+3}{z+1}.$$

*It follows that*
$${F}_{1}(z)={f}_{2}(z){f}_{2}(z+1){f}_{2}^{\text{'}}(z)=\frac{2{z}^{9}-{P}_{1}(z)}{{z}^{4}+{z}^{3}},\text{\hspace{0.17em}}{F}_{2}(z)={f}_{2}(z+1){f}_{2}^{\text{'}}(z)=\frac{2{z}^{6}-{P}_{2}(z)}{{z}^{3}+{z}^{2}},$$
*where P*_{1}(*z*); *P*_{2}(*z*) *are polynomials in z with* deg *P*_{1}(*z*) ≥ 8 *and* deg *P*_{2}(*z*) ≥ 5. *Clearly, f*_{2}(*z*) *satisfies*
$$\underset{r\to +\infty}{\mathrm{lim}\text{\hspace{0.17em}}sup}\frac{T(r,{f}_{2})}{T(r,{f}_{2}^{\text{'}})}=\underset{r\to +\infty}{\mathrm{lim}\text{\hspace{0.17em}}sup}\frac{3\text{\hspace{0.17em}}\mathrm{log}\text{\hspace{0.17em}}r}{3\text{\hspace{0.17em}}\mathrm{log}\text{\hspace{0.17em}}r}=1\text{\hspace{0.17em}}<+\infty .$$

*Thus, we have*
$$\delta (\infty ,{F}_{1})=\frac{5}{9}>\delta (\infty ,\text{\hspace{0.17em}}{f}_{2}^{\text{'}})=\frac{1}{3},$$
*and*
$$\delta (\infty ,{F}_{2})=\frac{1}{2}>\delta (\infty ,\text{\hspace{0.17em}}{f}_{2}^{\text{'}})=\frac{1}{3},$$
*showing the inequalities in Theorems 1.1 and 1.2 may hold.*

*Thus, Theorems 1.1 and 1.2 are sharp.*

*In addition, from the above examples, we can find that* $\delta (\infty ,{F}_{1})=\genfrac{}{}{0.1ex}{}{5}{9}<\delta (\infty ,\phantom{\rule{0.167em}{0ex}}{f}_{2})=\genfrac{}{}{0.1ex}{}{2}{3}$ and $\delta (\infty ,{F}_{2})=\genfrac{}{}{0.1ex}{}{1}{2}<\delta (\infty ,\phantom{\rule{0.167em}{0ex}}{f}_{2})=\genfrac{}{}{0.1ex}{}{2}{3}.$. *So, we give the following question: Under the conditions of Theorems 1.1 and 1.2*, *do F*_{1}; *F*_{2}*satisfy*
$$\delta (\infty ,{f}^{\prime})\le \delta (\infty ,\text{\hspace{0.17em}}{F}_{1})\le \delta (\infty ,f),$$
*and*
$$\delta (\infty ,{f}^{\prime})\le \delta (\infty ,\text{\hspace{0.17em}}{F}_{2})\le \delta (\infty ,f)?$$

We also get the following relations between δ(∞, *f*) and δ(∞, *F*_{i}), *i* = 1, 2.

*Let f*(*z*) *be a meromorphic function of finite order, and* c *be a non-zero complex constant. If* $\delta =\delta (\infty ,f)>\genfrac{}{}{0.1ex}{}{8}{m+6},$, *then δ*(∞, *F*_{1}) > 0.

*Let* *f*(*z*) *be a meromorphic function of finite order, and* c *be a non-zero complex constant. If* $\delta =\delta (\infty ,f)>\genfrac{}{}{0.1ex}{}{8}{m+5},$, *then δ*(∞, *F*_{2}) > 0.

*Let f*(*z*) *be a meromorphic function of finite order, and* c *be a non-zero complex constant. Set*
$${F}_{3}(z)=f{(z)}^{m}f(z+c).$$

*If* $\delta =\delta (\infty ,f)>\genfrac{}{}{0.1ex}{}{4}{m+3},$, *then δ*(∞, *F*)> 0.

The following ideas derive from Hayman [5], Laine-Yang [12], Zheng-Chen [13]. In 1959, Hayman [5] studied the value distribution of meromorphic functions and their derivatives, and obtained the following famous theorems.

([5]). *Let f*(*z*) *be a transcendental entire function. Then*

*for n* = 3 *and a* ≠ 0, Ψ(*z*) = *f*′(*z*) − *af*(*z*)^{n} *assumes all finite values infinitely often.*

*for n* = 2, Φ(*z*) = *f*′(*z*) *f*(*z*)^{n}*assumes all finite values except possibly zero infinitely often.*

Recently, some authors studied the zeros of *f*(*z*+*c*) *f*(*z*)^{n} − *a* and *f*(*z*+*c*) − *af*(*z*)^{n} − *b*, where *a*(≠0), *b* are complex constants or small functions. Some related results can be found in [12–17]. Especially, Laine-Yang [12] and Zheng-Chen [13] proved the following result, which is regarded as a difference counterpart of Theorem1. 8.

([12, 13]). *Let* *f*(*z*) *be a transcendental entire function of finite order, and c be a non-zero complex constant. Then*

*for n* ≥ 2, Φ_{1}(*z*) = *f*(*z*+*c*) *f*(*z*)^{n} *assumes every a* ∈ ℂ\{0} *infinitely often.*

*for n* ≥ 3, *a* ≠0, *Ψ*_{1}(*z*) = *f*(*z*+*c*)− *af*(*z*)^{n} *assumes every b* ∈ ℂ *infinitely often*.

In the following, we investigate the zeros of some differential-difference polynomials of a meromorphic function *f*(*z*) taking small function a(*z*) with respect to *f*(*z*), where and in the following a(*z*) is a non-zero small function of growth *S*(*r, f*), and obtain some theorems as follows.

*Let* *f*(*z*) *be a transcendental meromorphic function of finite order, and* c *be a non-zero complex constant. Set*
$${G}_{1}(z)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}f{(z)}^{m}\text{\hspace{0.17em}}f{(z+c)}^{n}f\prime (z).$$

*If m* ≥ *n* + 8 *or n* ≥ *m* + 8, *then G*_{1}(*z*) − *a*(*z*) *has infinitely many zeros.*

*An example shows that the conclusion can not hold if f*(*z*) *is of infinite order. Let f*(*z*) = 2*e*^{ez}, *a*(*z*) = *e*^{z}, *m* = 9, *n* = 1 *and e*^{c} = −10, *then*
$${G}_{1}(z)-a(z)=({2}^{m+n+{1}_{e}[(m+1)+n{e}^{c}]{e}^{z}}-1){e}^{z}=({2}^{11}-1){e}^{z},$$
*then G*_{1}(*z*) − *a*(*z*) *has finite many zeros.*

*Let f*(*z*) *be a transcendental meromorphic function of finite order, and* *c*_{1}, *c*_{2}, … *c*_{n} be non-zero complex constants. Set
$${G}_{2}(z)=f{(z)}^{m}{\displaystyle \prod}_{j=1}^{n}f{(z+{c}_{j})}^{sj}{\displaystyle \prod}_{i=1}^{k}{f}^{(i)}(z).$$

*If m* ≥ σ_{1} + 2*n* + *k*(*k* + 3) + 4, *where* σ_{1} = *s*_{1} + *s*_{2} + … + *s*_{n}, *then G*_{2}(*z*) − *a*(*z*) *has infinitely many zeros.*

Let *P*_{n}(*z*) = *a*_{n}*z*^{n} + *a*_{n−1}*z*^{n−1} + … + *a*_{1}*z* + *a*_{0} be a non-zero polynomial, where *a*_{0} *a*_{1} … *a*_{n} are complex constants and *t* is the number of the distinct zeros of *P*_{n}(*z*). Then we further obtain the following results.

*Let f*(*z*) *be a transcendental meromorphic function of finite order, and c be a non-zero complex constant. Set*
$${G}_{3}(z)=f{(z)}^{m}{P}_{n}(f(z+c)){\displaystyle \prod}_{i=1}^{k}{f}^{(i)}(z).$$

*If m* ≥ *n* + *t* + *k*(*k* + 3) + 4, *then G*_{3}(*z*) − *a*(*z*) *has infinitely many zeros.*

*Let f*(*z*) *be a transcendental meromorphic function of finite order, and c be a non-zero complex constant. Set*
$${G}_{4}(z)={P}_{m}(f(z))f{(z+c)}^{n}{\displaystyle \prod}_{i=1}^{k}{f}^{(i)}(z).$$

*If m* ≥ *n* + *t* + *k*(*k* + 3) + 4, *then G*_{4}(*z*) − *a*(*z*) *has infinitely many zeros.*

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