\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 143, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/143\hfil Growth of entire solutions]
{Growth of entire solutions of singular initial-value problem in several complex
variables}

\author[ D. Kumar, M. Harfaoui \hfil EJDE-2012/143\hfilneg]
{Devendra Kumar, Mohammed Harfaoui}  % in alphabetical order

\address{Devendra Kumar \newline
Department of Mathematics\\
Research and Post Graduate Studies, 
M. M. H. College, Model Town, 
Ghaziabad-201001, U. P., India}
\email{d\_kumar001@rediffmail.com}

\address{Mohammed Harfaoui \newline
University Hassan II Mohammedia \\
Laboratory of Mathematics, Criptography and Mechanical F. S. T. \\
BP 146, Mohammedia 20650 Morocco}
\email{mharfaoui04@yahoo.fr}

\thanks{Submitted June 22, 2012. Published August 20, 2012.}
\subjclass[2000]{30B10, 30D20}
\keywords{Entire function;
initial-value  problem; multinomials; order and type}

\begin{abstract}
 In this article, we characterize the order, type, lower
 order, and lower type of entire function solutions to a class of
 singular initial-value problems, in terms of multinomials for
 $n\geq 2$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

  Let $z_j=x_j+iy_j$ denote a complex variable, $1\le j\le n$. Let
 $z=(z_1,\dots,z_n)$, $z^{2k} = z_1^{2k_1},\dots, z_n^{2k_n}$ where
 $k$ is the vector $(k_1,\dots,k_n)$ with $k_j$ a nonnegative
 integer $(j=1,\dots,n)$ and let $\|k\| = k_1+\dots+k_n$. Let
 $\phi(z)$ be an entire function of $z_1^2,\dots,z_n^2$ in a
 domain $D$ that includes the origin and let
 $\Delta_j=D^2_{z_j}+\frac{\alpha_j}{z_j} D_{z_j}, \alpha_j\ge 0,
 j=1,\dots,n$. Also, let $a>-1$ and $\varepsilon_j=1$ if
 $j=1,\dots,m$ and $\varepsilon_j=-1$ if $j=m+1,\dots,n$. Now
 consider the representations of an entire function solutions of
 the problem
\begin{equation}
\Big(D^2_t+\frac{a}{t} D_t\Big) u(z,t) = \sum^n_{j=1}
    \varepsilon_j \Delta_j u(z,t)  \label{a1}
\end{equation}
with initial data
\[
      u(z,0) = \phi(z),\quad u_t(z,0)=0
\]
 in terms of a set of associated multinomials $\{R_k(z,t)\}$
 throughout $(z,t)$ space, $t$ real. These multinomials are
 solutions of \eqref{a1} corresponding to the choice of
 $\phi(z)=z^{2k}$ in \eqref{a1}.

 Let $G$ be a region in $\mathbb{R}^n$ (positive hyper octant) and let
 $G_R\subset \mathbb{C}^n$ denote the region obtained from $G$ by a
 similarity transformation about the origin, with ratio of
 similitude $R$.

 \begin{definition} \label{def1.1}\rm
 Let $\phi(z)=\sum^\infty_{\|k\|=0} a_k z^{2k_n}_n$ be an entire function of several complex variables.
 Then $\phi(z)$ is of growth $(\rho,T)$ if
 \begin{equation}
T = \limsup_{\|k\|\to\infty} \frac{\|2k\|}{e\rho}
 [ |a_k|d_k(G)]^{\rho/\|2k\|},\quad  (0<\rho<\infty)
 \label{a2}
\end{equation}
 where
\[
 d_k(G) = \max_{R\in G} (R^{2k});\quad  R^{2k} =
 R_1^{2k_1},\dots,R_n^{2k_n}.
\]
This implies the existence of a positive constant $M$ such that
 \[
|\phi(z)| \le Me^{T|z^2|^\rho} \quad \forall z\in \mathbb{C}^n.
\]
\end{definition}

 Using \eqref{a2},  for each $\varepsilon>0$ there exists a
 positive integer $k_0$ such that if $k\ge k_0$, then
 \begin{equation}
 |a_k|d_k(G) \le \big[ \frac{e\rho(T+\varepsilon)}
 {\|2k\|}\big]^{\|2k\|/rho}.
\label{a3}
\end{equation}
We can easily estimate, from \cite[(4.14)]{b1}, that
 \begin{equation}
 |R_k(z,t)| \le \Big(\frac{\|2k\|} {\rho  T}\Big)^{\|2k\|/rho} \overline{M} (\rho,T)
 e^{-\|2k\|/\rho}  e^{K|t|+\sum^n_{j=1} T_j|z^2_j|^{\rho_j}}, \label{a4}
\end{equation}
 where
 \[
\overline{M}(\rho,T) = \int^\infty_0 e^{-\sigma +
 T|\sigma^2|^\rho}
 d\sigma,
\]
and $K$ is the sum of the absolute values of the coefficients of
multinomial and $M(\rho,T)$ is a generic constant depending only
 on the $\rho^{^\prime}_js$ and $T^{^\prime}_js$.

 Now let
 \[ 
u(z,t) = \sum^\infty_{\|k\|=0} a_k R_k (z,t) 
\]
 or
 \begin{equation}
 |u(z,t)| \le \sum^N_{\|k\|=0} |a_k| |R_k(z,t)| +
 \sum^\infty_{\|k\|=N+1} |a_k| |R_k(z,t)|.\label{a5}
\end{equation}

 Using the bound \eqref{a5} on $|R_k(z,t)|$ and the estimate on
 $|a_k|d_k(G)$ from \eqref{a3}, we see that the bound on second
 sum in \eqref{a5} is given by
 \[
\frac{K(\rho,T)} {d_k(G)} e^{K|t|+\sum^n_{j=1}
 T_j|z^2_j|^{\rho_j}} \sum^\infty_{\|k\|=N+1}
 \Big(\frac{T+\varepsilon}{T}\Big)^{\|2k\|/\rho}.
\]
 Since the series of constants in \eqref{a5}  converges, it
 follows that the series $u(z,t) = \sum^\infty_{\|k\|=0} a_k
 R^k(z,t)$ converges for all $n$ complex variables
 $(z_1,\dots,z_n)$ and real $t$ and uniformly so in compact
 subsets of $(z,t)$ space.

 Now we can establish a theorem.

\begin{theorem} \label{thm1.1}
 Let $\phi(z) = \sum^\infty_{\|k\|=0} a_k  z^{2k_1}_1, \dots, z_n^{2k_n}$ 
be entire in $(z_1^2,\dots,z_n^2)$  and converge in a domain 
$G_r:z\in \mathbb{C}^n; |z|^2 = \max_{i\le j\le n} |z_j|^2<R^2,R>0$ 
is a fixed positive real. Then the series
 $u(z,t) = \sum^\infty_{\|k\|=0} a_k R_k(z,t)$ converges for all
 $n$-complex variables $(z_1,\dots,z_n)$ and real $t$ and
 uniformly so in compact subsets of $(z,t)$ space.
\end{theorem}

 Bragg and  Dettman \cite{b2} proved the following theorem.

\begin{theorem} \label{thmA} 
Let $\phi(x) =\sum^\infty_{\|k\|=0}
 a_kx^{2k}$ be analytic in $(x_1^2,\dots,x_n^2)$ and converge in a
 domain $D$ that includes the origin. Then the series
 $\sum^\infty_{\|k\|=0} a_kP_k(x,t)$ converges to an analytic
 solution of the problem \eqref{a1} replacing $z$ by $x$, at least
 in region $S$ where $S$ is defined by $(x,t)\in S$ if and only if
 \begin{equation}
|x_1|+|t|, \dots, |x_m|+|t|,
 \big(x_{m+1}^2+t^2\big)^{1/2}, \dots,
\big(x_n^2 + t^2\big)^{1/2} \in D.
\end{equation}
\end{theorem}

 We shall proceed to the complex transformation of above Theorem A
 in the following manner.

 Let $(z_1,\dots,z_n)$ be an element of $\mathbb{C}^n$ and $\mathbb{R}^{2n}$, the
 space of real coordinates. The transformation from real to the
 complex coordinates are given by $x_k = \frac{z_k+\overline{z}_k} {2}$,
 $y_k = \frac{z_k-\overline{z}_k} {2_i}$. We equip $\mathbb{C}^n$ with the
 Euclidean metric of $\mathbb{R}^{2n}$;
 \[ 
ds^2 = \sum^n_{k=1} (dx_k^2 + dy_k^2) = \sum^n_{k=1} dz_k . d_{\overline{z}_k}.
\]

 Let $z_k$ be a point on the domain $G_R$ for which $|a_k
 R_k(z_k,0)| = \sup_{z_k\in G_R} |a_k R_k $ $(z_k,0)| = C_k$. By a
 rotation, we can assume that $z^2_k = (x_k^2, 0,\dots,0)$. If
 $\widetilde{f}(w) = f(w^2,0,\dots,0)$ and $\widetilde{f}(w) =\sum^\infty_{l=0}
 a_l w^{2l}$ is the Taylor series expansion of $\widetilde{f}$ at the
 origin, then $\left|a_k x_k^{2k} \right| = C_k$ and therefore we
 have the following theorem.

\begin{theorem} \label{thm1.2} 
Let $\phi(z) = \sum^\infty_{\|k\|=0} a_k  z^{2k}$ be entire in 
$( z^2_1, \dots, z^2_n)$ and
 converge in a domain $G_R$ that includes the origin. Then the
 series $u(z,t) = \sum^\infty_{\|k\|+0} a_k R_k(z,t)$ converges to
 an entire solution of the problem \eqref{a1} at least in a region
 $S$ where $S$ is defined by $(z,t)\in S$ if and only if
 \[
|z_1|+|t|, \dots, |z_m+|t|, \;
(z^2_{m+1}+t^2)^{1/2}, \dots, (z^2_n +  t^2)^{1/2} \in G_R. 
\]
\end{theorem}

Let $\phi(z) =\sum^\infty_{\|k\| = 0} a_k z^{2k}$ be the power
 series expansion of the function $\phi(z)$. Then the maximum
 modulus of $u(z,t)$ and $\phi(z)$ are defined as in complex
 function theory  \cite[pp. 129, 132]{r1},
 \begin{gather*}
 M_{f,G} (R) = \max_{z\in G_R} |f(z)| , \\
 M_{u,S} (R) = \max_{(z,t)\in S} |u(z,t)|.
\end{gather*}
 Following the usual definitions of order and type of an entire
 function of $n$-complex variables $(z_1^2,\dots,z_n^2)$, the
 order $\rho$ and type $T$ of $u(z,t)$ are defined as in \cite{f1}
 \begin{gather}
 \rho(u) = \limsup_{R\to\infty} \frac{\log \log
 M_{u,S} (R)} {\log R}, \label{a6} \\
 T(u) = \limsup_{R\to \infty} \frac{\log M_{u,S}(R)}
 {R^{\rho(u)}}. \label{a7}
\end{gather}

 In this paper we characterize the order, lower order, type and
 lower type of entire function solutions of problem \eqref{a1}
 in terms of a set $\{R_k(z,t)\}$ of multinomials for $n\ge 2$.
 Multinomials of this type have been constructed by  Miles and
Yong \cite{m1} when $z=x$ and $m=n$ or $m=0$. In these cases
 \eqref{a1} reduces to either the generalized
 Euler-Poisson-Darboux or the generalized Beltrami equation. 
 Gilbert and  Howard \cite{g1,g2}  discussed analyticity
 properties of solutions of special cases of \eqref{a1}. 
 Bragg and  Dettman obtained representation of analytic
 solutions of problem \eqref{a1} for $z=x$ in terms of these
 multinomials for $n\ge 2$ \cite{b2} and for $n=1$ in \cite{b3}. It has been
 found \cite{b2} that $R_k(x,t), n\ge 2$, can be expressed as a
 convolution of $n$ polynomials $R_{k_j}(x_j,t), j=1,\dots ,n$. For
 $n=1$ the corresponding $R_k(x,t)$ are defined in terms of Jacobi
 polynomials. The Growth estimates for the solutions of
 \eqref{a1} in terms of multinomials $R_k(z,t)$ for $n\ge 2$ then
 permit the obtaining of global region of convergence from
 acknowledge of singularities of the given data function
 $\phi(z)$. It should be noted that the function $\phi(z)$
 is the analytic continuation of its restriction to the axis of
 symmetry; i.e., $\phi(z) = u(z,0)$. Using various techniques,
 the characterizations of order and type
 of entire function solutions of similar problems were obtained by
 McCoy \cite{m2,m3} Kumar \cite{k1,k2,k3} and others for $n=1$.
 However, non of them have considered the case for $n\ge 2$.


 \section{Auxiliary Results}

  In this section we shall prove some auxiliary results
 which will be used in the sequel.

\begin{lemma} \label{lem2.1} 
If $u(z,t) = \sum^\infty_{\|k\|=0} a_k  R_k(z,t)$ is an entire function 
solution of problem \eqref{a1} in  terms of a set $\{R_k(z,t)\}$
of multinomials corresponding to  given data function 
$\phi(z) = \sum^\infty_{\|k\|=0} a_k z^{2k}$  in \eqref{a1} then $\phi$ 
and $\phi^*$ are also entire functions  of $n-$complex variables 
$(z^2_1,\dots,z^2_n)$. Further,
 \begin{equation}
[N(\varepsilon)]^{-1} M_{\phi,G}(R) \le
 M_{u,S}(\varepsilon^{-1} R) \le C M_{\phi^*,G} (R) \label{b1}
\end{equation}
 where
\begin{gather*}
 \phi^*(z) = \sum^\infty_{\|k\|=0} |a_k| \Big\{
 \prod^n_{j=1} k_j^{p_j}\Big\} z_1^{2k_1}, \dots, z_n^{2k_n},\\
 N(\varepsilon) = \sup\{N(\varepsilon e^{i\theta},\xi): 0\le
 \theta \le 2\pi, -1\le \xi\le 1,0<\varepsilon <1\}
\end{gather*}
 and $C$ is a constant.
\end{lemma}

\begin{proof} 
From Theorem 1.1 and 1.2, bearing in mind with the
 relation of \cite[(3.1)]{b2}, we obtain
 \begin{align*}
 |u(z,t)| 
&\le  \sum^\infty_{\|k\|=0} |a_k| \Big\{\Gamma
 \left(\frac{a+1}{2n}\right)\Big\}^n 
\frac{2^m K^{n-m}} {\pi^{m/2}} \Big\{ \prod^m_{j=1} k_j \frac{\Gamma(k_j+(\alpha_j
 +1)/2)} {\Gamma(k_j - 1/2)}\Big\}\\
 &\quad\times \{|z_j|+|t|\}^{2k_j} \Big\{ \prod^n_{j=m+1} \frac{k_j^{q_j}
 k_j !} {\Gamma((k_j)+(a+1)/2n)} (z^2_j+t^2)^{k_j}\Big\} 
\end{align*}
where $q_j = \max((\alpha_j-1)/2, ((a+1)/2n)-1, -1/2)$, 
$j = m+1,\dots,n$.

 Using the relation $\Gamma(x+a)/\Gamma x \sim x^a$ as
 $x\to\infty$, we have
 \[
 \frac{\Gamma(k_j+(\alpha_j+1)/2)} {\Gamma(k_j-1/2)} 
\sim  (k_j-1/2)^{(\alpha_j+2)/2},\quad
\frac{k_j^{q_j} k_j !}  {\Gamma k_j + \frac{(a+1)}{2n}} 
\sim k_j^{q_j+1} (k_j)^{(a+1)/2n}
\]
and we see that there exist constants $C,p_1,\dots,p_n$ with 
$p_j = p_j(\alpha_j), j=1,\dots,m$ and $p_j = p_j(\alpha_j, a,n)$ for
 $j=m+1,\dots,n$ such that
 \begin{equation}
\begin{split}
|u(z,t)| &\le \sum^\infty_{\|k\|=0} |a_j|
 C\Big\{\prod^n_{j=1} k_j^{p_j}\Big\} (|z_1|+|t|)^{2k_1} \dots
 (|z_m|+|t|)^{2k_m}\\
&\quad\times (z_{m+1}^2+t^2)^{k_{m+1}} \dots (z_n^2 +
 t^2)^{k_n}. 
\end{split}\label{b2}
\end{equation}
Now, $|\phi(z)| \le \sum^\infty_{\|k\|=0} |a_k| |z_1|^{2k_1}
 \dots |z_n|^{2k_n}$, the series \eqref{b2} converges for
$z\in  G_R$. But for $z\in G_R$, the series
 \[
 \sum^\infty_{\|k\|=0} |a_k| \Big\{\prod^n_{j=1} k_j^{p_j}
 \Big\} |z_1|^{2k_1}\dots |z_n|^{2k_n}
\]
also converges. By Theorem 1.2, if $\phi(z)$ is entire in
 $(z_1^2,\dots, z_n^2)$, then $u(z,t)$ converges to an entire
 solution of problem \eqref{a1}. We see that
 \[
\lim_{\|k\|\to\infty} \Big[ |a_k| \prod^n_{j=1}
 k_j^{p_j}\Big]^{1/\|2k\|}
 = \lim_{\|k\|\to\infty} |a_k|^{\frac{1}{\|2k\|}} = 0.
 \]
Hence both $\phi$ and $\phi^*$ are entire.

Using \eqref{b2} we obtain
 \begin{equation}
 M_{u,S}(R) \le C \sum^\infty_{\|k\|=0} |a_k| \Big\{
    \prod^n_{j=1} k_j^{p_j}\Big\} R^{2k_n}%\\
         =C M_{\phi^*,G} (R) \label{b3}
\end{equation}
 where
\[
\phi^*(z) = \sum^\infty_{\|k\|=0} |a_k| \Big\{\prod^n_{j=1}
  k_j^{p_j}\Big\} z_1^{2k_1}, \dots, z_n^{2k_n}.
\]

Now for reverse relation, we have
\begin{gather*}
 \phi(z) =  \sum^\infty_{\|k\|=0} a_k z_1^{2k_1} \dots
  z_n^{2k_n}\\
\begin{aligned}
|\phi(z)|  &\leq  \sum^\infty_{\|k\|=0} |a_k| \Big\{
  \prod^n_{j=1} k_j^{p_j}\Big\} |z_1|^{2k_1} \dots |z_n|^{2k_n}\\
  &=  \sum^\infty_{\|k\|=0} |a_k| \Big\{\prod^n_{j=1}
  k_j^{p_j}\Big\} [|z_1+|t|]^{2k_1}\dots \{|z_{m}| +
  |t|\}^{2k_m}\\
  &\quad\times   \left[\left(z^2_{m+1}+t^2\right)^{1/2}\right]^{2k_{m+1}}\dots
  \left[(z^2_n+t^2)^{1/2}\right]^{2k_n}\\
  &\quad\times  \left[\frac{|z_1|} {|z_1|+|t|}\right]^{2k_1} \dots
  \left[\frac{|z_m|} {|z_m|+|t|}\right]^{2k_m}\\
  &\quad\times  \left[\frac{|z_{m+1}|} {(z_{m+1}^2+t^2)^{1/2}}\right]^{2k_{m+1}}
  \dots \left[ \frac{|z_n|}
  {(z^2_n+t^2)^{1/2}}\right]^{2k_n}.
\end{aligned}
\end{gather*}
This relation is valid globally, and leads to the estimates
\begin{gather*}
|\phi(z)| \le M_{u,S} (R) N(\varepsilon), \varepsilon =
  (|z|/R)^2 = \max_{1\le j\le n} \Big(\frac{|z_j|}
  {R_j}\Big)^2, \\
N(\varepsilon) = \sup \{|N(\varepsilon e^{i\theta}, \xi)|
  :0\le\theta\le 2\pi, -1\le \xi\le 1\}.
\end{gather*}
For $z=\varepsilon R e^{i\theta} (\varepsilon$ real,
  $0<\varepsilon<1\}$, we have
\[
 M_{\phi,G} (\varepsilon R) \le M_{u,S} (R) N(\varepsilon)
\]
  or
\begin{equation}
 [N(\varepsilon)]^{-1} M_{\phi,G} (R) \le M_{u,S}
  (\varepsilon^{-1} R). \label{b4}
\end{equation}
Combining \eqref{b3} and \eqref{b4} we obtain \eqref{b1}.
\end{proof}

\begin{lemma} \label{lem2.2}
Let $u(z,t)$ be an entire function solution
  of  \eqref{a1} in terms of a set $\{R_k(z,t)\}$ of
  multinomials corresponding to given data function $\phi(z)$ in
  \eqref{a1}. Then the orders and types of $u(z,t)$ and $\phi$
  respectively are identical.
\end{lemma}

  \begin{proof} Let $\phi(z) = \sum^\infty_{\|k\|=0} a_k
  z_1^{2k_1}\dots z_n^{2k_n}$ be an entire function of order
  $\rho(\phi)$ and type $T(\phi)$. Then it is well known 
\cite[ Thm.  1]{g3} that
\begin{gather}
 \rho(\phi) = \limsup_{\|k\|\to\infty} \big\{
  \frac{\|2k\| \log \|k\|} {-\log|a_k|}\big\}, \label{b5} \\
 (e\rho(\phi) T(\phi))^{1/\rho(\phi)} =
  \limsup_{\|k\|\to\infty} \big\{ \|2k\|^{1/\rho(\phi)}
  [|a_k|d_k(G)]^{1/\|2k\|}\big\}. \label{b6}
\end{gather}
Hence for the function $\phi^*(z) = \sum^\infty_{\|k\|=0} |a_k|
  \left\{\prod^n_{j=1} k_j^{p_j}\right\} z_1^{2k_1} \dots
  z_n^{2k_n}$, we have
\begin{align*}
 \frac{1}{\rho(\phi^*)} 
&=  \liminf_{\|k\|\to\infty}  \frac{\log 
[|a_k| \prod^n_{j=1} k_j^{p_j}]^{-1}}  {2\|k\| \log \|k\|} \\
  &=  \liminf_{\|k\| \to\infty} \frac{ \log|a_k|^{-1} -
  \log [\prod^n_{j=1} k_j^{p_j}]} {2\|k\| \log \|k\|}\\
  &=  \liminf_{\|k\|\to\infty} \frac{\log |a_k|^{-1}}
  {2\|k\|\log \|k\|}.
\end{align*}
Hence $\rho(\phi) = \rho(\phi^*)$. Since $\phi$ and $\phi^*$
  have same order, using \eqref{b6} we can easily show that
  $T(\phi) = T(\phi^*)$.

  Now using the relation \eqref{b1} with the definitions of order
  and type given by \eqref{a6} and \eqref{a7}, the proof is
  complete.
\end{proof}

\begin{lemma} \label{lem2.3} 
If $|a_k|/|a_{k'}|$, $\|k'\| = \|k\|+1$,
  forms a non-decreasing function of $k$ then
  $|\beta_k|/|\beta_{k'}|$ also forms a non-decreasing function
  of $k$, where
\begin{equation}
\begin{split}
\beta_k &= a_k
  \Big\{\Gamma\left(\frac{a+1}{2n}\right)\Big\}^n \frac{2^n
  K^{n-m}} {\pi^{m/2}}
 \Big\{ \prod^m_{j=1} k_j(k_j -  1/2)^{(\alpha_j+2)/2}\Big\}\\
&\quad\times \Big\{ \prod^n_{j=m+1} k_j^{(q_j+1+(a+1)/2n)}\Big\}. \label{b7}
\end{split}
\end{equation}
\end{lemma}

\begin{proof} 
We have
\begin{align*}
\frac{|\beta_k|} {|\beta_{k'}|}
& = a_k \Big\{\Gamma\left(\frac{a+1}{2n}\right)\Big\}^2 
\frac{2^n K^{n-m}} {\pi^{m/2}} 
\big\{ \prod^m_{j=1} k_j (k_j-1/2)^{(\alpha_j+2)/2}  \big\}
\\
&\quad\times \frac{\big\{ \prod^n_{j=m+1} k_j^{q_j +1+(a+1)/2n}\big\}}
  {a_{k+1}\big\{\Gamma\left(\frac{a+1}{2n}\right)\big\}^n
  \frac{2^n K^{n-m}} {\pi^{m/2}} \big\{ \prod^m_{j=1} (k_j+1)
  \left(k_j+\frac{1}{2}\right)^{(\alpha_j+2)/2}\big\}}\\
&\quad\times \frac{1}{\big\{\prod^n_{j=m+1} k_j^{q_j+1+(a+1)/2n}\big\}} 
\\
  &=  \frac{a_k}{a_{k+1}} 
\frac{\prod^m_{j=1}  k_j\left(k_j-\frac{1}{2}\right)^{(\alpha_j+2)/2} 
\big\{  \prod^n_{j=m+1} k_j^{(q_j+1+(a+1)/2n)}\big\}} 
{\prod^m_{j=1}  (k_j+1)(k_j+1/2)^{(\alpha_j+2)/2}, \prod^n_{j=m+1}
  (k_j+1)^{(q_j+1+(a+1)/2n)}}. 
\end{align*}
Let 
\[
 G(x) = \frac{\prod^m_{j=1}
  x_j(x_j-\frac{1}{2})^{(\alpha_j+2)/2} \prod^n_{j=m+1}
  x_j^{(q_j+1+(a+1)/2n)}} {\prod^m_{j=1} (x_j+1)
  (x_j+\frac{1}{2})^{(\alpha_j+2)/2} \prod^n_{j=m+1}
  (x_j+1)^{(q_j+1+(a+1)/2n)}} 
\]
\begin{align*}
 \log G(x) 
&=  \sum^m_{j=1} \log [  x_j(x_j-1/2)^{(\alpha_j+2)/2}] 
 + \sum^n_{j=m+1} \log  x_j^{(q_j+1+(a+1)/2n)}\\
&\quad - \sum^m_{j=1} \log (x_j+1) (x_j+\frac{1}{2})^{(\alpha_j+2)/2}
  - \sum^n_{j=m+1} \log (x_j+1)^{(q_j+1+(a+1)/2n)} 
\end{align*}
By logarithmic differentiation, we obtain
\begin{align*}
 \frac{G'(x)} {G(x)} 
&=  \sum^m_{j=1} \left(\frac{1}{x_j} +
  \frac{(\alpha_j+2)} {2(x_j-\frac{1}{2})} \right) +
  \sum^n_{j=m+1} \frac{q_j+1+\frac{(a+1)} {2n}} {x_j}\\
&\quad - \sum^m_{j=1} \frac{1}{x_j+1} \frac{(\alpha_j+2)}
  {2(x_j+\frac{1}{2})} - \sum^n_{j=m+1} \frac{q_j+1+\frac{(a+1)}
  {2n}} {x_j+1}. 
\end{align*}
Let 
\[
t(x_j) = \sum^m_{j=1} \frac{1}{x_j} + \frac{(\alpha_j+2)}
  {2(x_j-\frac{1}{2})} + \sum^n_{j=m+1} \frac{q_j+1+\frac{(a+1)}
  {2n}} {x_j}.
\]
Then $t(x_j) - t(x_{j+1})>0$ for any $x_j>0$. Hence $t(x_j)$ is
decreasing function and subsequently $G^{^\prime}(x_j)>0$ for
$x_j>0$. Hence $|\beta_k|/|\beta_{k^{^\prime}}|$ is
non-decreasing if $|a_k|/|a_{k^{^\prime}}|$ is non-decreasing.
\end{proof}

\section{Main Results}

\begin{theorem} \label{thm3.1}
 Let $u(z,t)$ be an entire function
  converges to solution of problem \eqref{a1} corresponding to
  given data function $\phi(z)$ in \eqref{a1} having order $\rho(u)$.
  Then
\begin{equation}
\rho(u) = \limsup_{\|k\|\to\infty} \frac{\|2k\|\log
  \|k\|} {-\log |\beta_k|} \label{c1}
\end{equation}
where $\beta_k$ is given by \eqref{b7}.
\end{theorem}

\begin{proof} It is well known \cite[Thm. 1]{g3} that if 
$f(z) =  \sum^\infty_{\|k\|=0} a_k z^{2k}$ be an entire function of order
  $\rho(f)$ then
  \begin{equation}
 \rho(f) = \limsup_{\|k\|\to\infty} \frac{\|2k\|\log
  \|k\|} {-\log|a_k|}. \label{c2}
\end{equation}
 Hence for the function $u(z,0) = \sum^\infty_{\|k\|=0} \beta_k
  z_1^{2k_1}\dots z_n^{2k_n}$, we have
\begin{align*}
\frac{1}{\rho(u)}
&=  \liminf_{\|k\|\to\infty}
  \frac{-\log |\beta_k|} {\|2k\| \log \|2k\|}\\
&=  \liminf_{\|k\|\to\infty} \frac{\log |a_k|^{-1} -
  \log \big[\big\{ \Gamma\left(\frac{(a+1)}{2n}\right)\big\}^n
  \frac{2^n K^{n-m}} {\pi^{m/2}} \left\{\prod^n_{j=1}
  k_j^{p_j}\right\}\big]} {\|2k\| \log \|2k\|}\\
 &=  \liminf_{\|k\|\to\infty} \frac{\log |a_k|^{-1}}
  {\|2k\| \log \|2k \|} -
  \frac{\log\left[\left\{\Gamma\left(\frac{a+1}{2n}\right)\right\}^n
  \frac{2^n K^{n-m}} {\pi^{m/2}} \left\{\prod^n_{j=1}
  k_j^{p_j}\right\}\right]^{1/\|2k\|}} {\log \|2k\|}\\
 &=  \liminf_{\|k\|\to\infty} \frac{\log |a_k|^{-1}}
  {\|2k\|\log \|2k\|}.
 \end{align*}
\end{proof}

Now using \eqref{c2} for data function $\phi(z)$, we get the
  required results.

\begin{theorem} \label{thm3.2} 
Let $u(z,t)$ be an entire function  converges to solution of \eqref{a1} 
corresponding to   given data function $\phi(z)$ in \eqref{a1} having 
type $T(u)$.   Then
  \[
 (e\rho(u) T(u))^{1/\rho(u)} =
  \limsup_{\|k\|\to\infty} \left\{ \|2k\|^{1/\rho(u)}
  \left[ |\beta_k| d_k (G)\right]^{1/\|2k\|}\right\},
  (0<\rho(u)<\infty). 
\]
\end{theorem}

  \begin{proof} 
For an entire function $f(z) =
  \sum^\infty_{\|k\|=0} a_k z^{2k}$,  Gol'dberg \cite[Thm. 1]{g3}
  obtained type in terms of the coefficients of its Taylor series
  expansion as
 \begin{equation}
 (e \rho(f) T(f))^{1/\rho(f)} =
  \limsup_{\|k\|\to\infty} \left\{ \|2k\|^{1/\rho(f)}
  [|a_k| d_k(G)]^{1/\|2k\|}, (0<\rho(f)<\infty)\right\}. \label{c3}
\end{equation}
It can be seen that
\begin{equation} [|\beta_k| d_k(G)]^{1/\|2k\|} \to [|a_k|
  d_k(G)]^{1/\|2k\|} \quad \texttt{as } \|k\| \to\infty.
  \label{c4}
\end{equation}
 Hence the result follows by using \eqref{c3} for data function
  $\phi(z)$ and taking into account the equation \eqref{c4}.
\end{proof}

In analogy with the definitions of order $\rho(u)$ and type
$T(u)$, we define lower order $\lambda(u)$ and lower type $t(u)$ as
\begin{gather*}
\lambda(u) =  \liminf_{R\to\infty} \frac{\log \log
  M_{u,S} (R)} {\log R}\\
  t(u) =  \liminf_{R\to\infty} \frac{\log M_{u,S} (R)}
  {R^{\rho(u)}}, 0<\rho(u)<\infty.
\end{gather*}

\begin{theorem} \label{thm3.3} 
Let $u(z,t)$ be an entire function  converges to the problem \eqref{a1} 
corresponding to data  function $\phi(z)$ in \eqref{a1} having lower 
order $\lambda(u)$. Then
  \begin{equation}
 \lambda(u) \ge \liminf_{\|k\|\to\infty} \frac{\|2k\|
  \log \|2k\|} {-\log |\beta_k|}. \label{c5}
\end{equation}
Also if $|\beta_k|/|\beta_{k'}|$, where $\|k'\| =
  \|k\|+1$, is a non-decreasing function of $k$, then equality
  holds in \eqref{c5}.
\end{theorem}

\begin{proof} For entire function 
$f(z) = \sum^\infty_{\|k=0}  a_k z_1^{2k_1} \dots z_n^{2k_n}$, if $|a_k|/|a_{k'}|$
forms a non-decreasing function of $k$ then we have  \cite[Thm. 1]{k4}
  \begin{equation}
\lambda(f) = \liminf_{\|k\|\to\infty} \frac{\|2k\|
  \log \|2k\|} { \log |a_k|^{-1}}. \label{c6}
\end{equation}
  Let $|\beta_k|/|\beta_{k'}|$ forms a non-decreasing
  function of $k$ for $k>k_0$. Applying Lemma 2.3 and \eqref{c6}
  to $u(z,0) = \sum^\infty_{\|k\|=0} \beta_k z_1^{2k_1} \dots
  z_n^{2k_n}$, we obtain
\[
 \frac{1}{\lambda(u)}
=  \limsup_{\|k\|\to\infty}  \frac{\log |a_k|^{-1} - \log \left[C \prod^n_{j=1}
  k_j^{p_j}\right]} {\|2k\| \log \|2k\|}
=  \limsup_{\|k\|\to\infty} \frac{\log |a_k|^{-1}} {
  \|2k\| \log \|2k\|}
\]
 Then $\lambda(u) = \lambda(\phi)$.
\end{proof}

In a similar manner we can prove the following theorem.

\begin{theorem} \label{thm3.4} 
Let $u(z,t)$ be an entire function
converging to a solution of \eqref{a1} corresponding to
data function $\phi(z)$ in \eqref{a1} having lower type $t(u)$. Then
  \begin{equation}
 t(u) \ge \liminf_{\|k\|\to\infty}
  \frac{\|2k\|} {e\rho(u)} |\beta_k|^{\rho(u)/\|2k\|}. \label{c7}
\end{equation}
Also, if $|\beta_k|/|\beta_{k'}|$, where
$\|k'\|  = \|k\|+1$, is a non-decreasing function of $k>k_0$, then
equality holds in \eqref{c7}.
\end{theorem}

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\end{document}