Measures of Growth of Entire Solutions of Generalized Axially Symmetric Helmholtz Equation

Abstract

For an entire solution of the generalized axially symmetric Helmholtz equation 𝜕²𝑢/𝜕𝑥²+𝜕²𝑢/𝜕𝑦²+(2𝜇/𝑦)(𝜕𝑢/𝜕𝑦)+𝑘²𝑢=0,𝜇>0, measures of growth such as lower order and lower type are obtained in terms of the Bessel-Gegenbauer coefficients. Alternative characterizations for order and type are also obtained in terms of the ratios of these successive coefficients.

1. Introduction

The solutions of the partial differential equation 𝜕²𝑢𝜕𝑥²+𝜕²𝑢𝜕𝑦²+2𝜇𝑦𝜕𝑢𝜕𝑦+𝑘²𝑢=0,𝜇>0, (1) are called the generalized axially symmetric Helmholtz equation functions (GASHEs). The GASHE function 𝑢, regular about the origin, has the following Bessel-Gegenbauer series expansion: 𝑢(𝑟,𝜃)=Γ2𝜇(𝑘𝑟)^∞−𝜇_𝑛=0𝑎_𝑛𝑛!Γ𝐽2𝜇+𝑛^𝜇+𝑛(𝑘𝑟)𝐶^𝜇_𝑛(cos𝜃), (2) where 𝑥=𝑟cos𝜃, 𝑦=𝑟sin𝜃,𝐽_𝜇+𝑛 are Bessel functions of first kind, and 𝐶^𝜇_𝑛 are Gegenbauer polynomials. A GASHE function 𝑢 is said to be entire if the series (2) converges absolutely and uniformly on the compact subsets of the whole complex plane. For 𝑢 being entire, it is known [1, page 214] that: limsup_𝑛→∞𝑎_𝑛Γ𝑛+𝜇+1^1/𝑛=0. (3)

Now we define 𝑀(𝑟,𝑢)=max_{0≤𝜃≤2𝜋}|𝑢(𝑟,𝜃)|. (4)

Following the usual definitions of order and type of an entire function of a complex variable 𝑧, the order 𝜌 and type 𝑇 of 𝑢 are defined as 𝜌(𝑢)=limsup_𝑟→∞loglog𝑀(𝑟,𝑢),𝑇(𝑢)log𝑟=limsup_𝑟→∞log𝑀(𝑟,𝑢)𝑟^𝜌(𝑢),0<𝜌(𝑢)<∞. (5)

Gilbert and Howard [2] have studied the order 𝜌(𝑢) of an entire GASHE function 𝑢 in terms of the coefficients 𝑎_𝑛’s occurring in the series expansion (2) (see also [1, Theorem 4.5.9]). It has been noticed that the coefficients characterizations for lower order and lower type of 𝑢 have not been studied so far. In this paper, we have made an attempt to bridge this gap. McCoy [3] studied the order and type of an entire function solutions of certain elliptic partial differential equation in terms of series expansion coefficients and approximation errors. Recently, Kumar [4, 5] obtained some bounds on growth parameters of entire function solutions of Helmholtz equation in 𝑅² in terms of Chebyshev polynomial approximation errors in sup-norm. In the present paper, we have considered the different partial differential equation from those of McCoy [3] and Kumar [4, 5] and obtained the growth parameters such as lower order and lower type of entire GASHE function in terms of the coefficients in its Bessel-Gegenbauer series expansion (2). Alternative characterizations for order and type are also obtained in terms of the ratios of these successive coefficients. Our approach and method are different from all these of the above authors.

2. Auxiliary Results

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

We prove the following lemma.

Lemma 1. If 𝑢 is an entire GASHE function, then for all 𝑟>0, 𝑟^∗>1, 𝐾,𝐾_∗<∞ and for all 𝑛, 𝐾_∗⁻¹𝑎_𝑛𝑘𝑛!2𝑟^𝑟_∗^𝑛Γ𝜇+(1/2)²𝜋2^2−𝜇≤𝑀(𝑟,𝑢)≤𝐾^∞_𝑛=0𝑎_𝑛𝑛!𝑘𝑟2^𝑛. (6)

Proof. First we prove right hand inequality. Using the relations 𝐽_𝜇(𝑟≤())𝑟/2^𝜇Γ,𝜇+1max_{0≤𝜃≤2𝜋}𝐶^𝜇_𝑛(≤Γcos𝜃)𝑛+2𝜇(Γ(Γ,𝑛+1))2𝜇 (7) in (2), we get 𝑀(𝑟,𝑢)≤^∞_𝑛=0𝑎_𝑛𝑛!𝑘𝑟2^𝑛Γ(𝑛+1)Γ𝑛+𝜇+1≤𝐾^∞_𝑛=0𝑎_𝑛𝑛!𝑘𝑟2^𝑛. (8)
Now to prove left hand inequality, we use the orthogonality property of Gegenbauer polynomials [1, page 173] and the uniform convergence of the series (2) as 𝑎_𝑛2^(2𝜇−1)Γ𝜇+1/2²Γ()𝑛+𝜇2𝜇𝑘𝑟^−𝜇𝐽_𝜇+𝑛()=𝑘𝑟_𝜋0sin^2𝜇𝜃𝐶^𝜇_𝑛(cos𝜃)𝑢(𝑟,𝜃)𝑑𝜃. (9)
From the series expansion of 𝐽_𝜇+𝑛(𝑘𝑟), we get 𝐽_𝜇+𝑛(𝑘𝑟)=𝑘𝑟𝑟^∞𝜇+𝑛_𝑚=0(−1)^𝑚(𝑘𝑟)^2𝑚2^2𝑚=𝑚!Γ𝑛+𝜇+𝑚+1𝑘𝑟𝑟^𝜇+𝑛1Γ×𝑛+𝜇+1_∞𝑚=0(−1)^𝑚(𝑘𝑟)^2𝑚Γ𝑛+𝜇+12^2𝑚𝑚!Γ𝑛+𝜇+𝑚+1 (10) and for 𝑛≥[(𝑘𝑟)²], where [𝑥] denotes the integral part of 𝑥, we have 𝐽_𝜇+𝑛()≥1𝑘𝑟2Γ𝑛+𝜇+1𝑘𝑟2^𝜇+𝑛. (11)
From (7), (9), and (11), for 𝑛≥[(𝑘𝑟)²], we now get 𝑎_𝑛𝑛!𝑘𝑟2^𝑛≤𝜋2^2−𝜇Γ𝑛+2𝜇Γ𝑛+𝜇+1Γ𝜇+1/2²(Γ(𝑛+1))²𝑛+𝜇𝑀(𝑟,𝑢). (12)
Since [Γ(𝑛+2𝜇)Γ(𝑛+𝜇+1)/(Γ(𝑛+1))²]^1/𝑛→1 as 𝑛→∞, we can choose constants 𝑘_∗<∞ and 𝑟_∗>1 such that (𝑛+𝜇)Γ(𝑛+2𝜇)Γ(𝑛+𝜇+1)/(Γ(𝑛+1))²≤𝐾_∗𝑟^𝑛_∗ for 𝑛≥1. Thus, for 𝑛≥[(𝑘𝑟)²], (12) gives that Γ𝜇+1/2²𝜋2^2−𝜇𝐾_∗⁻¹𝑎_𝑛𝑛!𝑘𝑟2𝑟_∗^𝑛≤𝑀(𝑟,𝑢). (13)
Hence the proof of Lemma 1 is complete.

We now define 𝑓(𝑧)=^∞_𝑛=0𝑎_𝑛(𝑘/2)^𝑛𝑧𝑛!^𝑛,𝑔(𝑧)=^∞_𝑛=0𝑎_𝑛𝑘𝑛!2𝑟_∗^𝑛𝑧^𝑛. (14)

Lemma 2. If 𝑢 is an entire GASHE function, then 𝑓 and 𝑔 are also entire functions of the complex variable 𝑧. Further Γ𝜇+1/2²𝐾_∗𝜋2^2−𝜇𝑚𝑟,𝑔≤𝑀(𝑟,𝑢)≤𝐾𝑀𝑟,𝑓, (15) where 𝑚(𝑟,𝑔)=max_𝑛[(|𝑎_𝑛|/𝑛!)(𝑘𝑟/2𝑟_∗)^𝑛] and 𝑀(𝑟,𝑓)=max_{|𝑧|≤𝑟}|𝑓(𝑧)|.

Proof. Let 𝑢 be an entire. In view of (3), we have lim_𝑛→∞𝑎_𝑛𝑘𝑛!2^𝑛1/𝑛=lim_𝑛→∞|𝑎_𝑛|𝑘𝑛!2𝑟_∗^𝑛1/𝑛=lim_𝑛→∞|𝑎_𝑛|Γ𝑛+𝜇+1^1/𝑛=0. (16)
Hence both 𝑓 and 𝑔 are entire. Inequalities in (15) follow from (6).

Lemma 3. Let 𝑓 and 𝑔 be entire functions of particular form defined by (14). Then orders and types of 𝑓 and 𝑔, respectively, are identical.

Proof. It is well known [6, pages 9–11] that if 𝜙(𝑧)=∑^∞_𝑛=0𝛼_𝑛𝑧^𝑛 is an entire function, then the order 𝜌(𝜙) and type 𝑇(𝜙) are given as 𝜌𝜙=limsup_𝑛→∞𝑛log𝑛log𝛼_𝑛⁻¹,𝑇𝜙=1𝑒𝜌𝜙limsup_𝑛→∞𝑛𝛼_𝑛^{𝜌(𝜙)/𝑛}. (17)(18)
Hence for the function 𝑓(𝑧)=∑^∞_𝑛=0(𝑘/2)^𝑛(|𝑎_𝑛|/𝑛!)𝑧^𝑛, we have 1𝜌𝑓=liminf_𝑛→∞log(𝑎_𝑛/𝑛!)(𝑘/2)^𝑛−1𝑛log𝑛=liminf_𝑛→∞log𝑎_𝑛⁻¹+𝑛log𝑛log𝑛!−()𝑛log𝑛𝑛log𝑘/2𝑛log𝑛=liminf_𝑛→∞log𝑎_𝑛⁻¹.𝑛log𝑛 (19)
Similarly, for 𝑔(𝑧)=∑^∞_𝑛=0(|𝑎_𝑛|/𝑛!)(𝑘/2𝑟_∗)^𝑛𝑧^𝑛, we have 1𝜌𝑔=liminf_𝑛→∞log𝑎_𝑛/𝑛!𝑘/2𝑟_∗^𝑛−1𝑛log𝑛=liminf_𝑛→∞log𝑎_𝑛⁻¹+log𝑛!−log𝑘/2𝑟_∗𝑛log𝑛=liminf_𝑛→∞log𝑎_𝑛⁻¹.𝑛log𝑛 (20)
It follows that 𝜌(𝑓)=𝜌(𝑔). Since 𝑓 and 𝑔 have the same order, using (18), we can easily see that 𝑇(𝑓)=𝑇(𝑔). Hence the proof is complete.

In analogy with the definitions of order and type, we define lower order 𝜆 and lower type 𝑡 as 𝜆(𝑢)=liminf_𝑟→∞loglog𝑀(𝑟,𝑢),𝑡(𝑢)log𝑟=liminf_𝑟→∞log𝑀(𝑟,𝑢)𝑟^𝜌(𝑢),0<𝜌(𝑢)<∞. (21)

Theorem 4. Let 𝑢 be an entire GASHE function of order 𝜌(𝑢), lower order 𝜆(𝑢), type 𝑇(𝑢), and lower type 𝑡(𝑢). If 𝑓 and 𝑔 are entire functions as defined above, then 𝜌𝑓(𝑢)𝑔,𝑇𝑓(𝑢)𝑔,𝜆𝑔(𝑢)𝑓,𝑡𝑔(𝑢)𝑓.=𝜌=𝜌=𝑇=𝑇≤𝜆≤𝜆≤𝑡≤𝑡 (22)(23)(24)(25)

Proof. Using (15) we get limsup_{𝑟→∞inf}loglog𝑚𝑟,𝑔log𝑟≤limsup_{𝑟→∞inf}loglog𝑀(𝑟,𝑢)log𝑟≤limsup_{𝑟→∞inf}loglog𝑀(𝑟,𝑢).log𝑟 (26)
In view of [6, page 13] for an entire function 𝜙 of finite order we have, log𝑀𝑟,𝜙≃log𝑚𝑟,𝑓as𝑟→∞. (27)
Now from above relation (26), we obtain 𝜌𝑔≤𝜌(𝑢)≤𝜌𝑓,𝜆𝑔≤𝜆(𝑢)≤𝜆𝑓. (28)
Since 𝜌(𝑔)=𝜌(𝑓), it proves (22) and (24).
Denoting by 𝜌 the common value of order of 𝑓, 𝑔, and 𝑢, we have from (15), limsup_𝑟→∞log𝑚𝑟,𝑔𝑟^𝜌≤limsup_𝑟→∞log𝑀(𝑟,𝑢)𝑟^𝜌≤limsup_𝑟→∞log𝑀𝑟,𝑓𝑟^𝜌. (29)
Hence by Lemma 3 we obtain (23). Similarly, we can prove (25).

Lemma 5. If (𝛽_𝑛/𝛽_𝑛+1) forms a nondecreasing function of 𝑛, then (𝛾_𝑛/𝛾_𝑛+1) and (𝛿_𝑛/𝛿_𝑛+1) also form a nondecreasing function of 𝑛, where 𝛽_𝑛=|𝑎_𝑛|/Γ(𝑛+𝜇+1),𝛾_𝑛=(|𝑎_𝑛|/𝑛!)(𝑘/2)^𝑛,𝛿_𝑛=(|𝑎_𝑛|/𝑛!)(𝑘/2𝑟_∗)^𝑛.

Proof. We have 𝛾_𝑛𝛾_𝑛+1=𝑎_𝑛𝑎_𝑛+1(𝑛+1)!𝑘𝑛!2⁻¹=𝛽_𝑛𝛽_𝑛+1(𝑛+1)Γ𝑛+𝜇+1Γ𝑘𝑛+𝜇+22⁻¹. (30)
Let 𝑝(𝑥)=(𝑥+1)Γ𝑥+𝜇+1Γ≃()𝑥+𝜇+2𝑥+1𝑥+𝜇+1⁻¹ (31)
[byΓ(x+a)/Γx∼x^aasx→∞]log𝑝(𝑥)=log(𝑥+1)−log𝑥+𝜇+1. (32)
By logarithmic differentiation, we get 𝑝^′(𝑥)𝑝(𝑥)=1−1𝑥+1.𝑥+𝜇+1 (33)
Let 𝑤(𝑥)=1/(𝑥+1),𝑤(𝑥)−𝑤(𝑥+𝜇)>0 for any 𝑥>0.
Hence 𝑤(𝑥) is a decreasing function and subsequently 𝑝^′(𝑥)>0 for 𝑥>0. Hence (𝛾_𝑛/𝛾_𝑛+1) is nondecreasing if (𝛽_𝑛/𝛽_𝑛+1) is nondecreasing. Similarly we can prove the result for (𝛿_𝑛/𝛿_𝑛+1).

3. Main Results

Now we prove the following theorem.

Theorem 6. Let 𝑢 be an entire GASHE function of order 𝜌(𝑢)(0<𝜌(𝑢)<∞), type 𝑇(𝑢), and lower type 𝑡(𝑢). Then liminf_𝑛→∞𝑘2𝑟_∗^𝜌(𝑢)𝑛𝜌(𝑢)𝛽_𝑛+1𝛽_𝑛^𝜌(𝑢)≤𝑡(𝑢)≤𝑇(𝑢)≤limsup_𝑛→∞𝑘2^𝜌(𝑢)𝑛𝜌(𝑢)𝛽_𝑛+1𝛽_𝑛^𝜌(𝑢). (34)
Further, if (𝛽_𝑛+1/𝛽_𝑛) forms a nondecreasing function of 𝑛 for all 𝑛>𝑛₀, then limsup_𝑛→∞𝑛𝜌(𝑢)𝛽_𝑛+1𝛽_𝑛^𝜌(𝑢)(𝑢2≤𝑒𝑇)𝑘^𝜌(𝑢),limsup_𝑛→∞log𝑛log𝛽_𝑛/𝛽_𝑛+1=𝜌(𝑢). (35)

Proof. If 𝜙(𝑧)=∑^∞_𝑛=0𝛼_𝑛𝑧^𝑛 is an entire function of order 𝜌(𝜙), type 𝑇(𝜙), and lower type 𝑡(𝜙), then we have [7, Theorem 1] liminf_𝑛→∞𝑛𝜌𝜙𝛼_𝑛+1𝛼_𝑛^𝜌(𝜙)≤𝑡𝜙≤𝑇𝜙≤limsup_𝑛→∞𝑛𝜌𝜙𝛼_𝑛+1𝛼_𝑛^𝜌(𝜙). (36)
Applying right hand inequality to 𝑓(𝑧)=∑𝛾_𝑛𝑧^𝑛, we get 𝑇𝑓≤limsup_𝑛→∞𝑛𝜌𝑓𝑎_𝑛+1𝑎_𝑛𝑛!()!𝑘𝑛+12^𝜌(𝑓)=limsup_𝑛→∞𝑛𝜌𝑎_𝑛+1Γ𝑛+𝜇+1Γ𝑛+𝜇+2𝑎_𝑛×Γ𝑛+𝜇+2Γ)𝑘𝑛+𝜇+1(𝑛+12^𝜌. (37)
Since Γ𝑛+𝜇+2Γ1𝑛+𝜇+1()∼𝑛+1𝑛+𝜇+1()𝑇(𝑢)𝑓𝑛+1→1as𝑛→∞,=𝑇≤limsup_𝑛→∞𝑘2^𝜌𝑛𝜌𝛽_𝑛+1𝛽_𝑛^𝜌. (38)
To prove left hand inequality in (34), we consider the entire function 𝑔(𝑧)=∑^∞_𝑛=0𝛿_𝑛𝑧^𝑛. Then liminf_𝑛→∞𝑛𝜌𝑎_𝑛+1()!𝑘𝑛+12𝑟_∗𝑛!𝑎_𝑛^𝜌≤𝑡𝑔=liminf_𝑛→∞𝑛𝜌𝑎_𝑛+1𝑎_𝑛Γ𝑛+𝜇+1Γ×Γ𝑛+𝜇+2𝑛+𝜇+2Γ()𝑘𝑛+𝜇+1𝑛+12𝑟_∗^𝜌=liminf_𝑛→∞𝑛𝜌𝛽_𝑛+1𝛽_𝑛𝑘2𝑟_∗^𝜌≤𝑡𝑔≤𝑡(𝑢). (39)
Thus the proof of (34) is complete. To prove (35), consider an entire function 𝜙(𝑧)=∑^∞_𝑛=0𝛼_𝑛𝑧^𝑛 of order 𝜌(𝜙) and type 𝑇(𝜙). If |𝛼_𝑛/𝛼_𝑛+1| forms a nondecreasing function of 𝑛 for 𝑛>𝑛₀, then we know ([8], [9, Theorem  2]) that 𝜌𝜙=limsup_𝑛→∞log𝑛log𝛼_𝑛/𝛼_𝑛+1. (40)
Further, we have [8, Theorem 3] limsup_𝑛→∞𝑛𝜌𝜙𝛼_𝑛+1𝛼_𝑛^𝜌(𝜙)≤𝑒𝑇𝜙. (41)
Now let us suppose that (𝛽_𝑛/𝛽_𝑛+1) forms a nondecreasing function of 𝑛 for 𝑛>𝑛₀. From Lemma 5, (𝛾_𝑛/𝛾_𝑛+1) also forms a nondecreasing function of 𝑛 for 𝑛>𝑛₀. Using (40) to 𝑓(𝑧)=∑^∞_𝑛=0𝛾_𝑛𝑧^𝑛, we get 𝜌𝑓=limsup_𝑛→∞log𝑛log𝛾_𝑛/𝛾_𝑛+1=limsup_𝑛→∞×𝛽log𝑛log^𝑛𝛽_𝑛+1()𝑘+log𝑛+1−log𝑛+𝜇+1−log2⁻¹=limsup_𝑛→∞log𝑛log𝛽_𝑛/𝛽_𝑛+1,𝛽log^𝑛/𝛽_𝑛+1𝑘2𝛽∼log^𝑛𝛽_𝑛+1as𝑛→∞. (42)𝜌𝑓=limsup_𝑛→∞log𝑛log𝛾_𝑛/𝛾_𝑛+1=limsup_𝑛→∞log𝑛log𝛽_𝑛/𝛽_𝑛+1,𝛽log^𝑛/𝛽_𝑛+1𝑘2𝛽∼log^𝑛𝛽_𝑛+1as𝑛→∞. (43)
Now using (41) for 𝑓(𝑧)=∑^∞_𝑛=0𝛾_𝑛𝑧^𝑛, we get limsup_𝑛→∞𝑛𝜌𝑓𝛽_𝑛+1𝛽_𝑛𝑛+𝜇+1()𝑘𝑛+12^𝜌(𝑓)≤𝑒𝑇𝑓. (44)
Since 𝜌(𝑓)=𝜌, 𝑇(𝑓)=𝑇, thus, we get limsup_𝑛→∞𝑛𝜌𝛽_𝑛+1𝛽_𝑛^𝜌2≤𝑒𝑘^𝜌𝑇. (45)
Hence the proof is complete.

Theorem 7. Let 𝑢 be an entire GASHE function of order 𝜌, 0<𝜌<∞, lower order 𝜆, and lower type 𝑡. If (𝛽_𝑛/𝛽_𝑛+1) forms a nondecreasing function of 𝑛 for 𝑛>𝑛₀, then 𝜆=liminf_𝑛→∞𝑛log𝑛log𝛽_𝑛⁻¹,𝑡=liminf_𝑛→∞𝑛𝛽𝑒𝜌^{𝑛𝜌/𝑛}. (46)(47)

Proof. For entire function 𝜙(𝑧)=∑^∞_𝑛=0𝛼_𝑛𝑧^𝑛, if |𝛼_𝑛/𝛼_𝑛+1| forms a nondecreasing function of 𝑛 for 𝑛>𝑛₀, then we have ([8], [10, Theorem  2]) 𝜆𝜙=liminf_𝑛→∞𝑛log𝑛log𝛼_𝑛⁻¹. (48)
Let (𝛽_𝑛/𝛽_𝑛+1) forms a nondecreasing function 𝑛 for 𝑛>𝑛₀. Applying Lemma 5 and (48) to 𝑓(𝑧)=∑^∞_𝑛=0𝛾_𝑛𝑧^𝑛, we get 𝜆𝑓=liminf_𝑛→∞𝑛log𝑛log(|𝑎_𝑛|/𝑛!)(𝑘/2)^𝑛−1=liminf_𝑛→∞𝑛log𝑛log𝛽_𝑛⁻¹. (49)
Similarly, using Lemma 5 and (48) for entire function 𝑔(𝑧)=∑^∞_𝑛=0𝛿_𝑛𝑧^𝑛, we have 𝜆𝑔=liminf_𝑛→∞𝑛log𝑛log|𝑎_𝑛|/𝑛!𝑘/(2𝑟_∗)^𝑛−1=liminf_𝑛→∞𝑛log𝑛log𝛽_𝑛⁻¹−log(𝑛+1)^𝜇−𝑛log𝑘/2𝑟_∗=liminf_𝑛→∞𝑛log𝑛log𝛽_𝑛⁻¹. (50)
The result (46) is now followed by (24) and above two relations for 𝜆(𝑓) and 𝜆(𝑔).
If 𝜙(𝑧)=∑^∞_𝑛=0𝛼_𝑛𝑧^𝑛 is an entire function of order 𝜌(𝜙), lower type 𝑡(𝜙), and |𝛼_𝑛/𝛼_𝑛+1| forms a nondecreasing function of 𝑛 for 𝑛>𝑛₀, then by a result of Shah [11], we have 𝑡𝜙=liminf_𝑛→∞𝑛𝜙𝛼𝑒𝜌^{𝑛𝜌(𝜙)/𝑛},𝑡𝑔=liminf_𝑛→∞𝑛𝑔𝛽𝑒𝜌^{𝑛𝜌(𝑔)/𝑛}. (51) Equation (47) now follows in view of (22) and (25). Hence the proof is complete.

Theorem 8. Let 𝑢 be an entire GASHE function of lower order 𝜆, and let |𝛼/𝛼_𝑛+1| forms a nondecreasing function of 𝑛 for 𝑛>𝑛₀. Then 𝜆=liminf_𝑛→∞log𝑛log𝛽_𝑛/𝛽_𝑛+1, (52)

Proof. For an entire function 𝜙(𝑧)=∑^∞_𝑛=0𝛼_𝑛𝑧^𝑛, from [12, Corollary, page 312], we get 𝜆𝜙=liminf_𝑛→∞𝑛log𝑛log𝛼_𝑛/𝛼_𝑛+1. (53) provided |𝛼_𝑛/𝛼_𝑛+1| forms a nondecreasing function of 𝑛 for 𝑛>𝑛₀. Applying this condition on {𝛽_𝑛}, we can easily show, as in Theorem 6, that 𝜆𝑓=liminf_𝑛→∞log𝑛log𝛽_𝑛/𝛽_𝑛+1. (54)
Applying (53) to 𝑔(𝑧)=∑^∞_𝑛=0𝛿_𝑛𝑧^𝑛 also, we have 𝜆𝑔=liminf_𝑛→∞log𝑛log𝛽_𝑛/𝛽_𝑛+1. (55)
The relation (48) now follows on using (24). Hence the proof is complete.

Acknowledgments

The authors are very much thankful to the reviewers for giving fruitful comments to improve the paper. This work is supported by University Grants Commission, New Delhi, India, under the Grant no. F. 11-27-2004 (SA-I).