\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 $$\Big(D^2_t+\frac{a}{t} D_t\Big) u(z,t) = \sum^n_{j=1} \varepsilon_j \Delta_j u(z,t) \label{a1}$$ 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 $$T = \limsup_{\|k\|\to\infty} \frac{\|2k\|}{e\rho} [ |a_k|d_k(G)]^{\rho/\|2k\|},\quad (0<\rho<\infty) \label{a2}$$ 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 $$|a_k|d_k(G) \le \big[ \frac{e\rho(T+\varepsilon)} {\|2k\|}\big]^{\|2k\|/rho}. \label{a3}$$ We can easily estimate, from \cite[(4.14)]{b1}, that $$|R_k(z,t)| \le \Big(\frac{\|2k\|} {\rho T}\Big)^{\|2k\|/rho} \overline{M} (\rho,T) e^{-\|2k\|/\rho} e^{K|t|+\sum^n_{j=1} T_j|z^2_j|^{\rho_j}}, \label{a4}$$ 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 $$|u(z,t)| \le \sum^N_{\|k\|=0} |a_k| |R_k(z,t)| + \sum^\infty_{\|k\|=N+1} |a_k| |R_k(z,t)|.\label{a5}$$ 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 $G_r:z\in \mathbb{C}^n; |z|^2 = \max_{i\le j\le n} |z_j|^20$ 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 $$|x_1|+|t|, \dots, |x_m|+|t|, \big(x_{m+1}^2+t^2\big)^{1/2}, \dots, \big(x_n^2 + t^2\big)^{1/2} \in D.$$ \end{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, $$[N(\varepsilon)]^{-1} M_{\phi,G}(R) \le M_{u,S}(\varepsilon^{-1} R) \le C M_{\phi^*,G} (R) \label{b1}$$ 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 $$\begin{split} |u(z,t)| &\le \sum^\infty_{\|k\|=0} |a_j| C\Big\{\prod^n_{j=1} k_j^{p_j}\Big\} (|z_1|+|t|)^{2k_1} \dots (|z_m|+|t|)^{2k_m}\\ &\quad\times (z_{m+1}^2+t^2)^{k_{m+1}} \dots (z_n^2 + t^2)^{k_n}. \end{split}\label{b2}$$ 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 $$M_{u,S}(R) \le C \sum^\infty_{\|k\|=0} |a_k| \Big\{ \prod^n_{j=1} k_j^{p_j}\Big\} R^{2k_n}%\\ =C M_{\phi^*,G} (R) \label{b3}$$ 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 $$[N(\varepsilon)]^{-1} M_{\phi,G} (R) \le M_{u,S} (\varepsilon^{-1} R). \label{b4}$$ 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 $$\begin{split} \beta_k &= a_k \Big\{\Gamma\left(\frac{a+1}{2n}\right)\Big\}^n \frac{2^n K^{n-m}} {\pi^{m/2}} \Big\{ \prod^m_{j=1} k_j(k_j - 1/2)^{(\alpha_j+2)/2}\Big\}\\ &\quad\times \Big\{ \prod^n_{j=m+1} k_j^{(q_j+1+(a+1)/2n)}\Big\}. \label{b7} \end{split}$$ \end{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 $$\rho(u) = \limsup_{\|k\|\to\infty} \frac{\|2k\|\log \|k\|} {-\log |\beta_k|} \label{c1}$$ 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 $$\rho(f) = \limsup_{\|k\|\to\infty} \frac{\|2k\|\log \|k\|} {-\log|a_k|}. \label{c2}$$ 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 $$(e \rho(f) T(f))^{1/\rho(f)} = \limsup_{\|k\|\to\infty} \left\{ \|2k\|^{1/\rho(f)} [|a_k| d_k(G)]^{1/\|2k\|}, (0<\rho(f)<\infty)\right\}. \label{c3}$$ It can be seen that $$[|\beta_k| d_k(G)]^{1/\|2k\|} \to [|a_k| d_k(G)]^{1/\|2k\|} \quad \texttt{as } \|k\| \to\infty. \label{c4}$$ 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 $$\lambda(u) \ge \liminf_{\|k\|\to\infty} \frac{\|2k\| \log \|2k\|} {-\log |\beta_k|}. \label{c5}$$ 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} $$\lambda(f) = \liminf_{\|k\|\to\infty} \frac{\|2k\| \log \|2k\|} { \log |a_k|^{-1}}. \label{c6}$$ 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 $$t(u) \ge \liminf_{\|k\|\to\infty} \frac{\|2k\|} {e\rho(u)} |\beta_k|^{\rho(u)/\|2k\|}. \label{c7}$$ 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} \begin{thebibliography}{99} \bibitem{b1} L. R. Bragg; \emph{Complex transformations of solutions of generalized initial value heat problems}, Rocky Mountain J. Math. 20 No. 3(1990), 677-705. \bibitem{b2} L. R. Bragg, J .W. Dettman; \emph{Multinomial representation of solutions of a class of singular initial value problems}, Proc. Amer. Math. Soc. 21(1969), 629-634. \bibitem{b3} L. R. Bragg, J.W. Dettman; \emph{Expansions of solutions of certain hyperbolic and elliptic problems in terms of Jacobi polynomials}, Duke Math. J. 36, No. 1 (1969), 129-144. \bibitem{f1} B. A. Fuks; \emph{Introduction to the Theory of Analytic Functions of Several Complex Variables}, Translation Math. Monographs \& Amer. Math. Soc. Providence, RI 1963. \bibitem{g1} R. P. Gilbert; \emph{An investigation of the analytic properties of solutions to the generalized axially symmetric reduced wave equation in $n+1$ variables, with an application to the theory of potential scattering}, SIAM J. Math. 16 (1968), 30-50. \bibitem{g2} R. P. Gilbert, H. C. Howard; \emph{On solutions of the generalized bi-axially symmetric Helmholts equation generated by integral operators}, J. Reine Angew. Math. 218 (1965), 109-120. \bibitem{g3} A. A. Gol'dberg; \emph{Elementary remarks on the formulas defining order and type of functions of several variable}, (Russian) Akad. Nauk. Armjan SSR, Dokl. 29 (1959), 145-151. \bibitem{k1} D. Kumar; \emph{On the $(p,q)$-growth of entire function solution of Helmholtz equation}, J. Nonlinear Sci. Appl. 4 (2011), No.1, 5-14. \bibitem{k2} D. Kumar; \emph{Growth and Chebyshev approximation of entire function solution of Helmholtz equation in $R^2$}, Europian J. Pure and Appl. Math. 3, No. 6 (2010), 1062-1069. \bibitem{k3} D. Kumar; \emph{Ultraspherical expansions of generalized bi-axially symmetric potentials and pseudo-analytic functions}, Complex Variables and Elliptic Equations, Vol. 53, Issue 1 (2008), 53-64. \bibitem{k4} S. Kumar, G.S. Srivastava; \emph{Maximum term and lower order of entire functions of several complex variables}, Bulletin of Math. Anal. and Appl. 3(1) (2011), 156-164. \bibitem{m1} E. P. Miles, Jr.; E.C. Young; \emph{Basic sets of polynomial for generalized Beltrami and Euler-Poisson-Darboux equations and their iterates}, Proc. Amer. Math. Soc. 18 (1567), 981-986. \bibitem{m2} P. A. McCoy; \emph{Polynomial approximation of generalized biaxisymmetric potentials}, J. Approx. Theory, 25 (1979), 153-168. \bibitem{m3} P. A. McCoy; \emph{Polynomial approximation and growth of generalized axisymmetric potentials}, Can. J. Math. 31, No. 1 (1979), 49-59. \bibitem{r1} A. R. Reddy; \emph{Approximation of an entire function}, J. Approx. Theory, 3 (1970), 128-137. \end{thebibliography} \end{document}