1. Introduction, Definitions, and Notations {#sec1}
===========================================

Let *f* be an entire function defined in the open complex plane and let $$\begin{matrix}
{M_{f}\left( r \right) = \max\left\{ {\left| {f\left| z \right|} \right|:\left| z \right| = r} \right\}} \\
\end{matrix}$$ be its maximum modulus function. If *f* is nonconstant then *M* ~*f*~(*r*) is strictly increasing and continuous and its inverse *M* ~*f*~ ^−1^(*r*) : (\|*f*(0)\|, *∞*) → (0, *∞*) exists and is such that $$\begin{matrix}
{\underset{s\rightarrow\infty}{\lim}M_{f}^{- 1}\left( s \right) = \infty.} \\
\end{matrix}$$ We use the standard notations and definitions in the theory of entire functions which are available in \[[@B11]\]. In the sequel we use the following notation: $$\begin{matrix}
{\begin{matrix}
{{\log}^{\lbrack 0\rbrack}x = x,\quad\quad{\log}^{\lbrack k\rbrack}x = \log\left( {{\log}^{\lbrack k - 1\rbrack}x} \right)} \\
\end{matrix}\text{for}\,\, k = 1,2,3,\ldots,\operatorname{}} \\
{\begin{matrix}
{{\exp}^{\lbrack 0\rbrack}x = x,\quad\quad{\exp}^{\lbrack k\rbrack}x = \exp\left( {{\exp}^{\lbrack k - 1\rbrack}x} \right)} \\
\end{matrix}\text{for}\,\, k = 1,2,3,\ldots.\operatorname{}} \\
\end{matrix}$$

The following definitions are well known.

Definition 1 .The order *ρ* ~*f*~ and the lower order *λ* ~*f*~ of an entire function *f* are defined as $$\begin{matrix}
{\rho_{f} = \lim\underset{r\rightarrow\infty}{\sup}\frac{{\log}^{\lbrack 2\rbrack}M_{f}\left( r \right)}{\log r},} \\
{\lambda_{f} = \lim\underset{r\rightarrow\infty}{\inf}\frac{{\log}^{\lbrack 2\rbrack}M_{f}\left( r \right)}{\log r}.} \\
\end{matrix}$$

Juneja et al. \[[@B3]\] defined the (*p*, *q*)th order and (*p*, *q*)th lower order of an entire function *f*, respectively, as follows: $$\begin{matrix}
{\rho_{f}\left( {p,q} \right) = \lim\underset{r\rightarrow\infty}{\sup}\frac{{\log}^{\lbrack p\rbrack}M_{f}\left( r \right)}{{\log}^{\lbrack q\rbrack}r},} \\
{\lambda_{f}\left( {p,q} \right) = \lim\underset{r\rightarrow\infty}{\inf}\frac{{\log}^{\lbrack p\rbrack}M_{f}\left( r \right)}{{\log}^{\lbrack q\rbrack}r},} \\
\end{matrix}$$ where *p*, *q* are any two positive integers with *p* ≥ *q*.

If *p* = *l* and *q* = 1 then we write *ρ* ~*f*~(*l*, 1) = *ρ* ~*f*~ ^\[*l*\]^ and *λ* ~*f*~(*l*, 1) = *λ* ~*f*~ ^\[*l*\]^.

Also for *p* = 2 and *q* = 1 we, respectively, denote *ρ* ~*f*~(2,1) and *λ* ~*f*~(2,1) by *ρ* ~*f*~ and *λ* ~*f*~.

In this connection we just recall the following definition.

Definition 2 (see \[[@B3]\]).An entire function *f* is said to have index-pair (*p*, *q*), *p* ≥ *q* ≥ 1, if *b* \< *ρ* ~*f*~(*p*, *q*) \< *∞* and *ρ* ~*f*~(*p* − 1, *q* − 1) is not a nonzero finite number, where *b* = 1 if *p* = *q* and *b* = 0 if *p* \> *q*. Moreover if 0 \< *ρ* ~*f*~(*p*, *q*) \< *∞*, then $$\begin{matrix}
{\rho_{f}\left( {p - n,q} \right) = \infty\quad\text{for}\,\, n < p,} \\
{\rho_{f}\left( {p,q - n} \right) = 0\quad\text{for}\,\, n < q,} \\
{\rho_{f}\left( {p + n,q + n} \right) = 1\quad\text{for}\,\, n = 1,2,\ldots.} \\
\end{matrix}$$ Similarly for 0 \< *λ* ~*f*~(*p*, *q*) \< *∞*, one can easily verify that $$\begin{matrix}
{\lambda_{f}\left( {p - n,q} \right) = \infty\quad\text{for}\,\, n < p,} \\
{\lambda_{f}\left( {p,q - n} \right) = 0\quad\text{for}\,\, n < q,} \\
{\lambda_{f}\left( {p + n,q + n} \right) = 1\quad\text{for}\,\, n = 1,2,\ldots.} \\
\end{matrix}$$

An entire function for which (*p*, *q*)th order and (*p*, *q*)th lower order are the same is said to be of regular (*p*, *q*)-growth. Functions which are not of regular (*p*, *q*)-growth are said to be of irregular (*p*, *q*)-growth.

Bernal \[[@B1]\] introduced the definition of relative order of *f* with respect to *g*, denoted by *ρ* ~*g*~(*f*) as follows: $$\begin{matrix}
{\rho_{g}\left( f \right)} \\
{\quad = \inf\left\{ {\mu > 0:M_{f}\left\{ r \right\} < M_{g}\left\{ {r\mu} \right\}\,\,\forall r > r_{0}\left\{ \mu \right\} > 0} \right\}} \\
{\quad = \lim\underset{r\rightarrow\infty}{\sup}\frac{\log M_{g}^{- 1}M_{f}\left( r \right)}{\log r}.} \\
\end{matrix}$$

The definition coincides with the classical one \[[@B10]\] if *g* = exp⁡.

Similarly one can define the relative lower order of *f* with respect to *g* denoted by *λ* ~*g*~(*f*) as follows: $$\begin{matrix}
{\lambda_{g}\left( f \right) = \lim\underset{r\rightarrow\infty}{\inf}\frac{\log M_{g}^{- 1}M_{f}\left( r \right)}{\log r}.} \\
\end{matrix}$$

In the case of relative order, it therefore seems reasonable to define suitably the relative (*p*, *q*)th order of entire functions. Lahiri and Banerjee \[[@B4]\] also introduced such definition in the following manner.

Definition 3 (see \[[@B4]\]).Let *p* and *q* be any two positive integers with *p* \> *q*. The relative (*p*, *q*)th order of *f* with respect to *g* is defined by $$\begin{matrix}
{\rho_{g}^{({p,q})}\left( f \right)} \\
{\quad = \inf\left\{ {\mu > 0:M_{f}\left\{ r \right. < M_{g}\left\{ {{\exp}^{\lbrack{p - 1}\rbrack}\left\{ {\mu{\log}^{\lbrack q\rbrack}r} \right.} \right.} \right.} \\
{\quad\quad\quad\quad\left. {\forall r > r_{0}\left. \mu \right\} > 0} \right\}} \\
{\quad = \lim\underset{r\rightarrow\infty}{\sup}\frac{{\log}^{\lbrack{p - 1}\rbrack}M_{g}^{- 1}M_{f}\left( r \right)}{{\log}^{\lbrack q\rbrack}r}.} \\
\end{matrix}$$ If *q* = 1, *k* ≥ 1, and *p* = *k* + 1 then *ρ* ~*g*~ ^(*p*,\ *q*)^(*f*) = *ρ* ~*g*~ ^*k*^(*f*). If *g* = exp⁡*z* then *ρ* ~*g*~ ^(*p*,\ *q*)^(*f*) = *ρ* ~*f*~(*p*, *q*).

Sánchez Ruiz et al. \[[@B7]\] gave a more natural definition of relative (*p*, *q*)th order of an entire function in light of index-pair which is as follows.

Definition 4 .Let *f* and *g* be any two entire functions with index-pairs (*m* ~1~, *q*) and (*m* ~2~, *p*), respectively, where *m* ~1~ = *m* ~2~ = *m* and *p*, *q*, and *m* are all positive integers such that *m* ≥ *p* and *m* ≥ *q*. Then the relative (*p*, *q*)th order of *f* with respect to *g* is defined as $$\begin{matrix}
{\rho_{g}^{({p,q})}\left( f \right)} \\
{  = \inf\left\{ {\mu > 0:M_{f}\left\{ r \right. < M_{g}} \right.} \\
{\quad\quad    \times \left\lbrack {{\exp}^{\lbrack p\rbrack}\left\{ {{\log}^{\lbrack m_{2}\rbrack}{\exp}^{\lbrack m_{1}\rbrack}\left\{ {\mu{\log}^{\lbrack q\rbrack}r} \right\}} \right\}} \right\rbrack} \\
{\quad\quad   \left. {\forall r > r_{0}\left. \mu \right\} > 0} \right\}} \\
{  = \inf\left\{ {\mu > 0:M_{f}\left\{ r \right. < M_{g}\left\{ {{\exp}^{\lbrack p\rbrack}\left\{ {\mu{\log}^{\lbrack q\rbrack}r} \right.} \right.} \right.} \\
{\quad\quad\quad\quad\left. {\forall r > r_{0}\left. \mu \right\} > 0} \right\}} \\
{  = \lim\underset{r\rightarrow\infty}{\sup}\frac{{\log}^{\lbrack p\rbrack}M_{g}^{- 1}M_{f}\left( r \right)}{{\log}^{\lbrack q\rbrack}r}.} \\
\end{matrix}$$Similarly one can define the relative (*p*, *q*)th lower order of an entire function *f* with respect to another entire function *g* denoted by *λ* ~*g*~ ^(*p*,\ *q*)^(*f*) where *p* and *q* are any two positive integers in the following way: $$\begin{matrix}
{\lambda_{g}^{({p,q})}\left( f \right) = \lim\underset{r\rightarrow\infty}{\inf}\frac{{\log}^{\lbrack p\rbrack}M_{g}^{- 1}M_{f}\left( r \right)}{{\log}^{\lbrack q\rbrack}r}.} \\
\end{matrix}$$

In fact [Definition 4](#deff4){ref-type="statement"} improves [Definition 3](#deff3){ref-type="statement"} ignoring the restriction *p* ≥ *q*.

In this paper we wish to prove some results related to the growth rates of entire functions on the basis of relative (*p*, *q*)th order and relative (*p*, *q*)th lower order with respect to another entire function extending some earlier results for any two positive integers *p* and *q*.

2. Lemmas {#sec2}
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In this section we present some lemmas which will be needed in the sequel.

Lemma 1 (see \[[@B9]\]).If *f* and *g* are any two entire functions with *g*(0) = 0. then $$\begin{matrix}
{M_{f \circ g}\left( r \right) \geq M_{g}\left( \frac{r}{2} \right)} \\
{for\,\, all\,\, sufficiently\,\, large\,\, values\,\, of\text{  }r \geqslant r_{0}.} \\
\end{matrix}$$

Lemma 2 (see \[[@B9]\]).Let *f* be entire and let *g* be a transcendental entire function of finite lower order. Then, for any *δ* \> 0, $$\begin{matrix}
{M_{f \circ g}\left( r^{1 + \delta} \right) \geq M_{f}\left( {M_{g}\left( r \right)} \right)} \\
{for\,\, all\,\, sufficiently\,\, large\,\, values\,\, of\text{  }r \geqslant r_{0}.} \\
\end{matrix}$$

Lemma 3 (see \[[@B5]\]).If *f* and *g* are any two entire functions with *g*(0) = 0. then, for any 0 \< *c* \< 1, $$\begin{matrix}
{M_{f \circ g}\left( r \right) \geq M_{f}\left( {cM_{g}\left( \frac{r}{2} \right)} \right)} \\
{for\,\, all\,\, sufficiently\,\, large\,\, values\,\, of\,\, r \geqslant r_{0},} \\
\end{matrix}$$

Lemma 4 (see \[[@B2]\]).If *f* and *g* are any two entire functions then for all sufficiently large values of *r*⩾*r* ~0~ $$\begin{matrix}
{M_{f \circ g}\left( r \right) \leq M_{f}\left( {M_{g}\left( r \right)} \right).} \\
\end{matrix}$$

3. Theorems {#sec3}
===========

In this section we present the main results of the paper.

Theorem 5 .Let *f* be an entire function and let *g* be any polynomial such that *f*∘*g* has got finite relative (*p*, *q*)th order with respect to *h* where *h* is a transcendental entire function and *p*, *q* are any two positive integers. Then *ρ* ~*h*~ ^(*p*,\ *q*)^(*f*) \< *∞*.

ProofGiven that *f*∘*g* is of finite relative (*p*, *q*)th order with respect to *h*, we have from [Definition 4](#deff4){ref-type="statement"}, for a suitable finite number *μ* \> 0 and for all sufficiently large values of *r*, that $$\begin{matrix}
{M_{f \circ g}\left( r \right) < M_{h}\left( {{\exp}^{\lbrack p\rbrack}\left( {\mu{\log}^{\lbrack q\rbrack}r} \right)} \right).} \\
\end{matrix}$$ Now let *m* be the order of the polynomial *g* so that $$\begin{matrix}
{g\left( z \right) = c_{1}z + c_{2}z^{2} + \cdots + c_{m}z^{m},\quad c_{m} \neq 0.} \\
\end{matrix}$$ Then by Cauchy\'s inequality we get from ([18](#EEq2){ref-type="disp-formula"}) that $$\begin{matrix}
{\left| c_{m} \right|r^{m} \leq M_{g}\left( r \right),\quad\quad\left| z \right| = r.} \\
\end{matrix}$$ Now given 0 \< *c* \< 1, in view of [Lemma 3](#lem3){ref-type="statement"} and from ([17](#EEq1){ref-type="disp-formula"}) it follows for all sufficiently large values of *r* that $$\begin{matrix}
{M_{f}\left( {c\left| c_{m} \right|\left( \frac{r}{2} \right)^{m}} \right) \leq M_{f \circ g}\left( r \right) \leq M_{h}\left( {{\exp}^{\lbrack p\rbrack}\left( {\mu{\log}^{\lbrack q\rbrack}r} \right)} \right).} \\
\end{matrix}$$ We rewrite the above to the equivalent for all sufficiently large values of *r* that $$\begin{matrix}
{M_{f}\left( r \right) \leq M_{h}\left( {{\exp}^{\lbrack p\rbrack}\left( {\mu{\log}^{\lbrack q\rbrack}\left( {\left( {c\left| c_{m} \right|} \right)^{- 1}2^{m}r^{1/m}} \right)} \right)} \right).} \\
\end{matrix}$$ Therefore from ([21](#EEq5){ref-type="disp-formula"}) we get for all sufficiently large values of *r* that $$\begin{matrix}
{M_{h}^{- 1}M_{f}\left( r \right) \leq {\exp}^{\lbrack p\rbrack}\left( {\mu{\log}^{\lbrack q\rbrack}\left( {\left( {c\left| c_{m} \right|} \right)^{- 1}2^{m}r^{1/m}} \right)} \right),} \\
{\text{i.e.},\,\,{\log}^{\lbrack p\rbrack}M_{h}^{- 1}M_{f}\left( r \right) \leq \mu{\log}^{\lbrack q\rbrack}\left( {\left( {c\left| c_{m} \right|} \right)^{- 1}2^{m}r^{1/m}} \right).} \\
\end{matrix}$$*Case*I. Assume *q* = 1. Then we have from ([22](#EEq6){ref-type="disp-formula"}) for all sufficiently large values of *r* that $$\begin{matrix}
{\frac{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}M_{f}\left( r \right)}{\log r} \leq \frac{\mu}{m}\frac{\log r + \left| {O\left| 1 \right|} \right|}{\log r},} \\
\end{matrix}$$ where *O*(1) stands for the constant expression, *m*log⁡((*c*\|*c* ~*m*~\|)^−1^2^*m*^). Then $$\begin{matrix}
{\lim\underset{r\rightarrow\infty}{\sup}\frac{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}M_{f}\left( r \right)}{\log r} \leq \frac{\mu}{m}\lim\underset{r\rightarrow\infty}{\sup}\frac{\log r + \left| {O\left| 1 \right|} \right|}{\log r},} \\
{\text{i.e.},\,\,\rho_{h}^{p}\left( f \right) \leq \frac{\mu}{m} < \infty.} \\
\end{matrix}$$*Case*II. Let us now assume *q* \> 1. Then we obtain from ([22](#EEq6){ref-type="disp-formula"}) for all sufficiently large values of *r* that $$\begin{matrix}
{\frac{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}M_{f}\left( r \right)}{{\log}^{\lbrack q\rbrack}r} \leq \mu{\,\,}\frac{{\log}^{\lbrack q\rbrack}r + \left| {O\left| 1 \right|} \right|}{{\log}^{\lbrack q\rbrack}r},} \\
\end{matrix}$$ where *O*(1) stands for a bounded quantity. Then $$\begin{matrix}
{\lim\underset{r\rightarrow\infty}{\sup}\frac{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}M_{f}\left( r \right)}{{\log}^{\lbrack q\rbrack}r} \leq \mu{\,\,}\lim\underset{r\rightarrow\infty}{\sup}\frac{{\log}^{\lbrack q\rbrack}r + \left| {O\left| 1 \right|} \right|}{{\log}^{\lbrack q\rbrack}r},} \\
{\text{i.e.},\,\,\rho_{h}^{({p,q})}\left( f \right) \leq \mu < \infty.} \\
\end{matrix}$$ Thus the theorem follows from ([24](#EEq7){ref-type="disp-formula"}) and ([26](#EEq8){ref-type="disp-formula"}).

In the forthcoming proofs we will assume the natural number *q* to be *q* \> 1, the reasonings being easily adapted for *q* = 1.

Theorem 6 .Let *f*, *g*, and *h* be any three transcendental entire functions and let *p* and *q* be two positive integers. If, for any *α*, *β* with 0 \< *α* \< 1, *β* \> 0, and *α*(*β* + 1) \> 1, it holds that the two limits *A*, *B* ∈ *R* ^+^ of some of eitherlim⁡sup⁡~*r*→*∞*~(log⁡^\[*p*\]^ *M* ~*h*~ ^−1^(*M* ~*g*~(*r*))/(log⁡^\[*q*\]^ *r*)^*α*^) = *A*, lim⁡inf⁡~*r*→*∞*~(log⁡^\[*p*\]^ *M* ~*h*~ ^−1^(*M* ~*f*~(*r*))/(log⁡^\[*p*\]^ *M* ~*h*~ ^−1^(*r*))^*β*+1^) = *B*,lim⁡inf⁡~*r*→*∞*~(log⁡^\[*p*\]^ *M* ~*h*~ ^−1^(*M* ~*g*~(*r*))/(log⁡^\[*q*\]^ *r*)^*α*^) = *A*, lim⁡sup⁡~*r*→*∞*~(log⁡^\[*p*\]^ *M* ~*h*~ ^−1^(*M* ~*f*~(*r*))/(log⁡^\[*p*\]^ *M* ~*h*~ ^−1^(*r*))^*β*+1^) = *B*, orlim⁡inf⁡~*r*→*∞*~(log⁡^\[*p*\]^ *M* ~*h*~ ^−1^(*M* ~*g*~(*r*))/(log⁡^\[*q*\]^ *r*)^*α*^) = *A*, lim⁡inf⁡~*r*→*∞*~(log⁡^\[*p*\]^ *M* ~*h*~ ^−1^(*M* ~*f*~(*r*))/(log⁡^\[*p*\]^ *M* ~*h*~ ^−1^(*r*))^*β*+1^) = *B* exist, then *ρ* ~*h*~ ^(*p*,\ *q*)^(*f*∘*g*) = *∞*.

Proof(i) The existence of *A* and *B* implies that given any *ɛ* \> 0, for sufficiently large values of *r*, $$\begin{matrix}
{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left( {M_{g}\left( r \right)} \right) \geq \left( {A - ɛ} \right)\left( {{\log}^{\lbrack q\rbrack}r} \right)^{\alpha},} \\
{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left( {M_{f}\left( r \right)} \right) \geq \left( {B - ɛ} \right)\left( {{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left( r \right)} \right)^{\beta + 1}.} \\
\end{matrix}$$ Since *M* ~*g*~(*r*) is a continuous, increasing, and unbounded function of *r*, we get from above for all sufficiently large values of *r* that $$\begin{matrix}
{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left( {M_{f}\left( M_{g}\left( r \right) \right)} \right)} \\
{\quad\quad \geq \left( {B - ɛ} \right)\left( {{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left( {M_{g}\left( r \right)} \right)} \right)^{\beta + 1}.} \\
\end{matrix}$$ Also *M* ~*h*~ ^−1^(*r*) is an increasing function of *r*; it follows from [Lemma 2](#lem2){ref-type="statement"}, ([27](#EEq9){ref-type="disp-formula"}), and ([28](#EEq10){ref-type="disp-formula"}) that given *δ* \> 0, for a sequence of values of *r* tending to infinity, the following holds: $$\begin{matrix}
{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}M_{f \circ g}\left( r^{1 + \delta} \right)} \\
{\quad\quad \geq {\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left( {M_{f}\left( {M_{g}\left( r \right)} \right)} \right)} \\
{\quad\quad \geq \left( {B - ɛ} \right)\left( {{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left( {M_{g}\left( r \right)} \right)} \right)^{\beta + 1}} \\
{\quad\quad \geq \left( {B - ɛ} \right)\left\lbrack {\left\lbrack {A - ɛ} \right\rbrack\left\lbrack {{\log}^{\lbrack q\rbrack}r} \right\rbrack\alpha} \right\rbrack^{\beta + 1},} \\
{\text{i.e.},\,\,{\,\,}\frac{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}M_{f \circ g}\left( r^{1 + \delta} \right)}{{\log}^{\lbrack q\rbrack}\left( r^{1 + \delta} \right)}} \\
{\quad\quad \geq \frac{\left( {B - ɛ} \right)\left( {A - ɛ} \right)^{\beta + 1}\left( {{\log}^{\lbrack q\rbrack}r} \right)^{\alpha{({\beta + 1})}}}{{\log}^{\lbrack q\rbrack}\left( r^{1 + \delta} \right)}.} \\
\end{matrix}$$ Hence $$\begin{matrix}
{\lim\underset{r\rightarrow\infty}{\sup}\frac{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}M_{f \circ g}\left( r^{1 + \delta} \right)}{{\log}^{\lbrack q\rbrack}\left( r^{1 + \delta} \right)}} \\
{\quad\quad \geq \lim\underset{r\rightarrow\infty}{\inf}\frac{\left( {B - ɛ} \right)\left( {A - ɛ} \right)^{\beta + 1}\left( {{\log}^{\lbrack q\rbrack}r} \right)^{\alpha{({\beta + 1})}}}{{\log}^{\lbrack q\rbrack}r + \left| {O\left| 1 \right|} \right|}} \\
\end{matrix}$$ for all sufficiently large values of *r*. Since *ɛ* \> 0 is arbitrary and *α*(*β* + 1) \> 1 it follows that $$\begin{matrix}
{\rho_{h}^{({p,q})}\left( {f \circ g} \right) = \infty.} \\
\end{matrix}$$ Under (ii) or (iii) a similar argument applies.

Theorem 7 .Let *f*, *g*, and *h* be any three transcendental entire functions and let *p* and *q* be two positive integers. If, for any *α*, *β* with *α* \> 1, 0 \< *β* \< 1, and *αβ* \> 1, it holds that the two limits *A*, *B* ∈ *R* ^+^ of eitherlim⁡sup⁡~*r*→*∞*~(log⁡^\[*p*\]^ *M* ~*h*~ ^−1^(*M* ~*g*~(*r*))/(log⁡^\[*q*\ +\ 1\]^ *r*)^*α*^) = *A*, lim⁡inf⁡~*r*→*∞*~(log⁡\[log⁡^\[*p*\]^ *M* ~*h*~ ^−1^(*M* ~*f*~(*r*))/log⁡^\[*p*\]^ *M* ~*h*~ ^−1^(*r*)\]/\[log⁡^\[*p*\]^ *M* ~*h*~ ^−1^(*r*)\]^*β*^) = *B*,lim⁡inf⁡~*r*→*∞*~(log⁡^\[*p*\]^ *M* ~*h*~ ^−1^(*M* ~*g*~(*r*))/(log⁡^\[*q*\ +\ 1\]^ *r*)^*α*^) = *A*, lim⁡sup⁡~*r*→*∞*~(log⁡\[log⁡^\[*p*\]^ *M* ~*h*~ ^−1^(*M* ~*f*~(*r*))/log⁡^\[*p*\]^ *M* ~*h*~ ^−1^(*r*)\]/\[log⁡^\[*p*\]^ *M* ~*h*~ ^−1^(*r*)\]^*β*^) = *B*, orlim⁡inf⁡~*r*→*∞*~(log⁡^\[*p*\]^ *M* ~*h*~ ^−1^(*M* ~*g*~(*r*))/(log⁡^\[*q*\ +\ 1\]^ *r*)^*α*^) = *A*, lim⁡inf⁡~*r*→*∞*~(log⁡\[log⁡^\[*p*\]^ *M* ~*h*~ ^−1^(*M* ~*f*~(*r*))/log⁡^\[*p*\]^ *M* ~*h*~ ^−1^(*r*)\]/\[log⁡^\[*p*\]^ *M* ~*h*~ ^−1^(*r*)\]^*β*^) = *B*exist, then *ρ* ~*h*~ ^(*p*,\ *q*)^(*f*∘*g*) = *∞*.

Proof(i) Given any *ɛ* \> 0, for a sequence of values of *r* tending to infinity, we get that $$\begin{matrix}
{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left( {M_{g}\left( r \right)} \right) \geq \left( {A - ɛ} \right)\left( {{\log}^{\lbrack{q + 1}\rbrack}r} \right)^{\alpha}} \\
\end{matrix}$$ and for all sufficiently large values of *r* that $$\begin{matrix}
{\log\left\lbrack \frac{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left\lbrack {M_{f}\left( r \right)} \right\rbrack}{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left\lbrack r \right\rbrack} \right\rbrack \geq \left( {B - ɛ} \right)\left\lbrack {{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left\lbrack r \right\rbrack} \right\rbrack^{\beta},} \\
{\text{i.e.},\,\,\frac{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left( {M_{f}\left( r \right)} \right)}{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left( r \right)} \geq \exp\left\lbrack {\left\lbrack {B - ɛ} \right\rbrack\left\lbrack {{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left\lbrack r \right\rbrack} \right\rbrack^{\beta}} \right\rbrack.} \\
\end{matrix}$$ Since *M* ~*g*~(*r*) is a continuous, increasing, and unbounded function of *r*, we get from above for all sufficiently large values of *r* that $$\begin{matrix}
\frac{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left( {M_{f}\left( M_{g}\left( r \right) \right)} \right)}{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left( {M_{g}\left( r \right)} \right)} \\
{\quad\quad \geq \exp\left\lbrack {\left\lbrack {B - ɛ} \right\rbrack\left\lbrack {{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left\lbrack {M_{g}\left\lbrack r \right\rbrack} \right\rbrack} \right\rbrack^{\beta}} \right\rbrack.} \\
\end{matrix}$$ Also *M* ~*h*~ ^−1^(*r*) is an increasing function of *r*; thus from [Lemma 2](#lem2){ref-type="statement"}, ([32](#EEq11){ref-type="disp-formula"}), and ([34](#EEq12){ref-type="disp-formula"}) it follows that, given that *δ* \> 0, for a sequence of values of *r* tending to infinity, $$\begin{matrix}
\frac{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}M_{f \circ g}\left( r^{1 + \delta} \right)}{{\log}^{\lbrack q\rbrack}\left( r^{1 + \delta} \right)} \\
{\quad \geq \frac{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left( {M_{f}\left( {M_{g}\left( r \right)} \right)} \right)}{{\log}^{\lbrack q\rbrack}\left( r^{1 + \delta} \right)}} \\
{\quad \geq \frac{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left( {M_{f}\left( {M_{g}\left( r \right)} \right)} \right)}{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left( {M_{g}\left( r \right)} \right)} \cdot \frac{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left( {M_{g}\left( r \right)} \right)}{{\log}^{\lbrack q\rbrack}r + \left| {O\left| 1 \right|} \right|}.} \\
\end{matrix}$$ Therefore $$\begin{matrix}
\frac{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}M_{f \circ g}\left( r^{1 + \delta} \right)}{{\log}^{\lbrack q\rbrack}\left( r^{1 + \delta} \right)} \\
{\quad \geq \,\exp\left\lbrack {\left\lbrack {B - ɛ} \right\rbrack\left\lbrack {{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left\lbrack {M_{g}\left\lbrack r \right\rbrack} \right\rbrack} \right\rbrack^{\beta}} \right\rbrack} \\
{\quad\quad \cdot \frac{\left( {A - ɛ} \right)\left( {{\log}^{\lbrack{q + 1}\rbrack}r} \right)^{\alpha}}{{\log}^{\lbrack q\rbrack}r + \left| {O\left| 1 \right|} \right|}} \\
{\quad \geq \exp\left\lbrack {\left\lbrack {B - ɛ} \right\rbrack\left\lbrack {A - ɛ} \right\rbrack^{\beta}\left\lbrack {{\log}^{\lbrack{q + 1}\rbrack}r} \right\rbrack^{\alpha\beta}} \right\rbrack} \\
{\quad\quad \cdot \frac{\left( {A - ɛ} \right)\left( {{\log}^{\lbrack{q + 1}\rbrack}r} \right)^{\alpha}}{{\log}^{\lbrack q\rbrack}r + \left| {O\left| 1 \right|} \right|}} \\
{\quad = \exp\left\lbrack {\left\lbrack {B - ɛ} \right\rbrack\left\lbrack {A - ɛ} \right\rbrack^{\beta}\left\lbrack {{\log}^{\lbrack{q + 1}\rbrack}r} \right\rbrack^{\alpha\beta - 1}{\log}^{\lbrack{q + 1}\rbrack}r} \right\rbrack} \\
{\quad\quad \cdot \frac{\left( {A - ɛ} \right)\left( {{\log}^{\lbrack{q + 1}\rbrack}r} \right)^{\alpha}}{{\log}^{\lbrack q\rbrack}r + \left| {O\left| 1 \right|} \right|}} \\
{\quad \geq \left( {{\log}^{\lbrack q\rbrack}r} \right)^{{({B - ɛ})}{({A - ɛ})}\beta{({{\log}^{\lbrack{q + 1}\rbrack}r})}^{\alpha\beta - 1}}} \\
{\quad\quad \cdot \frac{\left( {A - ɛ} \right)\left( {{\log}^{\lbrack{q + 1}\rbrack}r} \right)^{\alpha}}{{\log}^{\lbrack q\rbrack}r + \left| {O\left| 1 \right|} \right|}.} \\
\end{matrix}$$ Hence $$\begin{matrix}
{\lim\underset{r\rightarrow\infty}{\sup}\frac{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}M_{f \circ g}\left( r^{1 + \delta} \right)}{{\log}^{\lbrack q\rbrack}\left( r^{1 + \delta} \right)}} \\
{\quad\quad \geq \,\lim\underset{r\rightarrow\infty}{\inf}\left( {{\log}^{\lbrack q\rbrack}r} \right)^{{({B - ɛ})}{({A - ɛ})}\beta{({{\log}^{\lbrack{q + 1}\rbrack}r})}^{\alpha\beta - 1}}} \\
{\quad\quad\quad \cdot \frac{\left( {A - ɛ} \right)\left( {{\log}^{\lbrack{q + 1}\rbrack}r} \right)^{\alpha}}{{\log}^{\lbrack q\rbrack}r + \left| {O\left| 1 \right|} \right|}.} \\
\end{matrix}$$Since *ɛ* \> 0 is arbitrary and *α* \> 1, *αβ* \> 1, it follows that $$\begin{matrix}
{\rho_{h}^{({p,q})}\left( {f \circ g} \right) = \infty.} \\
\end{matrix}$$ Under (ii) or (iii) a similar argument may be used.

Theorem 8 .Let *f*, *g*, and *h* be any three transcendental entire functions such that 0 \< *λ* ~*h*~ ^(*p*,\ *q*)^  (*g*) ≤ *ρ* ~*h*~ ^(*p*,\ *q*)^(*g*) \< *∞* where *p* and *q* are any two positive integers. If the limit *A* ∈ *R* exists in eitherlim⁡sup⁡~*r*→*∞*~(log⁡^\[*p*\]^ *M* ~*h*~ ^−1^(*M* ~*f*~(*r*))/log⁡^\[*p*\]^ *M* ~*h*~ ^−1^(*r*)) = *A* orlim⁡inf⁡~*r*→*∞*~(log⁡^\[*p*\]^ *M* ~*h*~ ^−1^(*M* ~*f*~(*r*))/log⁡^\[*p*\]^ *M* ~*h*~ ^−1^(*r*)) = *A*, then $$\begin{matrix}
{\lambda_{h}^{({p,q})}\left( {f \circ g} \right) \leq A\lambda_{h}^{({p,q})}\left( g \right) \leq \rho_{h}^{({p,q})}\left( {f \circ g} \right) \leq A\rho_{h}^{({p,q})}\left( g \right).} \\
\end{matrix}$$

Proof(i) Since *M* ~*h*~ ^−1^(*r*) is an increasing function of *r*, it follows from Lemmas [2](#lem2){ref-type="statement"} and [4](#lem4){ref-type="statement"}, given *δ* \> 0, for all sufficiently large values of *r*, that $$\begin{matrix}
{M_{h}^{- 1}M_{f \circ g}\left( r^{1 + \delta} \right) \geq M_{h}^{- 1}\left\{ {M_{f}\left\{ {M_{g}\left\{ r \right\}} \right\}} \right\},} \\
\end{matrix}$$ $$\begin{matrix}
{M_{h}^{- 1}M_{f \circ g}\left( r \right) \leq M_{h}^{- 1}\left\{ {M_{f}\left\{ {M_{g}\left\{ r \right\}} \right\}} \right\},} \\
\end{matrix}$$ respectively.Therefore from ([40](#EEq13){ref-type="disp-formula"}) we get for all sufficiently large values of *r* that $$\begin{matrix}
{\frac{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}M_{f \circ g}\left( r^{1 + \delta} \right)}{{\log}^{\lbrack q\rbrack}r^{1 + \delta}} \geq \frac{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left\{ {M_{f}\left\{ {M_{g}\left\{ r \right\}} \right\}} \right\}}{{\log}^{\lbrack q\rbrack}r^{1 + \delta}}} \\
{\geq \frac{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left\{ {M_{f}\left\{ {M_{g}\left\{ r \right\}} \right\}} \right\}}{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left( {M_{g}\left( r \right)} \right)}} \\
{\quad \cdot \frac{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left( {M_{g}\left( r \right)} \right)}{{\log}^{\lbrack q\rbrack}r + \left| {O\left| 1 \right|} \right|}.} \\
\end{matrix}$$ From here it follows that $$\begin{matrix}
{\lim\underset{r\rightarrow\infty}{\sup}\frac{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}M_{f \circ g}\left( r \right)}{{\log}^{\lbrack q\rbrack}r}} \\
{\quad \geq \,\lim\underset{r\rightarrow\infty}{\sup}\frac{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left\{ {M_{f}\left\{ {M_{g}\left\{ r \right\}} \right\}} \right\}}{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left( {M_{g}\left( r \right)} \right)}} \\
{\quad\quad \times \lim\underset{r\rightarrow\infty}{\inf}\frac{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left( {M_{g}\left( r \right)} \right)}{{\log}^{\lbrack q\rbrack}r + \left| {O\left| 1 \right|} \right|}} \\
{\text{i.e.},\,\,\rho_{h}^{({p,q})}\left( {f \circ g} \right) \geq A\lambda_{h}^{({p,q})}\left( g \right).} \\
\end{matrix}$$ Similarly from ([41](#EEq14){ref-type="disp-formula"}) it follows for all sufficiently large values of *r* that $$\begin{matrix}
{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}M_{f \circ g}\left( r \right) \leq {\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left( {M_{f}\left( {M_{g}\left( r \right)} \right)} \right).} \\
\end{matrix}$$ Therefore $$\begin{matrix}
\frac{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}M_{f \circ g}\left( r \right)}{{\log}^{\lbrack q\rbrack}r} \\
{\quad \leq \frac{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left( {M_{f}\left( {M_{g}\left( r \right)} \right)} \right)}{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left( {M_{g}\left( r \right)} \right)} \cdot \frac{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left( {M_{g}\left( r \right)} \right)}{{\log}^{\lbrack q\rbrack}r}.} \\
\end{matrix}$$ Hence $$\begin{matrix}
{\lim{\underset{r\rightarrow\infty}{\inf}\frac{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}M_{f \circ g}\left( r \right)}{{\log}^{\lbrack q\rbrack}r}}} \\
{\quad \leq \,\lim\underset{r\rightarrow\infty}{\sup}\frac{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left\{ {M_{f}\left\{ {M_{g}\left\{ r \right\}} \right\}} \right\}}{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left( {M_{g}\left( r \right)} \right)}} \\
{\quad\quad \times \lim\underset{r\rightarrow\infty}{\inf}\frac{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left( {M_{g}\left( r \right)} \right)}{{\log}^{\lbrack q\rbrack}r},} \\
{\text{i.e.},\,\,\lambda_{h}^{({p,q})}\left( {f \circ g} \right) \leq A\lambda_{h}^{({p,q})}\left( g \right).} \\
\end{matrix}$$ Also from ([45](#EEq16){ref-type="disp-formula"}) we obtain for all sufficiently large values of *r* that $$\begin{matrix}
{\lim\underset{r\rightarrow\infty}{\sup}\frac{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}M_{f \circ g}\left( r \right)}{{\log}^{\lbrack q\rbrack}r}} \\
{\quad\quad \leq \lim\underset{r\rightarrow\infty}{\sup}\frac{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left\{ {M_{f}\left\{ {M_{g}\left\{ r \right\}} \right\}} \right\}}{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left( {M_{g}\left( r \right)} \right)}} \\
{\quad\quad\quad \times \lim\underset{r\rightarrow\infty}{\sup}\frac{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left( {M_{g}\left( r \right)} \right)}{{\log}^{\lbrack q\rbrack}r},} \\
{\text{i.e.},\,\,\rho_{h}^{({p,q})}\left( {f \circ g} \right) \leq A \cdot \rho_{h}^{({p,q})}\left( g \right).} \\
\end{matrix}$$ Then the thesis follows from ([43](#EEq15){ref-type="disp-formula"}), ([46](#EEq17){ref-type="disp-formula"}), and ([47](#EEq18){ref-type="disp-formula"}).(ii) follows with a similar argument.

Theorem 9 .Let *f*, *g*, and *h* be any three transcendental entire functions with *g*(0) = 0. If *p*, *q*, and *m* are any three positive integers with *m* \> *q*, then *ρ* ~*h*~ ^(p,\ *q*)^(*f*∘*g*) = *∞* under any of the following conditions: *ρ* ~*h*~ ^(*p*,\ *q*)^(*g*) = *∞*;min⁡(*ρ* ~*h*~ ^(*p*,\ *q*)^(*f*), *λ* ~*g*~(*m*, *q*)) \> 0;min⁡(*ρ* ~*g*~(*m*, *q*), *λ* ~*h*~ ^(*p*,\ *q*)^(*f*)) \> 0.

Proof(i) If *ρ* ~*h*~ ^(*p*,\ *q*)^(*g*) = *∞*, since *M* ~*h*~ ^−1^(*r*) is an increasing function of *r*, it follows from [Lemma 1](#lem1){ref-type="statement"}, for all sufficiently large values of *r*, that $$\begin{matrix}
{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left( {M_{f \circ g}\left( r \right)} \right) \geq {\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left( {M_{g}\left( \frac{r}{2} \right)} \right).} \\
\end{matrix}$$ Therefore $$\begin{matrix}
{\frac{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left( {M_{f \circ g}\left( r \right)} \right)}{{\log}^{\lbrack q\rbrack}r} \geq \frac{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left( {M_{g}\left( {r/2} \right)} \right)}{{\log}^{\lbrack q\rbrack}r}} \\
{\geq \frac{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left( {M_{g}\left( {r/2} \right)} \right)}{{\log}^{\lbrack q\rbrack}\left( {r/2} \right) + \left| {O\left| 1 \right|} \right|}.} \\
\end{matrix}$$ Then $$\begin{matrix}
{\lim\underset{r\rightarrow\infty}{\sup}\frac{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left( {M_{f \circ g}\left( r \right)} \right)}{{\log}^{\lbrack q\rbrack}}} \\
{\quad\quad \geq \lim\underset{r\rightarrow\infty}{\sup}\frac{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left( {M_{g}\left( {r/2} \right)} \right)}{{\log}^{\lbrack q\rbrack}\left( {r/2} \right) + \left| {O\left| 1 \right|} \right|},} \\
{\text{i.e.},\,\,\rho_{h}^{({p,q})}\left( {f \circ g} \right) \geq \rho_{h}^{({p,q})}\left( g \right) = \infty.} \\
\end{matrix}$$(ii) Suppose *ρ* ~*h*~ ^(*p*,\ *q*)^(*f*) \> 0 and *λ* ~*g*~(*m*, *q*) \> 0.As *M* ~*h*~ ^−1^(*r*) is an increasing function of *r*, we get from [Lemma 2](#lem2){ref-type="statement"} that given *δ* \> 0 and any *ɛ* \> 0, for all sufficiently large values of *r*, $$\begin{matrix}
{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}M_{f \circ g}\left( r^{1 + \delta} \right)} \\
{\quad\quad \geq {\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left( {M_{f}\left( {M_{g}\left( r \right)} \right)} \right)} \\
{\quad\quad \geq \left( {\rho_{h}^{({p,q})}\left( f \right) - ɛ} \right){\log}^{\lbrack q\rbrack}M_{g}\left( r \right)} \\
{\quad\quad \geq \left( {\rho_{h}^{({p,q})}\left( f \right) - ɛ} \right){\exp}^{\lbrack{m - q - 1}\rbrack}\left( {{\log}^{\lbrack{q - 1}\rbrack}r} \right)^{({\lambda_{g}{({m,q})} - ɛ})}.} \\
\end{matrix}$$ Thus $$\begin{matrix}
\frac{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}M_{f \circ g}\left( r^{1 + \delta} \right)}{{\log}^{\lbrack q\rbrack}r^{1 + \delta}} \\
{\quad \geq \frac{\left( {\rho_{h}^{({p,q})}\left( f \right) - ɛ} \right){\exp}^{\lbrack{m - q - 1}\rbrack}\left( {{\log}^{\lbrack{q - 1}\rbrack}r} \right)^{({\lambda_{g}{({m,q})} - ɛ})}}{{\log}^{\lbrack q\rbrack}r^{1 + \delta}}.} \\
\end{matrix}$$ Hence $$\begin{matrix}
{\lim\underset{r\rightarrow\infty}{\sup}\frac{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}M_{f \circ g}\left( r^{1 + \delta} \right)}{{\log}^{\lbrack q\rbrack}r^{1 + \delta}}} \\
{\geq \,\lim\underset{r\rightarrow\infty}{\inf}\frac{\left( {\rho_{h}^{({p,q})}\left( f \right) - ɛ} \right){\exp}^{\lbrack{m - q - 1}\rbrack}\left( {{\log}^{\lbrack{q - 1}\rbrack}r} \right)^{({\lambda_{g}{({m,q})} - ɛ})}}{{\log}^{\lbrack q\rbrack}r + \left| {O\left( 1 \right)} \right|},} \\
{\text{i.e.},\,\,\rho_{h}^{({p,q})}\left( {f \circ g} \right) = \infty.} \\
\end{matrix}$$ Under (iii) a similar argument to (i) applies.

In the line of [Theorem 9](#thm9){ref-type="statement"} one can easily prove the following result.

Theorem 10 .Let *f*, *g*, and *h* be any three transcendental entire functions with *g*(0) = 0. If *p*, *q*, and *m* are any three positive integers with *m* \> *q*, then *λ* ~*h*~ ^(*p*,\ *q*)^(*f*∘*g*) = *∞* if any of the following facts happens:*λ* ~*h*~ ^(*p*,\ *q*)^(*g*) = *∞*;min⁡(*λ* ~*h*~ ^(*p*,\ *q*)^(*f*), *λ* ~*g*~(*m*, *q*)) \> 0.

Theorem 11 .Let *f*, *g*, and *h* be any three transcendental entire functions such that *g*(0) = 0. If *p*, *q*, and *m* are any three positive integers with *m* \> *q* and any of the following two facts happensmin⁡(*ρ* ~*h*~ ^(*p*,\ *q*)^(*f*), *λ* ~*g*~(*m*, *q*)) \> 0 ormin⁡(*λ* ~*h*~ ^(*p*,\ *q*)^(*f*), *λ* ~*g*~(*m*, *q*)) \> 0, then $$\begin{matrix}
{\lim\underset{r\rightarrow\infty}{\sup}\frac{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left( {M_{f \circ g}\left( r \right)} \right)}{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left( {M_{f}\left( r \right)} \right)} = \infty.} \\
\end{matrix}$$

Proof(i) Since $$\begin{matrix}
{\lim\underset{r\rightarrow\infty}{\sup}\frac{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left( {M_{f \circ g}\left( r \right)} \right)}{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left( {M_{f}\left( r \right)} \right)}} \\
{\quad\quad \geq \,\lim\underset{r\rightarrow\infty}{\sup}\frac{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left( {M_{f \circ g}\left( r \right)} \right)}{{\log}^{\lbrack q\rbrack}r}} \\
{\quad\quad\quad \times \lim\underset{r\rightarrow\infty}{\inf}\frac{{\log}^{\lbrack q\rbrack}r}{{\log}^{\lbrack p\rbrack}M_{h}^{- 1}\left( {M_{f}\left( r \right)} \right)}} \\
{\quad\quad = \rho_{h}^{({p,q})}\left( {f \circ g} \right){\,\,}\frac{1}{\rho_{h}^{({p,q})}\left( f \right)}} \\
\end{matrix}$$ the result follows from [Theorem 9](#thm9){ref-type="statement"}.(ii) The proof can be carried out in the line of (i) and [Theorem 10](#thm10){ref-type="statement"}.

4. Conclusion {#sec4}
=============

After modifying the notion of relative order of higher dimensions in case of entire functions in \[[@B7]\], where a number of examples of relative order between functions were provided, in this paper we have obtained some growth properties of composite entire functions on the basis of relative (*p*, *q*)th order and relative (*p*, *q*)th lower order. In this process, [Theorem 5](#thm5){ref-type="statement"} and the first part of [Theorem 6](#thm6){ref-type="statement"} and Theorems [7](#thm7){ref-type="statement"} and [8](#thm8){ref-type="statement"} can be regarded as extensions of some results of \[[@B6]\].

The authors are grateful to the referee for his/her useful comments.

Conflict of Interests
=====================

The authors declare that there is no conflict of interests regarding the publication of this paper.

[^1]: Academic Editor: Tae-Sung Kim
