ABSTRACT

The purpose of this paper is to add some complements to the general theory of higher-order types of asymptotic variation developed in two previous papers so as to complete our elementary (but not too much!) theory in view of applications to the theory of finite asymptotic expansions in the real domain, the asymptotic study of ordinary differential equations and the like. The main results concern: 1) a detailed study of the types of asymptotic variation of an infinite series so extending the results known for the sole power series; 2) the type of asymptotic variation of a Wronskian completing the many already-published results on the asymptotic behaviors of Wronskians; 3) a comparison between the two main standard approaches to the concept of “type of asymptotic variation”: via an asymptotic differential equation or an asymptotic functional equation; 4) a discussion about the simple concept of logarithmic variation making explicit and completing the results which, in the literature, are hidden in a quite-complicated general theory.

Keywords:

Higher-Order Regularly-Varying Functions, Higher-Order Rapidly-Varying Functions, Smoothly-Varying Functions, Exponentially-Varying Functions, Logarithmically-Varying Functions, Asymptotic Differential Equations, Asymptotic Functional Equations, Asymptotic Variation of Wronskians

1. Introduction

This paper is a direct continuation of [1] [2] and contains some complements to the theory developed therein with the purpose of completing the general theory. This first section, besides a list of notations, contains a summary of the various involved classes of functions and their main characterizations.

­ §2 contains a detailed study of the possible types of asymptotic variation of a convergent infinite series ${\displaystyle {\sum }_{n=1}^{\infty }}{c}_n{\varphi }_n\left(x\right)$ where ${\left\{{\varphi }_n\left(x\right)\right\}}_n$ is an asymptotic sequence at $+\infty$ and each $\left\{{\varphi }_n\left(x\right)\right\}$ has a definite index of variation. Two quite different situations occur depending on whether

${\varphi }_1\left(x\right)\gg \cdots \gg {\varphi }_n(x)\gg \cdots ,x\to +\infty ,\text{or}{\varphi }_1\left(x\right)\ll \cdots \ll {\varphi }_n(x)\ll \cdots ,x\to +\infty .$

The first case is elementary whereas, in treating the second case, we give non-trivial extensions of one known result for power series.

­ §3 contains results concerning the type of asymptotic variation of a Wronskian whose arguments are functions with a definite index of asymptotic variation at $+\infty$ so completing the extensive study of the asymptotic behaviors of Wronskians developed in two previous papers. The obtained results are quite natural and are based on the asymptotic study of a Vandermonde determinant with a gap in the exponents, a study which parallels the analogous investigation for standard Vandermondians in the previous papers.

­ §4 contains a comparison between the two main standard approaches to the concept of “type of asymptotic variation”: via an asymptotic differential equation or an asymptotic functional equation. The theory developed in [1] [2] is necessarily based on asymptotic differential equations in order to define higher-order types of variation for differentiable functions, whereas the more general Karamata theory is based on asymptotic functional equations. We show that for a function with a monotonic derivative the two approaches coincide for each one of the studied classes of functions, a result already known for regular variation.

­ §5 contains a discussion about the concept of logarithmic variation starting from suitable asymptotic functional equations and showing, as in the previous section, the equivalence with corresponding asymptotic differential equations. Some of the results may be found in the literature but hidden in a quite-complicated general theory. The studied concept (namely, three related concepts) completes the list of the fundamental types of asymptotic variation in the way that we wished to systematize this theory.

­ §6 gives results on the inverse of a function with a definite type of exponential variation. The results require some calculations and are clarified by the concepts of logarithmic variation.

­ §7 contains some minor complements to the theory.

­ §8 contains the conclusions about the whole theory developed in three papers.

­ §9 contains a few bibliographical notes and a list of corrections for [1] [2] .

Here is a list of general notations used in [1] [2] .

­ $\mathbb{N}:=\left\{1,2,\cdots \right\}$ ; $\mathbb{Z}:=\left\{0,\pm 1,\pm 2,\cdots \right\}$ ;

­ $ℝ:=\text{realline}$ ; $\overline{ℝ}\equiv \text{extendedrealline}\mathrm{<mo>\; :\; </mo>}=ℝ\cup \left\{\pm \infty \right\}$ ;

­ $f\in A{C}^0\left(I\right)\equiv AC\left(I\right)\iff f$ is absolutely continuous on each compact subinterval of the interval I;

­ $f\in A{C}^k\left(I\right)\iff f^{\left(k\right)}\in AC\left(I\right)$ ;

­ For $f\in A{C}^k\left(I\right)$ we write “ ${\lim }_{x\to x_0}{f}^{\left(k+1\right)}\left(x\right)$” meaning that x runs through the points wherein $f^{\left(k+1\right)}$ exists as a finite number; $f\left(+\infty \right):={\lim }_{x\to +\infty }f\left(x\right)$.

­ The differentiation operators: $Df\left(x\right):=f^{\prime }\left(x\right);D^{k}f\left(x\right):=f^{(k)}\left(x\right)$.

­ The logarithmic derivative: $D_{\mathcal{l}}f:=f^{\prime }/f$.

­ Hardy’s notations:

“ $f\left(x\right)\ll g\left(x\right),x\to x_0$” or, equivalently “ $g\left(x\right)\gg f\left(x\right),x\to x_0$” stands for $f\left(x\right)=o\left(g\left(x\right)\right),x\to x_0$ ;

“ $f\left(x\right)\preccurlyeq g\left(x\right)\mathrm{,}x\to x_0$” or, equivalently “ $g\left(x\right)\succcurlyeq f\left(x\right)\mathrm{,}x\to x_0$” stands for $f\left(x\right)=O\left(g\left(x\right)\right),x\to x_0$.

­ The relation of “asymptotic similarity”, “ $f\left(x\right)\asymp g\left(x\right)\mathrm{,}x\to x_0$” means that

$c_1\left|g\left(x\right)\right|\le \left|f\left(x\right)\right|\le c_2\left|g\left(x\right)\right|$ $\forall x$ in a deleted neighborhood of $x_0$ ( $c_i=\text{constant}>0$ ). (1.1)

­ The relation of “asymptotic equivalence”:

$f\left(x\right)\sim g\left(x\right)\mathrm{,}x\to x_0$ stands for $f\left(x\right)=g\left(x\right)\left[1+o\left(1\right)\right],x\to x_0$.

­ The relation:

$f\left(x\right)=+\infty \left(g\left(x\right)\right)\mathrm{,}x\to x_0\left(x\in \mathcal{I}\right)\stackrel{\text{def}}{\iff }(\begin{array}{l}f\left(x\right)=h\left(x\right)g\left(x\right)\forall x\text{near}{x}_0\mathrm{,}\\ \underset{x\to x_0\mathrm{,}x\in \mathcal{I}}{\lim }h\left(x\right)=+\infty \mathrm{<mo>\; ;\; </mo>}\end{array}$ (1.2)

and a similar definition for notation $f\left(x\right)=-\infty \left(g\left(x\right)\right)\mathrm{,}x\to x_0\left(x\in \mathcal{I}\right)$. In particular:

$f\left(x\right)=\pm \infty \left(1\right)\mathrm{,}x\to x_0\left(x\in \mathcal{I}\right)\stackrel{\text{def}}{\iff }\underset{x\to x_0\mathrm{,}x\in \mathcal{I}}{\lim }f\left(x\right)=\pm \infty \mathrm{.}$ (1.3)

­ Factorial powers:

${\alpha }^{\underset{\_}{0}}\mathrm{<mo>\; :\; </mo>}=\mathrm{1\; <mo>\; ;\; </mo>}{\alpha }^{\underset{\_}{1}}\mathrm{<mo>\; :\; </mo>}=\alpha \mathrm{<mo>\; ;\; </mo>}{\alpha }^{\underset{\_}{k}}\mathrm{<mo>\; :\; </mo>}=\alpha \left(\alpha -1\right)\cdots \left(\alpha -k+1\right)\mathrm{<mo>\; ;\; </mo>}\alpha \in ℂ\mathrm{,}k\in \mathbb{N}\mathrm{<mo>\; ;\; </mo>}$ (1.4)

where ${\alpha }^{\underset{\_}{k}}$ is termed the “k-th falling ( $\equiv$ decreasing) factorial power of $\alpha$”. Notice that we have defined $0^{\underset{\_}{0}}:=1$.

­ Everywhere the symbol “ $\log x$” stands for “ ${\log }_{\text{e}}\left(x\right)$” := “the natural logarithm” of x.

­ Notation for the iterated natural logarithm:

$(\begin{array}{l}{\mathcal{l}}_k\left(x\right)\mathrm{<mo>\; :\; </mo>}=\underset{k}{\underset{︸}{\log (\log (\cdots (\log }}x)\cdots ))\mathrm{,}k\ge \mathrm{1,}\left(\text{defined}\text{for}x\text{large}\text{enough}\right)\mathrm{<mo>\; ;\; </mo>}{\mathcal{l}}_0\left(x\right)\mathrm{<mo>\; :\; </mo>}=x\mathrm{<mo>\; ;\; </mo>}\\ \frac{\text{d}}{\text{d}x}\left({\mathcal{l}}_k\left(x\right)\right)={\left({\displaystyle \underset{i=0}{\overset{k-1}{\prod }}}{\mathcal{l}}_i\left(x\right)\right)}^{-1},k\ge 1.\end{array}$ (1.5)

For the reader’s convenience we give a list of the special classes of functions characterized in [1] [2] mentioning only the main facts to be used in the present paper.

Classes of functions and their main characterizations.

(I) (Index of asymptotic variation). If $f\in AC\left[T\mathrm{,}+\infty \right)$,$f$ ultimately > 0, its index of asymptotic variation at $+\infty$ is defined as the value of the following limit (assumed to exist):

$\underset{x\to +\infty }{\lim }x{f}^{\prime }\left(x\right)/f\left(x\right)=\{\begin{array}{ll}0\hfill & \left(\text{slow variation at}+\infty \right)\mathrm{,}\hfill \\ \alpha \in ℝ\backslash \left\{0\right\}\hfill & \left(\text{regular variation at}+\infty \right)\mathrm{,}\hfill \\ \pm \infty \hfill & \left(\text{rapid variation at}+\infty \right)\mathrm{.}\hfill \end{array}$ (1.6)

(II) (Higher-order regular variation). A function $f\in A{C}^{n-1}\left[T\mathrm{,}+\infty \right)\mathrm{,}n\ge 1$ is termed “regularly varying at $+\infty$ (in the strong sense) of order n” if each of the functions $\left|f\right|\mathrm{,}\left|f^{\prime }\right|\mathrm{,}\cdots \mathrm{,}\left|f^{\left(n-1\right)}\right|$ never vanishes on a neighborhood of $+\infty$ and is regularly varying at $+\infty$ with its own index of variation. If this is the case we use notation

$f\in \left\{{\mathcal{R}}_{\alpha }\left(+\infty \right)\text{of}\text{order}n\right\}\mathrm{,}\alpha \mathrm{<mo>\; :\; </mo>}=“\text{the}\text{index}\text{of}f”\in ℝ\mathrm{.}$ (1.7)

If $f\in \left\{{\mathcal{R}}_{\alpha }\left(+\infty \right)\text{of}\text{order}n\right\}\mathrm{,}n\ge 1$, then relations

$\begin{array}{c}{f}^{\left(k\right)}\left(x\right)/f\left(x\right)=\alpha \left(\alpha -1\right)\cdots \left(\alpha -k+1\right)x^{-k}+o\left(x^{-k}\right)\\ \equiv {\alpha }^{\underset{\_}{k}}{x}^{-k}+o\left(x^{-k}\right)\mathrm{,}x\to +\infty \mathrm{,}1\le k\le n\mathrm{,}\end{array}$ (1.8)

hold true whichever $\alpha \in ℝ$ may be. The indexes of the derivatives are subject to the restrictions specified in ( [1]; Prop. 2.6, p. 796); in particular:

$f\in \left\{{\mathcal{R}}_0\left(+\infty \right)\text{of}\text{order}n\right\}\mathrm{,}n\ge 2\Rightarrow (\begin{array}{l}{f}^{\prime }\left(x\right)=o\left(x^{-1}f\left(x\right)\right)\\ f^{″}\left(x\right)={\alpha }_1{x}^{-1}{f}^{\prime }\left(x\right)\left[1+o\left(1\right)\right]\text{with}{\alpha }_1\le -\mathrm{1\; <mo>\; ;\; </mo>}\end{array}$ (1.9)

where ${\alpha }_1$ is the index of $f^{\prime }$ and the index of $f^{\left(k\right)}$ is “ ${\alpha }_1-k+1$” for $k\ge 2$. Notice that the last derivative involved in (1.8), i.e. $f^{\left(n\right)}$, may have an arbitrary sign if ${\alpha }^{\underset{\_}{n}}=0$.

(III) (Smooth variation). Relations in (1.8) characterize higher-order regular variation only for $\alpha \cancel{\in }\left\{\mathrm{0,1,}\cdots \mathrm{,}n-2\right\}$ ; as these relations are basic in the applications their validity defines the following concept.

A function $f\in A{C}^{n-1}\left[T\mathrm{,}+\infty \right)\mathrm{,}n\ge 1$,$f\left(x\right)\ne 0$ $\forall x$ large enough, is termed “smoothly varying at $+\infty$ of order n and index $\alpha$” if the relations in (1.8), referred to $\left|f\right|$, are satisfied. We denote this class by: { $S{\mathcal{R}}_{\alpha }\left(+\infty \right)$ of order n}. The following inclusions obtain:

$(\begin{array}{l}\left\{{\mathcal{R}}_{\alpha }\left(+\infty \right)\text{of}\text{order}n\right\}=\left\{S{\mathcal{R}}_{\alpha }\left(+\infty \right)\text{of}\text{order}n\right\}\text{if}n=1\text{or}\left\{n\ge \mathrm{2,}\alpha \ne \mathrm{0,1,}\cdots \mathrm{,}n-2\right\}\mathrm{<mo>\; ;\; </mo>}\\ \left\{{\mathcal{R}}_{\alpha }\left(+\infty \right)\text{of}\text{order}n\right\}⊊\left\{S{\mathcal{R}}_{\alpha }\left(+\infty \right)\text{of}\text{order}n\right\}\text{otherwise}\mathrm{<mo>\; ;\; </mo>}\end{array}$ (1.10)

the reason of the last strict inclusion being that some derivatives of a smoothly-varying function may vanish or change sign infinitely often. The following sets of asymptotic relations, for a fixed $\alpha \in ℝ$, are equivalent to each other:

$x^k{f}^{\left(k\right)}\left(x\right)/f\left(x\right)=\alpha \left(\alpha -1\right)\cdots \left(\alpha -k+1\right)+o\left(1\right)\mathrm{,}x\to +\infty \mathrm{,}1\le k\le n\mathrm{<mo>\; ;\; </mo>}$ (1.11)

$(\begin{array}{l}x{f}^{\prime }\left(x\right)/f\left(x\right)=\alpha +o\left(1\right),x\to +\infty ;\\ {\left(x{f}^{\prime }\left(x\right)/f\left(x\right)\right)}^{\left(k\right)}=o\left(x^{-k}\right),x\to +\infty ,1\le k\le n.\end{array}$ (1.12)

(IV) (Rapid variation of first order). A function $f\in A{C}^1\left[T\mathrm{,}+\infty \right)$ is called “rapidly varying at $+\infty$ of order 1 (in the strong restricted sense)” if:

$(\begin{array}{ll}f\left(x\right)\mathrm{,}{f}^{\prime }\left(x\right)\ne 0\hfill & \forall x\text{large enough}\mathrm{<mo>\; ;\; </mo>}\hfill \\ f\left(x\right)/f^{\prime }\left(x\right)=o\left(x\right)\mathrm{,}\hfill & x\to +\infty \mathrm{<mo>\; ;\; </mo>}\hfill \\ {\left(f\left(x\right)/f^{\prime }\left(x\right)\right)}^{\prime }=o\left(1\right)\mathrm{,}\hfill & x\to +\infty \mathrm{<mo>\; ;\; </mo>}\hfill \end{array}$ (1.13a)

or, equivalently, if:

$(\begin{array}{ll}f\left(x\right)\mathrm{,}{f}^{\prime }\left(x\right)\ne 0\hfill & \forall x\text{large enough}\mathrm{<mo>\; ;\; </mo>}\hfill \\ f^{″}\left(x\right)/f^{\prime }\left(x\right)\sim f^{\prime }\left(x\right)/f\left(x\right)\mathrm{,}\hfill & x\to +\infty \mathrm{<mo>\; ;\; </mo>}\hfill \end{array}$ (1.13b)

which imply $f^{″}\left(x\right)\ne 0$ for almost all x large enough.

(Rapid variation of higher order). A function $f\in A{C}^n\left[T\mathrm{,}+\infty \right)$ is called “rapidly varying at $+\infty$ of order $n\ge 2$ (in the strong restricted sense)” if all the functions $f\mathrm{,}{f}^{\prime }\mathrm{,}\cdots \mathrm{,}{f}^{\left(n-1\right)}$ are rapidly varying at $+\infty$ in the above-specified sense and this amounts to say that the following conditions hold true as $x\to +\infty$ :

$f^{\left(k\right)}\left(x\right)\ne 0\forall x\text{large enoughand}0\le k\le n\mathrm{<mo>\; ;\; </mo>}$ (1.14)

$f\left(x\right)/f^{\prime }\left(x\right)=o\left(x\right);f^{\prime }\left(x\right)/f^{″}\left(x\right)=o\left(x\right);\cdots ;f^{\left(n-1\right)}\left(x\right)/f^{\left(n\right)}\left(x\right)=o\left(x\right);$ (1.15)

${\left(f\left(x\right)/f^{\prime }\left(x\right)\right)}^{\prime }=o\left(1\right);{\left(f^{\prime }\left(x\right)/f^{″}\left(x\right)\right)}^{\prime }=o\left(1\right);\cdots ;{\left(f^{\left(n-1\right)}\left(x\right)/f^{\left(n\right)}\left(x\right)\right)}^{\prime }=o\left(1\right).$ (1.16)

If $f$ is rapidly varying at $+\infty$ of order $n\ge 2$ in the previous sense then all the functions $f\mathrm{,}{f}^{\prime }\mathrm{,}\cdots \mathrm{,}{f}^{\left(n-1\right)}$ belong to the same class, either ${\mathcal{R}}_{-\infty }\left(+\infty \right)$ or ${\mathcal{R}}_{+\infty }\left(+\infty \right)$, hence we shall use notation $f\in \left\{{\mathcal{R}}_{\pm \infty }\left(+\infty \right)\text{of}\text{order}n\right\}$ to denote that $f$ enjoys the properties in (1.14)-(1.15)-(1.16) plus the corresponding value $\pm \infty$ of the limit in (1.6). For an $f\in A{C}^n\left[T\mathrm{,}+\infty \right)$ satisfying (1.14) we have the characterizations that conditions in (1.16) hold true, i.e.

$f\in \left\{{\mathcal{R}}_{+\infty }\left(+\infty \right)\text{of}\text{order}n\right\}\cup \left\{{\mathcal{R}}_{-\infty }\left(+\infty \right)\text{of}\text{order}n\right\}\mathrm{,}$

if and only if the following equivalent sets of conditions are satisfied:

$(\begin{array}{l}{f}^{\prime }\left(x\right)/f\left(x\right)\sim f^{″}\left(x\right)/f^{\prime }\left(x\right)\sim \cdots \sim f^{\left(n\right)}\left(x\right)/f^{\left(n-1\right)}\left(x\right)\sim f^{\left(n+1\right)}\left(x\right)/f^{\left(n\right)}\left(x\right)\mathrm{,}\\ \text{i}\mathrm{.}\text{e}\mathrm{.}{D}_{\mathcal{l}}\left(f^{\left(k\right)}\left(x\right)\right)\sim D_{\mathcal{l}}\left(f\left(x\right)\right)\mathrm{,}x\to +\infty \mathrm{,}1\le k\le n\mathrm{<mo>\; ;\; </mo>}\end{array}$ (1.17)

$f^{\left(k+2\right)}\left(x\right)\sim {\left(f^{\left(k+1\right)}\left(x\right)\right)}^2/f^{\left(k\right)}\left(x\right)\mathrm{,}x\to +\infty \mathrm{,}0\le k\le n-\mathrm{1\; <mo>\; ;\; </mo>}$ (1.18)

It follows that even $f^{\left(n+1\right)}\left(x\right)\ne 0$ for almost all x large enough.

(V) (Types of exponential variation). If $f\in A{C}^{n-1}\left[T\mathrm{,}+\infty \right)$ then:

$\begin{array}{l}f\in \left\{{\mathcal{E}}_0\left(+\infty \right)\text{of}\text{order}n\right\}\\ \iff (\begin{array}{l}{f}^{\left(k\right)}\left(x\right)\ne 0\forall x\text{large}\text{enough}\text{and}0\le k\le n-\mathrm{1,}\\ f\left(x\right)\gg f^{\prime }\left(x\right)\gg \cdots \gg f^{\left(n-1\right)}\left(x\right)\gg f^{\left(n\right)}\left(x\right),x\to +\infty ;\end{array}\end{array}$ (1.19)

$\begin{array}{l}f\in \left\{{\mathcal{E}}_c\left(+\infty \right)\text{of}\text{order}n\right\}\mathrm{,}c\in ℝ\backslash \left\{0\right\}\\ \iff (\begin{array}{l}f\left(x\right)\ne 0\forall x\text{large}\text{enough}\mathrm{,}\\ f^{\left(k\right)}\left(x\right)\sim c^{k}f\left(x\right)\mathrm{,}x\to +\infty \mathrm{,}1\le k\le n\mathrm{<mo>\; ;\; </mo>}\end{array}\end{array}$ (1.20)

$\begin{array}{l}f\in \left\{{\mathcal{E}}_{\pm \infty }\left(+\infty \right)\text{of}\text{order}n\right\}\\ \iff (\begin{array}{l}f\left(x\right)\ne 0\forall x\text{large}\text{enough},\\ f^{\left(n\right)}\left(x\right)\gg f^{\left(n-1\right)}\left(x\right)\gg \cdots \gg f^{\prime }\left(x\right)\gg f\left(x\right),x\to +\infty ,\end{array}\end{array}$ (1.21)

wherein the correct index “ $+\infty$” or “ $-\infty$” is determined by the single limit “ ${\lim }_{x\to +\infty }{f}^{\prime }\left(x\right)/f\left(x\right)$”.

For $c=0$ there is no sign-restriction on the highest-order derivative $f^{\left(n\right)}$, whereas for $c\ne 0$ also $f^{\left(n\right)}$ turns out to be ultimately of one strict sign. More precisely, if $f\left(x\right)>0$ and $f\in \left\{{\mathcal{E}}_c\left(+\infty \right)\text{of}\text{order}n\right\}$ then:

$(\begin{array}{l}-\infty \le c<0\Rightarrow \underset{x\to +\infty }{\lim }{\left(-1\right)}^k{f}^{\left(k\right)}\left(x\right)=+\infty ,0\le k\le n;\\ 0<c\le +\infty \Rightarrow \underset{x\to +\infty }{\lim }{f}^{\left(k\right)}\left(x\right)=+\infty ,0\le k\le n.\end{array}$ (1.22)

Notice that in our definition of higher-order variation $f$ is allowed to be either >0 or <0, the essential point being that it ultimately assumes only one strict sign.

The reader must remember that this is a semiexpository paper like [1] [2] and, as such, some elementary or known facts are explicitly reported or proved to have an exposition self-contained and easily-read.

2. Types of Asymptotic Variation of Infinite Series

In [1] and [2] there are some results about the index of variation of a linear combination of functions belonging to one of the previously-studied classes; in this section we give some results about the type of asymptotic variation of an infinite series of such functions. We know from ( [1]; formula (2.27), p. 784) that

$\left\{{\varphi }_i\in {\mathcal{R}}_{{\alpha }_i}\left(+\infty \right),{\alpha }_i\in ℝ,c_i>0,1\le i\le n\right\}\Rightarrow {\displaystyle \underset{i=1}{\overset{n}{\sum }}}{c}_i{\varphi }_i\in {\mathcal{R}}_{\max {\alpha }_i}\left(+\infty \right),$ (2.1)

and similar results, with some restrictions, hold true when rapid variation is involved as stated in ( [1]; Prop. 2.3-(I), pp.788-789). Hence, when investigating the possible types of asymptotic variation of an infinite linear combination ${\displaystyle {\sum }_{i=1}^{\infty }}{c}_i{\varphi }_i\left(x\right),c_i>0$, there must be an essential difference between the two circumstances

${\varphi }_1\left(x\right)\gg \cdots \gg {\varphi }_n\left(x\right)\gg \cdots ,x\to +\infty ,$ (2.2)

${\varphi }_1\left(x\right)\ll \cdots \ll {\varphi }_n\left(x\right)\ll \cdots ,x\to +\infty .$ (2.3)

The simplest case is (2.2) and here are two elementary results extending Proposition 2.3-(II) in ( [1]; p. 789), with no restriction on the signs of the coefficients $c_i$. The C ¹-regularity assumption simplifies the exposition.

Proposition 2.1. Let the functions ${\varphi }_n\in C^1\left[T\mathrm{,}+\infty \right)\mathrm{,}n\in \mathbb{N}$, form the asymptotic scale (2.2), let ${\varphi }_1\left(x\right)\ne 0\forall x$ and let ${\left\{c_n\right\}}_n$ be a given sequence of arbitrary real numbers with $c_1\ne 0$.

(I) Assume the following further conditions:

$\left|{\varphi }_n\left(x\right)\right|\le A_n\left|{\varphi }_1\left(x\right)\right|q\left(x\right)\forall x\ge T\forall n\ge 2;{\displaystyle \underset{n=1}{\overset{\infty }{\sum }}}\left|c_n\right|A_n<+\infty ;$ (2.4)

$\left|{{\varphi }^{\prime }}_n\left(x\right)\right|\le B_n\left|{{\varphi }^{\prime }}_1\left(x\right)\right|r\left(x\right)\forall x\ge T\forall n\ge 2;{\displaystyle \underset{n=1}{\overset{\infty }{\sum }}}\left|c_n\right|B_n<+\infty ;$ (2.5)

where both $q\mathrm{,}r$ are suitable nonnegative functions such that

$q\mathrm{,}r\text{bounded}\text{on}\left[T\mathrm{,}+\infty \right)\text{and}=o\left(1\right)\mathrm{,}x\to +\infty \mathrm{.}$ (2.6)

Then the two series

${\displaystyle \underset{n=1}{\overset{\infty }{\sum }}}{c}_n{\varphi }_n(x),{\displaystyle \underset{n=1}{\overset{\infty }{\sum }}}{c}_n{{\varphi }^{\prime }}_n\left(x\right)$ (2.7)

are absolutely and uniformly convergent on each bounded interval of $\left[T\mathrm{,}+\infty \right)$. If $f\left(x\right)$ is the sum of the first series then $f\in C^1\left[T\mathrm{,}+\infty \right)$,$f^{\prime }\left(x\right)={\displaystyle {\sum }_{n=1}^{\infty }}{c}_n{{\varphi }^{\prime }}_n\left(x\right)$ and

$f^{\prime }\left(x\right)/f\left(x\right)\sim {{\varphi }^{\prime }}_1\left(x\right)/{\varphi }_1\left(x\right)\mathrm{,}x\to +\infty \mathrm{.}$ (2.8)

(II) Assume all the conditions in part (I) with the exception that (2.5) is now replaced by

$\left|{{\varphi }^{\prime }}_n\left(x\right)/{\varphi }_n\left(x\right)\right|\le C\left|{{\varphi }^{\prime }}_1\left(x\right)/{\varphi }_1\left(x\right)\right|\forall x\ge T\forall n\ge \mathrm{2,}$ (2.9)

or, more generally, by

$\left|{{\varphi }^{\prime }}_n\left(x\right)/{\varphi }_n\left(x\right)\right|\le C_n\left|{{\varphi }^{\prime }}_1\left(x\right)/{\varphi }_1\left(x\right)\right|\forall x\ge T\forall n\ge 2,{\displaystyle \underset{n=1}{\overset{\infty }{\sum }}}\left|c_n\right|A_n{C}_n<+\infty .$ (2.10)

Then all the conclusions in part (I) still hold true. In the special case

${\varphi }_n\in {\mathcal{R}}_{{\alpha }_n}\left(+\infty \right),{\alpha }_1>\cdots >{\alpha }_n>\cdots ,\left(\text{even}{\alpha }_1=+\infty \right),$ (2.11)

relations in (2.2) are automatically satisfied ( [1]; Prop. 2.3-(III), p. 789), and the sequence ${\left\{{{\varphi }^{\prime }}_n\left(x\right)\right\}}_n$ is an asymptotic scale as well provided that “ ${\alpha }_n\ne 0\forall n$” because of relations “ ${{\varphi }^{\prime }}_n\left(x\right)\sim {\alpha }_n{x}^{-1}{\varphi }_n\left(x\right)$”.

Remarks. Conditions in (2.4), (2.5), (2.10) are a kind of uniformity respectively for the infinite families of asymptotic relations: $c_n{\varphi }_n\left(x\right)=o\left({\varphi }_1\left(x\right)\right)$,$c_n{{\varphi }^{\prime }}_n\left(x\right)=o\left({{\varphi }^{\prime }}_1\left(x\right)\right)$,$c_n{{\varphi }^{\prime }}_n\left(x\right)/{\varphi }_n\left(x\right)=O\left({{\varphi }^{\prime }}_1\left(x\right)/{\varphi }_1\left(x\right)\right)$,$n\ge 2$.

These conditions cannot be dispensed with and even in a simple case such as

${\varphi }_n\in {\mathcal{R}}_{{\alpha }_n}\left(+\infty \right);{\alpha }_1>{\alpha }_2\ge \cdots \ge {\alpha }_n\ge \cdots ;{\alpha }_n\in ℝ\backslash \left\{0\right\},$ (2.12a)

the asymptotic relations

${{\varphi }^{\prime }}_n\left(x\right)/{\varphi }_n\left(x\right)\sim \left({\alpha }_n/{\alpha }_1\right)\left({{\varphi }^{\prime }}_1\left(x\right)/{\varphi }_1\left(x\right)\right)\mathrm{,}x\to +\infty \mathrm{,}$ (2.12b)

do not in themselves grant (2.10) as shown by the counterexample of

${\varphi }_n\left(x\right)\mathrm{<mo>\; :\; </mo>}=x^{1/n}{\left(\log x\right)}^n\mathrm{,}x\ge \mathrm{1\; <mo>\; ;\; </mo>}{{\varphi }^{\prime }}_n\left(x\right)/{\varphi }_n\left(x\right)=\left(\frac{1}{n}+\frac{n}{\log x}\right)\left({{\varphi }^{\prime }}_1\left(x\right)/{\varphi }_1\left(x\right)\right)\mathrm{.}$ (2.13)

Proof. For part (I) the estimates

${\displaystyle \underset{n=k}{\overset{\infty }{\sum }}}\left|c_n{\varphi }_n\left(x\right)\right|\le \left({\displaystyle \underset{n=k}{\overset{\infty }{\sum }}}\left|c_n\right|A_n\right)\left|{\varphi }_1\left(x\right)\right|q\left(x\right)=o\left({\varphi }_1\left(x\right)\right),k\ge 2;$ (2.14)

${\displaystyle \underset{n=k}{\overset{\infty }{\sum }}}\left|c_n{{\varphi }^{\prime }}_n\left(x\right)\right|\le \left({\displaystyle \underset{n=k}{\overset{\infty }{\sum }}}\left|c_n\right|B_n\right)\left|{{\varphi }^{\prime }}_1\left(x\right)\right|r\left(x\right)=o\left({{\varphi }^{\prime }}_1\left(x\right)\right),k\ge 2;$ (2.15)

imply the assertions concerning the convergence of the two series due to the local boundedness of ${\varphi }_1\mathrm{,}{{\varphi }^{\prime }}_1\mathrm{,}q\mathrm{,}r$, and moreover:

$f\left(x\right)=c_1{\varphi }_1\left(x\right)+{\displaystyle \underset{n=2}{\overset{\infty }{\sum }}}{c}_n{\varphi }_n\left(x\right)\sim c_1{\varphi }_1\left(x\right);f^{\prime }\left(x\right)=c_1{\varphi }_1\left(x\right)+{\displaystyle \underset{n=2}{\overset{\infty }{\sum }}}{c}_n{{\varphi }^{\prime }}_n\left(x\right)\sim c_1{{\varphi }^{\prime }}_1\left(x\right).$ (2.16)

For part (II) instead of (2.15) we now have:

$\begin{array}{c}{\displaystyle \underset{n=k}{\overset{\infty }{\sum }}}\left|c_n{{\varphi }^{\prime }}_n\left(x\right)\right|={\displaystyle \underset{n=k}{\overset{\infty }{\sum }}}\left|c_n{\varphi }_n\left(x\right)\right|\left|\frac{{{\varphi }^{\prime }}_n\left(x\right)}{{\varphi }_n\left(x\right)}\right|\le \left|\frac{{{\varphi }^{\prime }}_1\left(x\right)}{{\varphi }_1\left(x\right)}\right|{\displaystyle \underset{n=k}{\overset{\infty }{\sum }}}{C}_n\left|c_n{\varphi }_n\left(x\right)\right|\\ \le \left({\displaystyle \underset{n=k}{\overset{\infty }{\sum }}}\left|c_n\right|A_n{C}_n\right)\left|{{\varphi }^{\prime }}_1\left(x\right)\right|q\left(x\right);\end{array}$ (2.17)

for $k\ge 2$ and the subsequent conclusions are still valid. $\square$

An elementary example. For any function $\varphi$ such that:

$\varphi \in C^1\left[T,+\infty \right);\varphi \left(x\right)=o\left(1\right),x\to +\infty ;0<\varphi \left(x\right)\le M\forall x\ge T;$ (2.18)

we have the estimates:

$\begin{array}{l}{\varphi }_n\left(x\right):={\left(\varphi \left(x\right)\right)}^n={\left(\varphi \left(x\right)\right)}^{n-1}\varphi \left(x\right)\le M^{n-1}\varphi \left(x\right),n\ge 1;\\ \left|{{\varphi }^{\prime }}_n\left(x\right)/{\varphi }_n\left(x\right)\right|=n\left|{\varphi }^{\prime }\left(x\right)/\varphi \left(x\right)\right|,n\ge 2.\end{array}$ (2.19)

Whatever $M>0$ and $k\in \mathbb{N}$ :

$\{\begin{array}{l}{f}_k\left(x\right):={\displaystyle \underset{n\ge k}{\sum }}\frac{1}{n!}{\left(\varphi \left(x\right)\right)}^n=\exp \left(\varphi \left(x\right)\right)-1-\varphi \left(x\right)-\frac{1}{2!}{\left(\varphi \left(x\right)\right)}^2-\cdots -\frac{1}{\left(k-1\right)!}{\left(\varphi \left(x\right)\right)}^{k-1}~\frac{1}{k!}{\left(\varphi \left(x\right)\right)}^k,x\to +\infty ;\\ {f^{\prime }}_k\left(x\right)={\displaystyle \underset{n\ge k}{\sum }}\frac{1}{\left(n-1\right)!}{\left(\varphi \left(x\right)\right)}^{n-1}{\varphi }^{\prime }\left(x\right)~\frac{1}{\left(k-1\right)!}{\left(\varphi \left(x\right)\right)}^{k-1}{\varphi }^{\prime }\left(x\right),x\to +\infty .\end{array}$ (2.20)

And if $0<M<1$ and $k\in \mathbb{N}$ :

$\{\begin{array}{l}{g}_k\left(x\right):={\displaystyle \underset{n\ge k}{\sum }}{\left(\varphi \left(x\right)\right)}^n=\frac{1}{1-\varphi \left(x\right)}-1-\varphi \left(x\right)-{\left(\varphi \left(x\right)\right)}^2-\cdots -{\left(\varphi \left(x\right)\right)}^{k-1}~{\left(\varphi \left(x\right)\right)}^k,x\to +\infty ;\\ {g^{\prime }}_k\left(x\right)={\displaystyle \underset{n\ge k}{\sum }}n{\left(\varphi \left(x\right)\right)}^{n-1}{\varphi }^{\prime }\left(x\right)~k{\left(\varphi \left(x\right)\right)}^{k-1}{\varphi }^{\prime }\left(x\right),x\to +\infty .\end{array}$ (2.21)

In both cases “ ${f^{\prime }}_k\left(x\right)/f_k\left(x\right)~k{{\varphi }^{\prime }}_k\left(x\right)/{\varphi }_k\left(x\right)$” and “ ${g^{\prime }}_k\left(x\right)/g_k\left(x\right)~k{{\varphi }^{\prime }}_k\left(x\right)/{\varphi }_k\left(x\right)$”, hence:

$\varphi \in {\mathcal{R}}_{\alpha }\left(+\infty \right)\mathrm{,}-\infty \le \alpha \le 0\Rightarrow f_k\mathrm{,}{g}_k\in {\mathcal{R}}_{k\alpha }\left(+\infty \right)\mathrm{,}\left(k\in \mathbb{N}\right)\mathrm{,}$ (2.22)

the circumstance “ $\alpha >0$” being inconsistent with condition “ $\varphi =o\left(1\right)$”.

A less elementary example. Consider the sequence of “modified iterated logarithms”:

$(\begin{array}{l}{\bar{\mathcal{l}}}_1\left(x\right):=\log x,{\bar{\mathcal{l}}}_n\left(x\right):=\log \left(1+{\bar{\mathcal{l}}}_{n-1}\left(x\right)\right),n\ge 2;{\bar{\mathcal{l}}}_n\in C^{\infty }\left[1,+\infty \right);\\ \frac{\text{d}}{\text{d}x}{\bar{\mathcal{l}}}_n\left(x\right)={\left[x\left(1+{\bar{\mathcal{l}}}_{n-1}\left(x\right)\right)\left(1+{\bar{\mathcal{l}}}_{n-2}\left(x\right)\right)\cdots \left(1+{\bar{\mathcal{l}}}_1\left(x\right)\right)\right]}^{-1},n\ge 2;\end{array}$ (2.23)

which satisfy “ ${\bar{\mathcal{l}}}_n\left(x\right)\mathrm{,}\frac{\text{d}}{\text{d}x}{\bar{\mathcal{l}}}_n\left(x\right)>0$ for $x>1$” and:

$\begin{array}{l}{\bar{\mathcal{l}}}_1\left(x\right)\gg {\bar{\mathcal{l}}}_2\left(x\right)\gg \cdots \gg {\bar{\mathcal{l}}}_n\left(x\right)\gg \cdots ;\\ \frac{\text{d}}{\text{d}x}{\bar{\mathcal{l}}}_1\left(x\right)\gg \frac{\text{d}}{\text{d}x}{\bar{\mathcal{l}}}_2\left(x\right)\gg \cdots \gg \frac{\text{d}}{\text{d}x}{\bar{\mathcal{l}}}_n\left(x\right)\gg \cdots ,x\to +\infty .\end{array}$ (2.24)

To get useful estimates for our aim we must take $x\ge x_0>1$ ; for instance, using the elementary inequality “ $\log \left(1+x\right)<x$ for $x>0$” we get the following estimates for $x\ge e$ :

$\{\begin{array}{l}{\bar{\mathcal{l}}}_2\left(x\right)\equiv {\mathcal{l}}_2\left(x\right)+\log \left(1+{\left(\log x\right)}^{-1}\right)<{\mathcal{l}}_2\left(x\right)+{\left(\log x\right)}^{-1},x\ge e;\\ {\bar{\mathcal{l}}}_3\left(x\right)\le {\bar{\mathcal{l}}}_2\left(x\right)<{\mathcal{l}}_2\left(x\right)+{\left(\log x\right)}^{-1},x\ge e;\\ ⋮\\ {\bar{\mathcal{l}}}_n\left(x\right)<{\mathcal{l}}_2\left(x\right)+{\left(\log x\right)}^{-1}\le {\mathcal{l}}_2\left(x\right)+1,x\ge e,n\ge 2;\end{array}$ (2.25)

whence:

$\{\begin{array}{l}0<{\bar{\mathcal{l}}}_n\left(x\right)/{\bar{\mathcal{l}}}_1\left(x\right)\le \left[{\mathcal{l}}_2\left(x\right)+1\right]{\left(\log x\right)}^{-1}\equiv q\left(x\right),\left(A_n\equiv 1\right),x\ge e,n\ge 2;\\ 0<\frac{\text{d}}{\text{d}x}{\bar{\mathcal{l}}}_n\left(x\right)/\frac{\text{d}}{\text{d}x}{\bar{\mathcal{l}}}_1\left(x\right)={\left[\left(1+{\bar{\mathcal{l}}}_{n-1}\left(x\right)\right)\left(1+{\bar{\mathcal{l}}}_{n-2}\left(x\right)\right)\cdots \left(1+{\bar{\mathcal{l}}}_1\left(x\right)\right)\right]}^{-1}\\ \le {\left(1+{\bar{\mathcal{l}}}_1\left(x\right)\right)}^{-1}\equiv r\left(x\right),\left(B_n\equiv 1\right),x\ge e,n\ge 2.\end{array}$ (2.26)

It follows that for any sequence ${\left\{c_n\right\}}_n$ such that ${\displaystyle {\sum }_{n\ge 1}}\left|c_n\right|<+\infty$, both series

$f\left(x\right):={\displaystyle \underset{n=1}{\overset{\infty }{\sum }}}{c}_n{\bar{\mathcal{l}}}_n\left(x\right),f^{\prime }\left(x\right):={\displaystyle \underset{n=1}{\overset{\infty }{\sum }}}{c}_n\frac{\text{d}}{\text{d}x}{\bar{\mathcal{l}}}_n\left(x\right)$ (2.27)

are absolutely convergent on $\left[e,+\infty \right)$ and uniformly convergent on each bounded interval and $f^{\prime }\left(x\right)/f\left(x\right)\sim \frac{\text{d}}{\text{d}x}{\bar{\mathcal{l}}}_1\left(x\right)/{\bar{\mathcal{l}}}_1\left(x\right)\mathrm{,}x\to +\infty$, i.e. $f$ is slowly varying.

Let us now examine the case (2.3), the classical case being that of a power series with an infinite radius of convergence; here coefficients of nonconstant signs may generate entire functions with no definite type of asymptotic variation at $+\infty$ such as the trigonometric functions, hence in this case we must restrict our study to positive coefficients. A problem solved in American Mathematical Monthly, [3] , states that if the function $f\left(x\right):={\displaystyle {\sum }_{n=0}^{\infty }}{c}_n{x}^n$, with $c_n\ge 0$, is defined for all x and $k\in \mathbb{N}$ and is not a polynomial then $f\in {\mathcal{R}}_{+\infty }\left(+\infty \right)$, a result that can be extended to infinite series of regularly-varying functions.

Proposition 2.2. Assumptions:

${\varphi }_n\in C^1\left[T\mathrm{,}+\infty \right)\text{and}{\varphi }_n\in {\mathcal{R}}_{{\alpha }_n}\left(+\infty \right)\mathrm{,}{\alpha }_n\in ℝ\backslash \left\{0\right\}\mathrm{,}\forall n\in \mathbb{N}\mathrm{<mo>\; ;\; </mo>}$ (2.28)

${\varphi }_n\left(x\right)>0,{{\varphi }^{\prime }}_n\left(x\right)\ne 0,c_n>0\forall x\in \left[T,+\infty \right)\forall n\in \mathbb{N};$ (2.29)

$\begin{array}{l}f\left(x\right):={\displaystyle \underset{n=1}{\overset{\infty }{\sum }}}{c}_n{\varphi }_n\left(x\right)\text{and}{f}^{\prime }\left(x\right)={\displaystyle \underset{n=1}{\overset{\infty }{\sum }}}{c}_n{{\varphi }^{\prime }}_n\left(x\right)\text{uniformly}\text{convergent}\\ \text{on}\text{each}\text{compact}\text{interval};\end{array}$ (2.30)

$x{{\varphi }^{\prime }}_n\left(x\right)={\alpha }_n{\varphi }_n\left(x\right)\left[1+A_n\left(x\right)\right]\text{with}(\begin{array}{l}\left|A_n\left(x\right)\right|\le A\left(x\right)=o\left(1\right)\mathrm{,}x\to +\infty \\ {\alpha }_n\ne 0\end{array}\mathrm{,}\forall n\in \mathbb{N}\mathrm{<mo>\; ;\; </mo>}$ (2.31)

${\displaystyle \underset{n=1}{\overset{\infty }{\sum }}}{c}_n\left|{\alpha }_n\right|{\varphi }_n\left(x\right)\text{convergent}\text{on}\text{each}\text{compact}\text{interval}.$ (2.32)

We also assume (2.3) which we express by saying that ${\left\{{\varphi }_n\left(x\right)\right\}}_n$ forms an “inverted asymptotic scale at $+\infty$”; this implies “ ${\left\{{\alpha }_n\right\}}_n$ nondecreasing” and we put $\alpha :={\lim }_n{\alpha }_n$ ; hence ${\alpha }_n\le \alpha \forall n$. We separate the cases: “ $-\infty <\alpha <0$ ; $\alpha =0$ ; $0<\alpha \le +\infty$”. In the first two cases monotonicity and restriction ${\alpha }_n\ne 0$ imply “ ${\alpha }_n<0\forall n$”; whereas in the third case we may suppose, without loss of generality, that “ ${\alpha }_n>0\forall n$” due to the fact that for the sum of a finite number of terms we have ${\displaystyle {\sum }_{n=1}^p}{\varphi }_n\left(x\right)\in {\mathcal{R}}_p\left(+\infty \right)$ and we then split the given series “ ${\displaystyle {\sum }_{n=1}^{\infty }}={\displaystyle {\sum }_{n=1}^p}+{\displaystyle {\sum }_{n>p}}$” and apply the results below. These agreements on the signs of ${\alpha }_n$ imply that:

$\left|{\displaystyle \underset{n=1}{\overset{\infty }{\sum }}}{c}_{n}x{{\varphi }^{\prime }}_n\left(x\right)\right|={\displaystyle \underset{n=1}{\overset{\infty }{\sum }}}{c}_{n}x\left|{{\varphi }^{\prime }}_n\left(x\right)\right|;\left|{\displaystyle \underset{n=1}{\overset{\infty }{\sum }}}{c}_n{\alpha }_n{\varphi }_n\left(x\right)\right|={\displaystyle \underset{n=1}{\overset{\infty }{\sum }}}{c}_n\left|{\alpha }_n\right|{\varphi }_n\left(x\right).$ (2.33)

Thesis:

$x{f}^{\prime }\left(x\right)=\left({\displaystyle \underset{n=1}{\overset{\infty }{\sum }}}{c}_n{\alpha }_n{\varphi }_n\left(x\right)\right)\cdot \left[1+r\left(x\right)\right],\text{with}\left|r\left(x\right)\right|\le A\left(x\right);$ (2.34)

$f\in {\mathcal{R}}_{\alpha }\left(+\infty \right)\text{whichever}\text{the}\text{value}\text{of}\alpha \mathrm{<mo>\; :\; </mo>}-\infty <\alpha \le +\infty \mathrm{.}$ (2.35)

Remark. Of course, instead of (2.30) it is enough to assume that ${\displaystyle \sum }{c}_n{{\varphi }^{\prime }}_n\left(x\right)$ is uniformly convergent on each compact interval and that ${\displaystyle \sum }{c}_n{\varphi }_n\left(x\right)$ converges for at least one value of x; but in the present context it may sometimes be more convenient to use the convergence of ${\displaystyle \sum }{c}_n{\varphi }_n\left(x\right)$ in order to prove the convergence of ${\displaystyle \sum }{c}_n{{\varphi }^{\prime }}_n\left(x\right)$ thanks to relations in (2.31).

Proof. From (2.31)-(2.32) we get (2.34) as:

$(\begin{array}{l}{\displaystyle \underset{n=1}{\overset{\infty }{\sum }}}{c}_{n}x{{\varphi }^{\prime }}_n\left(x\right)={\displaystyle \underset{n=1}{\overset{\infty }{\sum }}}{c}_n{\alpha }_n{\varphi }_n\left(x\right)\left[1+A_n\left(x\right)\right]={\displaystyle \underset{n=1}{\overset{\infty }{\sum }}}{c}_n{\alpha }_n{\varphi }_n\left(x\right)+{\displaystyle \underset{n=1}{\overset{\infty }{\sum }}}{c}_n{\alpha }_n{A}_n{\varphi }_n\left(x\right);\\ {\displaystyle \underset{n=1}{\overset{\infty }{\sum }}}{c}_n\left|{\alpha }_n{A}_n\right|{\varphi }_n\left(x\right)\le A\left(x\right){\displaystyle \underset{n=1}{\overset{\infty }{\sum }}}{c}_n\left|{\alpha }_n\right|{\varphi }_n\left(x\right)=o\left({\displaystyle \underset{n=1}{\overset{\infty }{\sum }}}{c}_n{\alpha }_n{\varphi }_n\left(x\right)\right).\end{array}$ (2.36)

Case “ $\mathrm{0\; <mo>\; <\; </mo>}\alpha \le +\infty$”, which implies “ ${{\varphi }^{\prime }}_n\left(x\right)>0\forall x\forall n$”. First step:

$\begin{array}{l}\underset{x\to +\infty }{\bar{\lim }}\frac{f\left(x\right)}{x{f}^{\prime }\left(x\right)}=\underset{x\to +\infty }{\bar{\lim }}\frac{{\displaystyle \underset{n\ge 1}{\sum }}{c}_n{\varphi }_n\left(x\right)}{{\displaystyle \underset{n\ge 1}{\sum }}{c}_{n}x{{\varphi }^{\prime }}_n\left(x\right)}=\underset{x\to +\infty }{\bar{\lim }}\frac{{\displaystyle \underset{n\ge 1}{\sum }}{c}_n{\varphi }_n\left(x\right)}{{\displaystyle \underset{n\ge 1}{\sum }}{c}_n{\alpha }_n{\varphi }_n\left(x\right)\left[1+A_n\left(x\right)\right]}\ge \cdots \\ \left(\text{as}0<1+A_n\left(x\right)\le 1+A\left(x\right)\text{for}x\text{large}\text{enough}\right)\\ \cdots \ge \underset{x\to +\infty }{\bar{\lim }}\frac{{\displaystyle \underset{n\ge 1}{\sum }}{c}_n{\varphi }_n\left(x\right)}{\left[1+A\left(x\right)\right]{\displaystyle \underset{n\ge 1}{\sum }}{c}_n{\alpha }_n{\varphi }_n\left(x\right)}\ge \cdots \left(\text{as}0<{\alpha }_n\le \alpha \right)\cdots \\ \cdots \ge \underset{x\to +\infty }{\bar{\lim }}\frac{1}{\left[1+A\left(x\right)\right]\alpha }=\frac{1}{\alpha }\text{if}\alpha <+\infty .\end{array}$ (2.37)