To prove our theorems, we will require some lemmas as follows.

**Lemma 2.1** [16]

*Let* f(z) *be a transcendental entire function*, \nu (r,f) *be the central index of* f(z). *Then there exists a set* E\subset (1,+\mathrm{\infty}) *with finite logarithmic measure*, *we choose* *z* *satisfying* |z|=r\notin [0,1]\cup E *and* |f(z)|=M(r,f), *we get*

\frac{{f}^{(j)}(z)}{f(z)}={\left\{\frac{\nu (r,f)}{z}\right\}}^{j}(1+o(1))\phantom{\rule{1em}{0ex}}\mathit{\text{for}}j\in N.

**Lemma 2.2** [17]

*Let* f(z) *be an entire function of finite order* \sigma (f)=\sigma <\mathrm{\infty}, *and let* \nu (r,f) *be the central index of* *f*. *Then*, *for any* *ε* (>0), *we have*

\underset{r\to \mathrm{\infty}}{lim\hspace{0.17em}sup}\frac{log\nu (r,f)}{logr}=\sigma .

**Lemma 2.3** [18]

*Let* *f* *be a transcendental entire function*, *and let* E\subset [1,+\mathrm{\infty}) *be a set having finite logarithmic measure*. *Then there exists* \{{z}_{n}={r}_{n}{e}^{i{\theta}_{n}}\} *such that* |f({z}_{n})|=M({r}_{n},f), {\theta}_{n}\in [0,2\pi ), {lim}_{n\to \mathrm{\infty}}{\theta}_{n}={\theta}_{0}\in [0,2\pi ), {r}_{n}\notin E *and if* 0<\sigma (f)<\mathrm{\infty}, *then*, *for any given* \epsilon >0 *and sufficiently large* {r}_{n},

{r}_{n}^{\sigma (f)-\epsilon}<\nu ({r}_{n},f)<{r}_{n}^{\sigma (f)+\epsilon}.

**Lemma 2.4** [16]

*Let* P(z)={b}_{n}{z}^{n}+{b}_{n-1}{z}^{n-1}+\cdots +{b}_{0} *with* {b}_{n}\ne 0 *be a polynomial*. *Then*, *for every* \epsilon >0, *there exists* {r}_{0}>0 *such that for all* r=|z|>{r}_{0} *the inequalities*

(1-\epsilon )|{b}_{n}|{r}^{n}\le |P(z)|\le (1+\epsilon )|{b}_{n}|{r}^{n}

*hold*.

**Lemma 2.5** *Let* f(z) *and* A(z) *be two entire functions with* 0<\sigma (f)=\sigma (A)=\sigma <\mathrm{\infty}, 0<\tau (A)<\tau (f)<\mathrm{\infty}, *then there exists a set* E\subset [1,+\mathrm{\infty}) *that has infinite logarithmic measure such that for all* r\in E *and a positive number* \kappa >0, *we have*

\frac{M(r,A)}{M(r,f)}<exp\{-\kappa {r}^{\sigma}\}.

*Proof* By definition, there exists an increasing sequence \{{r}_{m}\}\to \mathrm{\infty} satisfying (1+\frac{1}{m}){r}_{m}<{r}_{m+1} and

\underset{m\to \mathrm{\infty}}{lim}\frac{logM({r}_{m},f)}{{r}_{m}^{\sigma}}=\tau (f).

(2)

For any given *β* (\tau (A)<\beta <\tau (f)), there exists some positive integer {m}_{0} such that for all m\ge {m}_{0} and for any given *ε* (0<\epsilon <\tau (f)-\beta), we have

logM({r}_{m},f)>(\tau (f)-\epsilon ){r}_{m}^{\sigma}.

(3)

Thus, there exists some positive integer {m}_{1} such that for all m\ge {m}_{1}, we have

{\left(\frac{m}{m+1}\right)}^{\sigma}>\frac{\beta}{\tau (f)-\epsilon}.

(4)

From (2)-(4), for all m\ge {m}_{2}=max\{{m}_{0},{m}_{1}\} and for any r\in [{r}_{m},(1+\frac{1}{m}){r}_{m}], we have

\begin{array}{rl}M(r,f)& \ge M({r}_{m},f)>exp\left\{(\tau (f)-\epsilon ){r}_{m}^{\sigma}\right\}\\ \ge exp\left\{(\tau (f)-\epsilon ){\left(\frac{m}{m+1}r\right)}^{\sigma}\right\}>exp\left\{\beta {r}^{\sigma}\right\}.\end{array}

(5)

Set E={\bigcup}_{m={m}_{2}}^{\mathrm{\infty}}[{r}_{m},(1+\frac{1}{m}){r}_{m}], then

{m}_{l}E=\sum _{m={m}_{2}}^{\mathrm{\infty}}{\int}_{{r}_{m}}^{(1+\frac{1}{m}){r}_{m}}\frac{dt}{t}=\sum _{m={m}_{2}}^{\mathrm{\infty}}log(1+\frac{1}{m})=\mathrm{\infty}.

From the definition of type of entire function, for any sufficiently small \epsilon >0, we have

M(r,A)<exp\left\{(\tau (A)+\epsilon ){r}^{\sigma}\right\}.

(6)

By (5) and (6), set \kappa =\beta -\tau (A)-\epsilon, for all r\in E, we have

\frac{M(r,A)}{M(r,f)}<exp\{-(\beta -\tau (A)-\epsilon ){r}^{\sigma}\}={e}^{-\kappa {r}^{\sigma}}.

Thus, this completes the proof of this lemma. □

**Lemma 2.6** [[19], Theorem 2.1]

*Let* f(z) *be a meromorphic function of finite order* *σ*, *and let* *η* *be a fixed nonzero complex number*, *then*, *for each* \epsilon >0, *we have*

m(r,\frac{f(z+c)}{f(z)})+m(r,\frac{f(z)}{f(z+c)})=O\left({r}^{\sigma -1+\epsilon}\right).

**Lemma 2.7** [[19], Corollary 2.5]

*Let* f(z) *be a meromorphic function with order* \sigma =\sigma (f), \sigma <+\mathrm{\infty}, *and let* *η* *be a fixed nonzero complex number*, *then*, *for each* \epsilon >0, *we have*

T(r,f(z+\eta ))=T(r,f)+O\left({r}^{\sigma -1+\epsilon}\right)+O(logr).

**Lemma 2.8** [1, 20]

*Let* g:(0,+\mathrm{\infty})\to R, h:(0,+\mathrm{\infty})\to R *be monotone increasing functions such that* g(r)\le h(r) *outside of an exceptional set* *E* *with finite linear measure*, *or* g(r)\le h(r), r\notin H\cup (0,1], *where* H\subset (1,\mathrm{\infty}) *is a set of finite logarithmic measure*. *Then*, *for any* \alpha >1, *there exists* {r}_{0} *such that* g(r)\le h(\alpha r) *for all* r\ge {r}_{0}.