# Growth Analysis of Wronskians in terms of Slowly Changing Functions

Authors: Sanjib Kumar Datta; Tanmay Biswas; Sultan Ali
Journal: Journal of Complex Analysis
 (2013)

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## Abstract

In the paper we establish some new results depending on the comparative growth properties of composite entire or meromorphic functions using generalisedL∗-order and generalised L∗-type and Wronskians generated by one of the factors.


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## Body

## 1. Introduction, Definitions, and Notations

We denote byℂ the set of all finite complex numbers. Let f be a meromorphic function defined on ℂ. We use the standard notations and definitions in the theory of entire and meromorphic functions which are available in [1] and [2]. In the sequel we use the following notation: log[k]x=log(log[k-1]x) for k=1,2,3,… and log[0]x=x.The following definitions are well known.Definition 1.
A meromorphic functiona=a(z) is called small with respect to f if T(r,a)=S(r,f).Definition 2.
Leta1,a2,….ak be linearly independent meromorphic functions and small with respect to f. We denote by L(f)=W(a1,a2,…,ak;f) the Wronskian determinant of a1,a2,…,ak,f that is,
(1)L(f)=|a1a2...akfa1′a2′...ak′f′.....................a1(k)a2(k)...ak(k)f(k)|.Definition 3.
Ifa∈ℂ∪{∞}, the quantity
(2)δ(a;f)=1-limsupr→∞N(r,a;f)T(r,f)=liminfr→∞m(r,a;f)T(r,f)
is called the Nevanlinna deficiency of the value “a.”From the second fundamental theorem it follows that the set of values ofa∈ℂ∪{∞}, the quaintity for which δ(a;f)>0 is countable and ∑a≠∞δ(a;f)+δ(∞;f)≤2 (cf. [1, page43]). If, in particular, ∑a≠∞δ(a;f)+δ(∞;f)=2, we say that f has the maximum deficiency sum.LetL=L(r) be a positive continuous function increasing slowly, that is, L(ar)~L(r) as r→∞ for every positive constant a. Singh and Barker [3] defined it in the following way.Definition 4 (see [3]).
A positive continuous functionL(r) is called a slowly changing function if, for ε(>0),
(3)1kε≤L(kr)L(r)≤kε,forr≥r(ε)and
uniformly for k(≥1).
If further,L(r) is differentiable, the above condition is equivalent to
(4)limr→∞rL′(r)L(r)=0.Somasundaram and Thamizharasi [4] introduced the notions of L-order and L-type for entire functions. The more generalised concept for L-order and L-type for entire and meromorphic functions are L*-order and L*-type, respectively. In the line of Somasundaram and Thamizharasi [4], for any positive integer m one may define the generalised L*-order ρf[m]L* (generalised L*-lower order λf[m]L*) and generalised L*-type σf[m]L* in the following manner.Definition 5.
The generalisedL*-order ρf[m]L* and the generalised L*-lower order λf[m]L* of an entire function f are defined as
(5)ρf[m]L*=limsupr→∞log[m+1]M(r,f)log[reL(r)],λf[m]L*=liminfr→∞log[m+1]M(r,f)log[reL(r)].
Whenf is meromorphic, it can be easily verified that
(6)ρf[m]L*=limsupr→∞log[m]T(r,f)log[reL(r)],λf[m]L*=liminfr→∞log[m]T(r,f)log[reL(r)].Definition 6.
The generalisedL*-type σf[m]L* of an entire function f is defined as follows:
(7)σf[m]L*=limsupr→∞log[m]M(r,f)[reL(r)]ρf[m]L*,0<ρf[m]L*<∞.
For meromorphic f,
(8)σf[m]L*=limsupr→∞log[m-1]T(r,f)[reL(r)]ρf[m]L*,0<ρf[m]L*<∞.Form=1, we may get the classical cases {cf. [4]} of Definitions 5 and 6, respectively.Lakshminarasimhan [5] introduced the idea of the functions of L-bounded index. Later Lahiri and Bhattacharjee [6] worked on the entire functions of L-bounded index and of nonuniform L-bounded index. Since the natural extension of a derivative is a differential polynomial, in this paper we prove our results for a special type of linear differential polynomials, namely, the Wronskians. In the paper we establish some new results depending on the comparative growth properties of composite entire or meromorphic functions using generalised L*-order and generalised L*-type and wronskians generated by one of the factors.


## 2. Lemmas

In this section we present some lemmas which will be needed in the sequel.Lemma 7 (see [7]).
Iff  be meromorphic and g  be entire then for all sufficiently large values of r,
(9)T(r,f∘g)≤{1+o(1)}T(r,g)logM(r,g)T(M(r,g),f).Lemma 8 (see [8]).
Letf  be meromorphic and g entire and suppose that 0<μ<ρg≤∞. Then for a sequence of values of r tending to infinity,
(10)T(r,f∘g)≥T(exp(rμ),f).Lemma 9 (see [9]).
Letf be a transcendental meromorphic function having the maximum deficiency sum. Then
(11)limr→∞T(r,L(f))T(r,f)=1+k-kδ(∞;f).Lemma 10 (see [10]).
Iff be a transcendental meromorphic function having the maximum deficiency sum. Then the generalised L*-order (generalised L*-lower order) of L(f) and that of f are the same.Lemma 11.
Letf be a transcendental meromorphic function having the maximum deficiency sum. Then
(12)σL(f)[m]L*=(1+k-kδ(∞;f))·σf[m]L*,form=1,σL(f)[m]L*=σf[m]L*,otherwise.Proof.
By Lemmas9 and 10 we get that
(13)σL(f)L*=limsupr→∞T(r,L(f))[reL(r)]ρL(f)L*=limr→∞T(r,L(f))T(r,f)·limsupr→∞T(r,f)[reL(r)]ρfL*=(1+k-kδ(∞;f))·σfL*.
Also by Lemma 9,  limr→∞(log[m-1]T(r,L(f)))/(log[m-1]T(r,f)) exists and is equal to 1 for m>1.
Therefore(14)σL(f)[m]L*=limsupr→∞log[m-1]T(r,L(f))[reL(r)]ρL(f)L*=limr→∞log[m-1]T(r,L(f))log[m-1]T(r,f)·limsupr→∞log[m-1]T(r,f)[reL(r)]ρfL*=σf[m]L*.
Thus the lemma follows.


## 3. Theorems

In this section we present the main results of the paper.Theorem 12.
Letf be a transcendental meromorphic function having the maximum deficiency sum and g  entire such that 0<λf[m]L*≤ρf[m]L*<∞ where m is any positive integer. Then for any positive real number A,
(15)limsupr→∞log[m+1]T(exp(rA),f∘g)log[m]T(exp(rμ),L(f))+K(r,g;L)=∞,
where 0<μ<ρg and
(16)K(r,g;L)={0ifrμ=o{L(exp(exp(μrA)))}asr→∞L(exp(exp(μrA)))otherwise.Proof.
Let0<μ<μ′<ρg. Using the definition of generalised L*-lower order we obtain in view of Lemma 8 for a sequence of values of r tending to infinity that
(17)log[m]T(exp(rA),f∘g)≥log[m]T(exp(exp(rA))μ′,f)i.e.,log[m]T(exp(rA),f∘g)≥(λf[m]L*-ε)·log{exp(exp(rA))μ′·expL(exp(exp(rA))μ′)}i.e.,log[m]T(exp(rA),f∘g)≥(λf[m]L*-ε)·{(exp(rA))μ′+L(exp(exp(rA))μ′)}i.e.,log[m]T(exp(rA),f∘g)≥(λf[m]L*-ε)·{(exp(rA))μ′(1+L(exp(exp(rA))μ′)(exp(rA))μ′)}i.e.,log[m+1]T(exp(rA),f∘g)≥O(1)+μ′logexp(rA)+log{1+L(exp(exp(rA))μ′)(exp(rA))μ′}i.e.,log[m+1]T(exp(rA),f∘g)≥O(1)+μ′rA+log{1+L(exp(exp(rA))μ′)(exp(rA))μ′}i.e.,log[m+1]T(exp(rA),f∘g)≥O(1)+μ′rA+log[1+L(exp(exp(μ′rA)))exp(μ′rA)]i.e.,log[m+1]T(exp(rA),f∘g)≥O(1)+μ′rA+L(exp(exp(μrA)))-log[exp{L(exp(exp(μrA)))}]+log[1+L(exp(exp(μ′rA)))exp(μ′rA)]i.e.,log[m+1]T(exp(rA),f∘g)≥O(1)+μ′rA+L(exp(exp(μrA)))+log[1exp{L(exp(exp(μrA)))}+log+L(exp(exp(μ′rA)))exp{L(exp(exp(μrA)))}·exp(μ′rA)]i.e.,log[m+1]T(exp(rA),f∘g)≥O(1)+μ′r(A-μ)·rμ+L(exp(exp(μrA))).
Again in view of Lemma 10 we have for all sufficiently large values of r that
(18)log[m]T(exp(rμ),L(f))≤(ρL(f)[m]L*+ε)log{exp(rμ)eL(exp(rμ))}i.e.,log[m]T(exp(rμ),L(f))≤(ρL(f)[m]L*+ε){logexp(rμ)+L(exp(rμ))}i.e.,log[m]T(exp(rμ),L(f))≤(ρL(f)[m]L*+ε){rμ+L(exp(rμ))}i.e.,log[m]T(exp(rμ),L(f))-(ρL(f)[m]L*+ε)L(exp(rμ))(ρL(f)[m]L*+ε)≤rμ.
Now from  (17) and (18) it follows for a sequence of values of r tending to infinity that
(19)log[m+1]T(exp(rA),f∘g)≥O(1)+(μ′r(A-μ)ρf[m]L*+ε)[(ρf[m]L*+ε)log[m]T(exp(rμ),L(f))O(1)+(μ′r(A-μ)ρf[m]L*+ε)-(ρf[m]L*+ε)L(exp(rμ))]+L(exp(exp(μrA)))(20)i.e.,log[m+1]T(exp(rA),f∘g)log[m]T(exp(rμ),L(f))≥L(exp(exp(μrA)))+O(1)logT(exp(rμ),L(f))+μ′r(A-μ)ρf[m]L*+ε×{1-(ρf[m]L*+ε)L(exp(rμ))log[m]T(exp(rμ),L(f))}.
Again from (19) we get for a sequence of values of r tending to infinity that
(21)log[m+1]T(exp(rA),f∘g)log[m]T(exp(rμ),L(f))+L(exp(exp(μrA)))≥O(1)-μ′r(A-μ)L(exp(rμ))log[m]T(exp(rμ),L(f))+L(exp(exp(μrA)))+(μ′r(A-μ)/(ρf[m]L*+ε))log[m]T(exp(rμ),L(f))log[m]T(exp(rμ),L(f))+L(exp(exp(μrA)))+L(exp(exp(μrA)))log[m]T(exp(rμ),L(f))+L(exp(exp(μrA)))i.e.,log[m+1]T(exp(rA),f∘g)log[m]T(exp(rμ),L(f))+L(exp(exp(μrA)))≥(O(1)-μ′r(A-μ)L(exp(rμ)))/(L(exp(exp(μrA))))(log[m]T(exp(rμ),L(f))/L(exp(exp(μrA))))+1+(((μ′r(A-μ)ρf[m]L*+ε)log[m]T(exp(rμ),L(f))L(exp(exp(μrA))))×(1+L(exp(exp(μrA)))log[m]T(exp(rμ),L(f)))-1((μ′r(A-μ)ρf[m]L*+ε)log[m]T(exp(rμ),L(f))L(exp(exp(μrA)))))+11+log[m]T(exp(rμ),L(f))/L(exp(exp(μrA))).Case I. If rμ=o{L(exp(exp(μrA)))} then it follows from  (20) that
(22)limsupr→∞log[m+1]T(exp(rA),f∘g)log[m]T(exp(rμ),L(f))=∞.Case II. rμ≠o{L(exp(exp(μrA)))} then the following two subcases may arise.
Subcase (a). If (exp(exp(μrA)))=o {log[m]T(exp(rμ),L(f))}, then we get from (21) that
(23)limsupr→∞log[m+1]T(exp(rA),f∘g)log[m]T(exp(rμ),L(f))+L(exp(exp(μrA)))=∞.Subcase (b). If L(exp(exp(μrA)))~log[m]T  (exp(rμ),L(f)) then
(24)limr→∞L{exp(exp(μrA))}log[m]T(exp(rμ),L(f))=1,
and we obtain from (21) that
(25)limsupr→∞log[m+1]T(exp(rA),f∘g)log[m]T(exp(rμ),L(f))+L(exp(exp(μrA)))=∞.
Combining Cases I and II we obtain that
(26)limsupr→∞log[m+1]T(exp(rA),f∘g)log[m]T(exp(rμ),L(f))+K(r,g;L)=∞,
where
(27)K(r,g;L)={0ifrμ=o{L(exp(exp(μrA)))}asr→∞L(exp(exp(μrA)))otherwise.
This proves the theorem.Theorem 13.
Letf be a meromorphic function and g  transcendental entire such that λf[m]L*>0, ρgL*<∞ and ∑a≠∞δ(a;g)+δ(∞;g)=2 where m≥1. Then for any positive real number A,
(28)limsupr→∞log[m+1]T(exp(rA),f∘g)logT(exp(rμ),L(g))+K(r,f;L)=∞,
where 0<μ<ρg and
(29)K(r,f;L)={0ifrμ=o{L(exp(exp(μrA)))}asr→∞L(exp(exp(μrA)))otherwise.The proof is omitted because it can be carried out in the line of Theorem12.Theorem 14.
Letf be a transcendental meromorphic function with ∑a≠∞δ(a;f)+δ(∞;f)=2 and g  entire such that λf[m]L*>0,0<ρf[m]L*<∞ and 0<σgL*<∞ where m is any positive integer. If L(M(r,g))=o{rαeαL(r)} as r→∞ and for some positive α<ρgL*, then
(30)limsupr→∞log[m]T(r,f∘g)log[m]T(exp(reL(r))ρgL*,L(f))≤ρf[m]L*·σgL*λf[m]L*.Proof.
SinceT(r,g)≤log+M(r,g) and by Lemma 7 we get for all sufficiently large values of r that
(31)logT(r,f∘g)≤log{1+o(1)}+logT(M(r,g),f)i.e.,log[m]T(r,f∘g)≤o(1)+log[m]T(M(r,g),f)i.e.,log[m]T(r,f∘g)log[m]T(exp(reL(r))ρgL*,L(f))≤o(1)+log[m]T(M(r,g),f)log[m]T(exp(reL(r))ρgL*,L(f))=o(1)+log[m]T(M(r,g),f)log{M(r,g)eL(M(r,g))}·logM(r,g)+L(M(r,g))[reL(r)]ρgL*·log{exp(reL(r))ρgL*}log[m]T(exp(reL(r))ρgL*,L(f))(32)i.e.,limsupr→∞log[m]T(r,f∘g)log[m]T(exp(reL(r))ρgL*,L(f))≤limsupr→∞log[m]T(M(r,g),f)log{M(r,g)eL(M(r,g))}·limsupr→∞logM(r,g)+L(M(r,g))[reL(r)]ρgL*·limsupr→∞log{exp(reL(r))ρgL*}log[m]T(exp(reL(r))ρgL*,L(f)).
As α<ρgL*, we can choose ε(>0) in such a way that α<ρgL*-ε and since L(M(r,g))=o(rαeαL(r)) as r→∞, we obtain that
(33)limr→∞L(M(r,g))[reL(r)]ρgL*-ε=0.
Now from (32) and (33) and in view of Lemma 10 it follows that
(34)limsupr→∞log[m]T(r,f∘g)log[m]T(exp(reL(r))ρgL*,L(f))≤ρf[m]L*·σgL*·1λL(f)[m]L*i.e.,limsupr→∞log[m]T(r,f∘g)log[m]T(exp(reL(r))ρgL*,L(f))≤ρf[m]L*·σgL*·1λf[m]L*.
Thus the theorem is established.In the line of Theorem14 the following theorem can be proved.Theorem 15.
Letf be meromorphic and g transcendental entire having the maximum deficiency sum such that λgL*>0, 0<ρf[m]L*<∞ and 0<σgL*<∞ where m≥1. If L(M(r,g))=o{rαeαL(r)} as r→∞ and for some positive α<ρgL*, then
(35)limsupr→∞log[m]T(r,f∘g)logT(exp(reL(r))ρgL*,L(g))≤ρf[m]L*·σgL*λgL*.
The proof is omitted.Theorem 16.
Letf be a transcendental meromorphic function such that ρf[n]L*<∞ with n≥1 and ∑a≠∞δ(a;f)+δ(∞;f)=2. Also let g be entire. If λf∘g[m]L*=∞ for m≥1 then
(36)limr→∞log[m]T(r,f∘g)log[n]T(r,L(f))=∞.Proof.
Let us suppose that the conclusion of the theorem does not hold.Then we can find a constantβ>0 such that for a sequence of values of r tending to infinity
(37)log[m]T(r,f∘g)≤βlog[n]T(r,L(f)).
Again from the definition of ρL(f)L* it follows that for all sufficiently large values of r and in view of Lemma 10,
(38)log[n]T(r,L(f))≤(ρL(f)[n]L*+ε)log(reL(r))i.e.,log[n]T(r,L(f))≤(ρf[n]L*+ε)log(reL(r)).
Thus from (37) and (38) we have for a sequence of values of r tending to infinity that
(39)log[m]T(r,f∘g)≤β(ρf[n]L*+ε)log(reL(r))i.e.,log[m]T(r,f∘g)log(reL(r))≤β(ρf[n]L*+ε)log(reL(r))log(reL(r))i.e.,liminfr→∞log[m]T(r,f∘g)logr=λf∘g[m]L*<∞.
This is a contradiction. This proves the theorem.Remark 17.
Theorem16 is also valid with “limit superior" instead of “limit" if λf∘g[m]L*=∞ is replaced by ρf∘g[m]L*=∞ and the other conditions remain the same.Corollary 18.
Under the assumptions of Theorem16 or Remark 17,
(40)limsupr→∞log[m-1]T(r,f∘g)log[n-1]T(r,L(f))=∞.Proof.
From Theorem16 or Remark 17 we obtain for all sufficiently large values of r and for K>1,
(41)log[m]T(r,f∘g)>Klog[n]T(r,L(f)),i.e.,log[m-1]T(r,f∘g)>{log[n-1]T(r,L(f))}K,
from which the corollary follows.Remark 19.
The conditionλf∘g[m]L*=∞ in Theorem 16 and Corollary 18 is necessary which is evident from the following example.Example 20.
Letf=expz, g=z,m=n=1, and L(r)=(1/p)exp(1/r) where p is any positive real number.
Then(42)ρfL*=1<∞,λf∘gL*=1<∞,∑a≠∞‍δ(a;f)+δ(∞;f)=2.
Now taking a1=1 and a2=⋯=ak=0 we get from Definition 2 that
(43)L(f)=|a1fa1′f′|=|1expz0expz|=expz.
Now
(44)T(r,f∘g)=T(r,expz)=rπ,T(r,L(f))=T(r,expz)=rπ.
Therefore
(45)limr→∞logT(r,f∘g)logT(r,L(f))=limr→∞logr+O(1)logr+O(1)=1,limr→∞T(r,f∘g)T(r,L(f))=limr→∞(r/π)(r/π)=1,
which is a contradiction.Remark 21.
Consideringf=expz, g=z, m=n=1, A=1, L(r)=(1/p)exp(1/r) for any positive real number p and taking a1=1, a2=⋯=ak=0 in Definition 2, one can easily verify that the condition ρf∘g[m]L*=∞ in Remark 17 and Corollary 18 is essential.Theorem 22.
Letf be meromorphic and g  transcendental entire such that ρg[n]L*<∞ with n≥1 and λf∘g[m]L*=∞ where m is any positive integer. Also let ∑a≠∞δ(a;g)+δ(∞;g)=2. Then
(46)limr→∞log[m]T(r,f∘g)log[n]T(r,L(g))=∞.We omit the proof of Theorem22 because it can be carried out in the line of Theorem 16.Remark 23.
Theorem22 is also valid with “limit superior" instead of “limit" if λf∘g[m]L*=∞ is replaced by ρf∘g[m]L*=∞ and the other conditions remain the same.In the line of Corollary18 we may easily verify the following.Corollary 24.
Under the assumptions of Theorem22 or Remark 19,
(47)limsupr→∞log[m-1]T(r,f∘g)log[n-1]T(r,L(g))=∞.Remark 25.
Takingf=z,g=expz, m=n=1, A=1, L(r)=(1/p)exp(1/r) for any positive real number p and choosing a1=1, a2=⋯=ak=0 in Definition 2 we may establish the necessity of the conditions λf∘g[m]L*=∞ and ρf∘g[m]L*=∞, respectively, in Theorem 22, Remark 23, and Corollary 24.Theorem 26.
Letf be meromorphic function and g  entire function having the maximum deficiency sum such that (i)0<ρf[m]L*<∞ and (ii)σgL*<∞ where m is any positive integer. Then
(
a
) if L(M(r,g))=o{T(r,L(g))} then
(48)liminfr→∞log[m]T(r,f∘g)T(r,L(g))+L(M(r,g))≤ρf[m]L*(1+k-kδ(∞;g)),
and (b) if T(r,L(g))=o{L(M(r,g))} then
(49)liminfr→∞log[m]T(r,f∘g)T(r,L(g))+L(M(r,g))≤ρf[m]L*.Proof.
SinceT(r,g)≤log+M(r,g) in view of Lemma 7 we obtain for all sufficiently large values of r that
(50)logT(r,f∘g)≤log{1+o(1)}+logT(M(r,g),f)i.e.,log[m]T(r,f∘g)≤o(1)+log[m]T(M(r,g),f)i.e.,log[m]T(r,f∘g)≤o(1)+(ρf[m]L*+ε)×{logM(r,g)+L(M(r,g))}.
Using the definition of L*-type we obtain from (50) for all sufficiently large values of r that
(51)log[m]T(r,f∘g)≤o(1)+(ρf[m]L*+ε)(σgL*+ε){reL(r)}ρgL*+(ρf[m]L*+ε)L(M(r,g)).
Again from the definition of L*-type and in view of Lemmas 10 and 11 we get for a sequence of values of r tending to infinity that
(52)T(r,L(g))≥(σL(g)L*-ε){reL(r)}ρL(g)L*i.e.,T(r,L(g))≥{(1+k-kδ(∞;g))σgL*-ε}{reL(r)}ρgL*i.e.,{reL(r)}ρgL*≤T(r,L(g))((1+k-kδ(∞;g))σgL*-ε).
Now from (51) and (52) it follows for a sequence of values of r tending to infinity that
(53)log[m]T(r,f∘g)≤o(1)+(ρf[m]L*+ε)(σgL*+ε)×T(r,L(g))((1+k-kδ(∞;g))σgL*-ε)+(ρf[m]L*+ε)L(M(r,g))i.e.,log[m]T(r,f∘g)T(r,L(g))+L(M(r,g))≤o(1)T(r,L(g))+L(M(r,g))+(((ρf[m]L*+ε)(σgL*+ε)((1+k-kδ(∞;g))σgL*-ε))×(1+L(M(r,g))T(r,L(g)))-1((ρf[m]L*+ε)(σgL*+ε)((1+k-kδ(∞;g))σgL*-ε)))+(ρf[m]L*+ε)1+(T(r,L(g))/L(M(r,g))).
If L(M(r,g))=o{T(r,L(g))} then from (53) we get that
(54)liminfr→∞log[m]T(r,f∘g)T(r,L(g))+L(M(r,g))≤(ρf[m]L*+ε)(σgL*+ε)(1+k-kδ(∞;g))(σgL*-ε).
Since ε(>0) is arbitrary it follows from above that
(55)liminfr→∞log[m]T(r,f∘g)T(r,L(g))+L(M(r,g))≤ρf[m]L*(1+k-kδ(∞;g)).
Thus the first part of Theorem 26 follows.
Again ifT(r,L(g))=o{L(M(r,g))} then from (53) it follows that
(56)liminfr→∞log[m]T(r,f∘g)T(r,L(g))+L(M(r,g))≤(ρf[m]L*+ε).
As ε(>0) is arbitrary we obtain from the above that
(57)liminfr→∞log[m]T(r,f∘g)T(r,L(g))+L(M(r,g))≤ρf[m]L*.
Thus the second part of Theorem 26 follows.Theorem 27.
Letf be a transcendental meromorphic function having the maximum deficiency sum and g  entire such that (i)0<ρf[m]L*<∞, (ii)ρf[m]L*=ρgL*, (iii)σgL*<∞ and (iv)0<σf[m]L*<∞ where m≥1. Then
(
a
) if L(M(r,g))=o{rαeαL(r)} as r→∞ and for some positive α<ρf[m]L* then
(58)liminfr→∞logT(r,f∘g)T(r,L(f))+L(M(r,g))≤ρfL*·σgL*(1+k-kδ(∞;f))·σfL*,liminfr→∞log[m]T(r,f∘g)log[m-1]T(r,L(f))+L(M(r,g))≤ρf[m]L*·σgL*σf[m]L*,form>1,
and (b) if log[m-1]T(r,L(f))=o{L(M(r,g))} then
(59)liminfr→∞log[m]T(r,f∘g)log[m-1]T(r,L(f))+L(M(r,g))≤ρf[m]L*.Proof.
In view of condition(ii) we obtain from (51) for all sufficiently large values of r that
(60)log[m]T(r,f∘g)≤o(1)+(ρf[m]L*+ε)×(σgL*+ε){reL(r)}ρf[m]L*+(ρf[m]L*+ε)L(M(r,g)).
Again from the definition of L*-type and in view of Lemma 10 we get for a sequence of values of r tending to infinity that
(61)log[m-1]T(r,L(f))≥(σL(f)[m]L*-ε){reL(r)}ρL(f)[m]L*i.e.,log[m-1]T(r,L(f))≥(σL(f)[m]L*-ε){reL(r)}ρf[m]L*i.e.,{reL(r)}ρf[m]L*≤log[m-1]T(r,L(f))(σL(f)[m]L*-ε).
Now from (60) and (61) it follows for a sequence of values of r tending to infinity that
(62)log[m]T(r,f∘g)≤o(1)+(ρf[m]L*+ε)(σgL*+ε)×log[m-1]T(r,L(f))(σL(f)[m]L*-ε)+(ρf[m]L*+ε)L(M(r,g))i.e.,log[m]T(r,f∘g)log[m-1]T(r,L(f))+L(M(r,g))≤o(1)log[m-1]T(r,L(f))+L(M(r,g))+(((ρf[m]L*+ε)(σgL*+ε)(σL(f)[m]L*-ε))×(1+L(M(r,g))log[m-1]T(r,L(f)))-1((ρf[m]L*+ε)(σgL*+ε)(σL(f)[m]L*-ε)))+(ρf[m]L*+ε)1+(log[m-1]T(r,L(f))/L(M(r,g))).
If L(M(r,g))=o{log[m-1]T(r,L(f))} then from (62) we get that
(63)liminfr→∞log[m]T(r,f∘g)log[m-1]T(r,L(f))+L(M(r,g))≤(ρf[m]L*+ε)(σgL*+ε)(σL(f)[m]L*-ε).
Since ε(>0) is arbitrary, it follows from the above that
(64)liminfr→∞log[m]T(r,f∘g)log[m-1]T(r,L(f))+L(M(r,g))≤ρf[m]L*·σgL*σL(f)[m]L*.
Now in view of Lemma 11 we get from the above that
(65)liminfr→∞logT(r,f∘g)T(r,L(f))+L(M(r,g))≤ρfL*·σgL*(1+k-kδ(∞;f))·σfL*,form=1,liminfr→∞log[m]T(r,f∘g)log[m-1]T(r,L(f))+L(M(r,g))≤ρf[m]L*·σgL*σf[m]L*,form>1.
Thus the first part of Theorem 27 follows.
Again iflog[m-1]T(r,L(f))=o{L(M(r,g))} then from (62) it follows that
(66)liminfr→∞log[m]T(r,f∘g)log[m-1]T(r,L(f))+L(M(r,g))≤(ρf[m]L*+ε).
As ε(>0) is arbitrary, we obtain from the above that
(67)liminfr→∞log[m]T(r,f∘g)log[m-1]T(r,L(f))+L(M(r,g))≤ρf[m]L*.
Thus the second part of Theorem27 follows.



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*Source: 395067-2013-01-17.xml*