# On Generalized Growth of Analytic Functions Solutions of Linear Homogeneous Partial Differential Equation of Second Order.

1. Introduction

Hu and Yang [1,2] studied the behavior of meromorphic solutions of the following homogeneous linear partial differential equations of the second order:

[mathematical expression not reproducible] (1)

They showed that these solutions are closely related to Bessel functions and Bessel polynomials for (t,z) [member of] [C.sup.2]. Several authors such as Berenstein and Li [3], Hu and Yang [4], Hu and Li [5], Li [6], and Li and Saleeby [7] have investigated the global solutions of some first-order partial differential equations. McCoy [8] and Kumar [9] studied the approximation of pseudoanalytic functions on the disk and obtained some coefficient and Bernstein-type growth theorems. Kapoor and Nautiyal [10] and Kumar and Basu [11] characterized the order and type of solutions (not necessarily entire solutions) of certain linear partial differential equations in terms of rates of decay of approximation errors in various norms. These solutions are related to Jacobi polynomials. Wang et al. [12] obtained the growth parameters order and type of entire function solutions of the partial differential equation

t [[[partial derivative].sup.2]u/[partial derivative][t.sup.2] + ([delta] + 1 - t) [[partial derivative]u/[partial derivative]t] + z [[partial derivative]u/[partial derivative]z] = 0 (2)

for a real [delta] > 0. These solutions are related to Laguerre polynomials. In this paper, our aim is to characterize the generalized growth parameters order and type of solutions of (2) which are analytic on bidisk of finite radius R. The generalized growth parameters have been studied by several authors such as Seremeta [13], Shah [14], Srivastava and Kumar [15], and Vakarchuk and Zhir [16,17]. Wang et al. [12] proved that the PDE (2) has an entire solution u = f(t, z) on [C.sup.2], if and only if u = f(t,z) has a series expansion f(t,z) = [[summation].sup.[infinity].sub.n=0][a.sub.n][L.sub.n]([delta], t)[z.sup.n]. Here

[mathematical expression not reproducible] (3)

are the Laguerre polynomials. Bernstein theorem identifies a real analytic function on the closed unit disk as the restriction of an analytic function defined on an open disk of radius R > 1 by computing R from the sequence of minimal errors generated from optimal polynomials approximates. The disk [D.sub.R] of maximum radius on which analytic function f(t,z) exists is denoted by f(t,z) [member of] A([D.sub.R] x [D.sub.R]). If f is an entire function, it has no singularities in the finite positive [C.sup.2] plane and write f [member of] A([C.sup.2]). Let [alpha] and [beta] be two positive, strictly increasing, and differentiable functions from (0, [infinity]) [right arrow] (0, [infinity]), which satisfies the following conditions for every [gamma] > 0:

[mathematical expression not reproducible] (4)

here d(u) denotes the differential of u. Now we define the generalized order and generalized type of an entire f(t, z) [member of] A([C.sup.2]) by

[mathematical expression not reproducible] (5)

where

[mathematical expression not reproducible] (6)

In view of the concept introduced by MacLane [18] to the measures of order and type for an analytic function on a disk [absolute value of z] < R, we normalize above definitions relative to the boundary under the transformation r [right arrow] R/(R - r). Thus an analytic function f(t, z) [member of] A([D.sub.R] x [D.sub.R]) with radial limits is said to be of generalized regular growth ([[rho].sub.0]([alpha], [beta]), [T.sub.o]([alpha], [beta])) if it satisfies

[mathematical expression not reproducible] (7)

where [[rho].sub.0]([alpha], [beta]) is referred to as the ([alpha], [beta])-order of f(t,z) provided that 0 < [[rho].sub.0]([alpha], [beta]) < [infinity] and [T.sub.o]([alpha], [beta]) is referred to as the ([alpha], [beta])-type.

Example A. f(t,z) = exp{(1/(1 - f))(1/(1 - z))} [member of] A([D.sub.1] x [D.sub.1]) has ([alpha], [beta])-order 2 and ([alpha], [beta])-type 1 for [alpha](x) = log x and [beta](x) = x.

In a neighborhood of origin, the function f(t, z) has the local expansion

f(t, z) = [[infinity].summation over (n=0)][[w.sub.n](t)/n!][z.sup.n], (8)

where [w.sub.n](t) = ([[partial derivative].sup.n]f/[partial derivative][z.sup.n])(t,0) is an entire solution of the ordinary differential equation

t [[d.sup.2]w/d[t.sup.2]] + ([delta] + 1 - t) [dw/dt] + nw = 0, (9)

where [w.sub.n](t)[z.sup.n] = n![a.sub.n][L.sub.n]([delta], t)[z.sup.n]. Following the method of Frobenius [19], a second independent solution of (9) can be obtained as

[X.sub.n]([delta], t) = q[L.sub.n]([delta], t) log t + [[infinity].summation over (i=0)][p.sub.i][t.sup.i], (10)

where q [not equal to] 0 and [p.sub.i] are constants. So there exist [a.sub.n] and [b.sub.n] satisfying

[w.sub.n](t) = n![a.sub.n][L.sub.n]([delta], t) + [b.sub.n][X.sub.n]([delta], t). (11)

Because of the singularity of [X.sub.n]([delta], t) at t = 0, it leads to [b.sub.n] = 0. Thus

f(t, z) = [[infinity].summation over (n=0)][a.sub.n][L.sub.n]([delta], t)[z.sup.n]. (12)

Since f(0, z) = [[summation].sup.[infinity].sub.n=0][a.sub.n][L.sub.n]([delta], 0)[z.sup.n], we have the estimate [absolute value of ([w.sub.n](0)[z.sup.n])] = [absolute value of (n![a.sub.n][L.sub.n]([delta],0)[z.sup.n])] ~ n![absolute value of [a.sub.n])]([n.sup.[delta]]/[GAMMA]([delta] + 1))[r.sup.n] for large n. Set

[mathematical expression not reproducible] (13)

2. Auxiliary and Main Results

First we prove the following lemma.

Lemma 1. Let f(t, z) = [[summation].sup.[infinity].sub.n=0][a.sub.n][L.sub.n]([delta], t)[z.sup.n] and f [member of] A([D.sub.R] x [D.sub.R]). For every 1 < r < R, we set

[mathematical expression not reproducible] (14)

then

[[bar.[rho]].sub.o]([alpha], [beta]) [less than or equal to] [mu]([alpha], [beta]),

[[rho]].sub.o]([alpha], [beta]) [less than or equal to] [[bar.[rho]].sub.o]([alpha], [beta]). (15)

Proof. From (13), for r sufficiently close to R, we have

log([[n.sup.[delta]]/[GAMMA]([delta] + 1)] [absolute value of [a.sub.n]][R.sup.n]) [less than or equal to] [n/[[beta].sup.-1]([alpha](n)/[bar.[mu]]], [bar.[mu]] = [mu] + [epsilon] (16)

or

log([[n.sup.[delta]]/[GAMMA]([delta] + 1)] [absolute value of [a.sub.n]][R.sup.n]) [less than or equal to] n log(r/R) + [n/[[beta].sup.-1]([alpha](n)/[bar.[mu]]]. (17)

Using the result

[mathematical expression not reproducible] (18)

with the method of Calculus that, for every r > 1 and [mu] > 0, the maximum of the function

x log (r/R) + [x/[[beta].sup.-1]([alpha](x)/[bar.[mu]] (19)

is reached at

x = [[alpha].sup.-1] {1 - d log([[beta].sup.-1]([alpha](x)/[mu]))/d (log x)/log (R/r)]}, (20)

we get

x = (1 + o(1))[[alpha].sup.-1]([bar.[mu]][beta](R/(R - r))), r [right arrow] [infinity]. (21)

Now we have

[mathematical expression not reproducible] (22)

Using the property of [alpha], we get

[alpha](log [bar.M](r, r, f))/[beta](R/(R - r))] [less than or equal to] [[bar.[mu]]. (23)

Applying the limit supremum as r [right arrow] R, we obtain

[[bar.[rho]].sub.o]([alpha], [beta]) [less than or equal to] [mu]([alpha], [beta]). (24)

Now consider

f(t, z) = [[infinity].summation over (n=0)][a.sub.n][L.sub.n]([delta], 0) [z.sup.n]; (25)

putting r = [square root of (r x R)][square root of (r/R)] in the above, we get

M(r, r, f) [less than or equal to] [[infinity].summation over (n=0)][absolute value of [a.sub.n\]][absolute value of [L.sub.n]([delta], t))] [([square root of (r x R)]).sup.n] [([square root of (r/R)]).sup.n],

(r/R) < 1, (26)

or

M(r, r, f) [less than or equal to] [[infinity].summation over (n=0)]sup([absolute value of [a.sub.n]][absolute value of [L.sub.n]([delta], t))] [([square root of (r x R)]).sup.n] ([([square root of (r/R)]).sup.n]), (27)

or

[mathematical expression not reproducible] (28)

or

[mathematical expression not reproducible] (29)

Proceeding to limits, we obtain

[[rho].sub.o]([alpha],[beta]) [less than or equal to] [[bar.[rho]].sub.o] ([alpha],[beta]). (30)

Combining (24) and (30), we get

[[rho].sub.o]([alpha], [beta]) [less than or equal to] [mu]([alpha], [beta]). (31)

Theorem 2. Let f(t, z) = [[summation].sup.[infinity].sub.n=0][a.sub.n][L.sub.n]([delta], t)[z.sup.n] such that

[mathematical expression not reproducible] (32)

then f is the restriction of analytic function in ([D.sub.R] x [D.sub.R]) (R > 1) and its ([alpha], [beta])-order [delta]([alpha], [beta]) = [mu]([alpha], [beta]).

Proof. Since f(0, z) = [[summation].sup.[infinity].sub.n=0][a.sub.n][L.sub.n]([delta], 0)[z.sup.n] is an analytic function in [D.sub.R], we have

[mathematical expression not reproducible] (33)

Using [L.sub.n]([delta], 0) = [n.sup.[delta]]/[GAMMA]([delta] + 1), we get

[mathematical expression not reproducible] (34)

which is necessary and sufficient condition for f [member of] A([D.sub.R] x [D.sub.R]). So, for every 1 < r < R, the series [[summation].sup.[infinity].sub.n=0][a.sub.n][L.sub.n]([delta],t)[z.sup.n] is convergent in [D.sub.R] x [D.sub.R] when [[summation].sup.[infinity].sub.n=0][a.sub.n][L.sub.n]([delta], t)[z.sup.n] is analytic in [D.sub.R] x [D.sub.R].

Now we have to prove that [mu]([alpha], [beta]) is the ([alpha], [beta])-order of f(t,z).

In order to complete the proof by Lemma 1, it is only to show that [rho]([alpha], [beta]) [greater than or equal to] [mu]([alpha], [beta]). In view of the definition of [rho]([alpha], [beta]), we have, for every [delta] > 0, that there exists 1 < [r.sub.[epsilon]] < R such that, for every [r.sub.[epsilon]] < r < R,

log M(r, r, f) [less than or equal to] [[alpha].sup.-1] [([rho]([alpha], [beta]) + [epsilon]) [beta](R/(R - r))]. (35)

Using Cauchy's estimate of analytic functions, we have

[absolute value of ([[partial derivative].sup.n]f/[partial derivative][z.sup.n]](0,0))] [less than or equal to] n[r.sup.-n] M(r, r, f) (36)

with the coefficients formula of the Taylor expansion,

[[partial derivative].sup.n]f/[partial derivative][z.sup.n](0,0) = [a.sub.n][L.sub.n]([delta],0) n!; (37)

we get M(r, r, f) [greater than or equal to] [absolute value of [a.sub.n]][absolute value of ([L.sub.n]([delta], 0))][r.sup.n]. Since [absolute value of ([L.sub.n]([delta], 0))] [greater than or equal to] 1, it gives

[absolute value of [a.sub.n]][r.sup.n] [less than or equal to] M(r, r, f). (38)

Now using (35) in (38), we get

[mathematical expression not reproducible] (39)

The minimum value of right-hand side is estimated at

[mathematical expression not reproducible] (40)

Using the properties of functions [alpha] and [beta],

[alpha](x/[[beta].sup.-1](c[alpha](x))) [equivalent] (1 + o(1))[alpha](x); (41)

for c > 0, x [right arrow] [infinity], and the properties of logarithm, we get

log([absolute value of [a.sub.n][[n.sup.[delta]]/[GAMMA]([delta] + 1)][R.sup.n]) [less than or equal to] [C.sub.1](n/[[beta].sup.-1]([alpha](n)/([rho] + [epsilon]))), (42)

where [C.sub.1] is a constant. Hence,

[beta]([C.sub.1]n/log([absolute value of [a.sub.n]]([n.sup.[delta]]/[GAMMA]([delta] + 1))[R.sup.n])) [less than or equal to] [alpha](n)/([rho] + [epsilon]). (43)

Proceeding to limit supremum as n [right arrow] [infinity], we obtain

[mu]([alpha], [beta]) [less than or equal to] [rho]([alpha], [beta]). (44)

Hence the proof is completed.

Example B. f(t,z) = [[summation].sup.[infinity].sub.m+n=0][(1 + m).sup.l][(1 + n).sup.l] exp{[m.sup.[mu]][n.sup.[mu]]}[t.sup.m][z.sup.n] [member of] A([D.sub.1] x [D.sub.1]) has ([alpha], [beta])-order ([mu]/(1 - [mu])) for [alpha](x) = log x and [beta](x) = x, where l is the positive integer.

Let f(t, z) = [[summation].sup.[infinity].sub.n=0][a.sub.n][L.sub.n]([delta], t)[z.sup.n] be analytic function of ([alpha], [beta])-order [rho] = [rho]([alpha], [beta]) and write

[mathematical expression not reproducible] (45)

Lemma 3. Let f(t,z) = [[summation].sup.[infinity].sub.n=0][a.sub.n][L.sub.n]([delta],t)[z.sup.n]. For every 1 < r < R,

[mathematical expression not reproducible] (46)

then

[sigma]([alpha], [beta]) [less than or equal to] [[bar.[sigma]].sub.1] ([alpha], [beta]). (47)

Proof. Following the same reasoning as in the proof of Lemma 1, we obtain

[mathematical expression not reproducible] (48)

Applying the limit supremum as r [right arrow] R, we get

[sigma]([alpha], [beta]) [less than or equal to] [[bar.[sigma]].sub.1]([alpha], [beta]). (49)

Theorem 4. Let f(t,z) = [[summation].sup.[infinity].sub.n=0][a.sub.n][L.sub.n]([delta], t)[z.sup.n] be of finite generalized ([alpha], [beta])-order [rho]([alpha], [beta]) and

[mathematical expression not reproducible] (50)

Then f is the restriction of an analytic function in ([D.sub.R] x [D.sub.R]) (R > 1) and its ([alpha], [beta])-type [sigma]([alpha], [beta]) = T([alpha], [beta]).

Proof. We have proven in Theorem 2 that f is the restriction of an analytic function in ([D.sub.R] x [D.sub.R]). Now, in order to complete the proof, first we shall prove that [sigma]([alpha], [beta]) [less than or equal to] T([alpha],[beta]). In view of the definition of T, for every [epsilon] > 0, there exists n [greater than or equal to] [n.sub.[epsilon]]:

[mathematical expression not reproducible] (51)

since

log([absolute value of [a.sub.n]][[n.sup.[delta]]/[GAMMA]([delta] + 1)][r.sup.n]) [less than or equal to] n log(r/R) + log([absolute value of [a.sub.n]][[n.sup.[delta]]/[GAMMA]([delta] + 1)][R.sup.n]). (52)

From inequality (51), we get

log([absolute value of [a.sub.n]][[n.sup.[delta]]/[GAMMA]([delta] + 1)][r.sup.n]) [less than or equal to] n log (r/R) + (n/[[[beta].sup.-1][([alpha](n)/([bar.T])].sup.1/[rho]]). (53)

For every 1 < r < R and r sufficiently close to R, we put

[phi](x, r) = x log (r/R) + (x/[[beta].sup.-1][[([alpha](x)/([bar.T]))].sup.1/[rho]]). (54)

The maximum value of [phi](x, r) is reached at

x = [x.sub.r]

= [[alpha].sup.-]{[[[bar.T][beta][1 - d log ([[beta].sup.-1][([alpha](x)[bar.T]).sup.1/[rho]]/d(log x)/log(R/r)]].sup.[rho]]}. (55)

Using the relation

log(r/R) = log ([r - R/R] + 1) ~ [r - R/R] (56)

(as (r - R)/R [right arrow] 0) and

[absolute value of (d(log([[beta].sup.-1]([alpha][(x)/[bar.T]).sup.1/[rho]]))/d(log x))] [less than or equal to] [k.sub.o], (57)

where [k.sub.o] is a positive constant, it gives

[x.sub.r] = (1 + o(1)) [[alpha].sup.-1]([bar.T][([beta](R/R - r)).sup.[rho]]). (58)

Now, from relation (53), we have

log ([absolute value of [a.sub.n]][[n.sup.[delta]]/[GAMMA]([delta] + 1)][r.sup.n]) [less than or equal to] sup [phi](x, r) (59)

or

[mathematical expression not reproducible] (60)

Since R/(R - r) > 1, it gives

log ([absolute value of [a.sub.n]][[n.sup.[delta]]/[GAMMA]([delta] + 1)][r.sup.n]) [less than or equal to] [k.sub.1][[alpha].sup.-1] ([bar.T][([beta](R/R - r)).sup.[rho]]). (61)

Then

logM(r, r, f) [less than or equal to] [k.sub.1][[alpha].sup.-1] ([bar.T][([beta](R/R - r)).sup.[rho]]), (62)

or

[alpha](log M(r, r, f))/[([beta](R/(R - r))).sup.[rho]] [less than or equal to] [bar.T], (63)

or

[sigma]([alpha], [beta]) [less than or equal to] T. (64)

The inequality obviously holds for T = [infinity]. Now we shall prove that [sigma]([alpha], [beta]) [greater than or equal to] T([alpha],[beta]). Let [sigma]([alpha],[beta]) < [infinity]. In view of definition of [sigma]([alpha], [beta]), we have, for every [epsilon] > 0, that there exist 1 < [r.sub.[epsilon] < R, such that, for every r > [r.sub.[epsilon]] (R > r > [r.sub.[epsilon]] > 1),

log M(r, r, f) [less than or equal to] [[alpha].sup.-1]([bar.[sigma]][([beta](R/R - r)).sup.[rho]]), [bar.[sigma]] = [sigma] + [epsilon]. (65)

Now, using (51), we get

[mathematical expression not reproducible] (66)

[mathematical expression not reproducible] (67)

or

[1/n]log([absolute value of [a.sub.n]][[n.sup.[delta]]/[GAMMA]([delta] + 1)[R.sup.n]) [less than or equal to] (1 + o(1)) log (R/r), (68)

as n [right arrow] [infinity]. Now, using log(.R/r) ~ (R - r)/r as r [right arrow] R, we have

[1/n]log ([absolute value of [a.sub.n]][[n.sup.[delta]]/[GAMMA]([delta] + 1)[R.sup.n]) [less than or equal to] (1 + o(1)) log ((R/r) - 1). (69)

Set

[r/R] = [[beta].sup.-1][[([alpha](n)/([bar.[sigma]]))].sup.1/[rho]]/1 + [[beta].sup.-1] [[([alpha](n)/([bar.[sigma]]))].sup.1/[rho]], (70)

in above inequality, we get

[1/n] log ([absolute value of [a.sub.n]][[n.sup.[delta]]/[GAMMA]([delta] + 1)[R.sup.n]) [less than or equal to] (1/[[beta].sup.-1][[([alpha](n)/([bar.[sigma]]))].sup.1/[rho]] (71)

or

[alpha](n)/[beta](n/log([absolute value of [a.sub.n]]([n.sup.[delta]]/[GAMMA]([delta] - 1))[R.sup.n])) [less than or equal to] [bar.[sigma]] = [sigma] + [epsilon]. (72)

Applying the limit supremum as n [right arrow] [infinity], we obtain

[sigma]([alpha], [beta]) [greater than or equal to] T ([alpha],[beta]). (73)

The result is obviously true for [sigma]([alpha], [beta]) = [infinity]. This completes the proof.

Example C. [mathematical expression not reproducible].

https://doi.org/10.1155/2017/8138309

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

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Devendra Kumar (1,2)

(1) Department of Mathematics, Faculty of Sciences, Al-Baha University, P.O. Box 1988, Al-Aqiq, Al-Baha 65431, Saudi Arabia

(2) Department of Mathematics, Research and Post Graduate Studies, MMH College, Model Town, Ghaziabad 201 001, India

Correspondence should be addressed to Devendra Kumar; d_kumar001@rediffmail.com

Received 20 August 2017; Revised 22 October 2017; Accepted 5 November 2017; Published 15 November 2017