\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 156, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/156\hfil Hyers-Ulam stability ]
{Hyers-Ulam stability for Gegenbauer differential  equations}

\author[S.-M. Jung \hfil EJDE-2013/156\hfilneg]
{Soon-Mo Jung}  % in alphabetical order

\address{Soon-Mo Jung \newline
 Mathematics Section,
 College of Science and Technology,
 Hongik University, 339-701 Sejong, South Korea}
\email{smjung@hongik.ac.kr}

\thanks{Submitted June 19, 2013. Published July 8, 2013.}
\subjclass[2000]{39B82, 41A30, 34A30, 34A25, 34A05}
\keywords{Gegenbauer differential equation; Hyers-Ulam stability;
\hfill\break\indent  power series method; second order differential equation}

\begin{abstract}
 Using the power series method, we  solve the non-homogeneous
 Gegenbauer differential equation
 $$
 ( 1 - x^2 )y''(x) + n(n-1)y(x) = \sum_{m=0}^\infty a_m x^m.
 $$
 Also we prove the Hyers-Ulam stability for the Gegenbauer
 differential equation.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks

\section{Introduction}

Let $Y$  be a normed linear space and  $I$ be an open subinterval of
$\mathbb{R}$.
If for any function $f : I \to Y$ satisfying the differential
inequality
$$
\big\| a_n(x)y^{(n)}(x) + a_{n-1}(x)y^{(n-1)}(x) + \cdots +
       a_1(x)y'(x) + a_0(x)y(x) + h(x) \big\| \leq \varepsilon
$$
for all $x \in I$ and for some $\varepsilon \geq 0$, there
exists a solution $f_0 : I \to Y$ of the differential equation
$$
a_n(x)y^{(n)}(x) + a_{n-1}(x)y^{(n-1)}(x) + \cdots +
a_1(x)y'(x) + a_0(x)y(x) + h(x) = 0
$$
such that $\| f(x) - f_0(x) \| \leq K(\varepsilon)$ for any
$x \in I$, where $K(\varepsilon)$ depends on $\varepsilon$ only,
then we say that the above differential equation satisfies the
Hyers-Ulam stability (or the local Hyers-Ulam stability if the
domain $I$ is not the whole space $\mathbb{R}$).
We may apply these terminologies for other differential
equations.
For more detailed definition of the Hyers-Ulam stability,
refer the reader to \cite{czerwik0,hir,jung2}.


Apparently Ob\a{l}oza \cite{ob1,ob2} was the first author who investigated the
Hyers-Ulam stability of linear differential equations.
Here, we cite a result by Alsina and Ger \cite{ag}:
If a differentiable function $f : I \to \mathbb{R}$ is a
solution of the differential inequality
$| y'(x) - y(x) | \leq \varepsilon$, where $I$ is an open
subinterval of $\mathbb{R}$, then there exists a solution
$f_0 : I \to \mathbb{R}$ of the differential equation
$y'(x) = y(x)$ such that $| f(x) - f_0(x) | \leq 3\varepsilon$
for any $x \in I$.
This result by Alsina and Ger was generalized by Takahasi,
Miura and Miyajima \cite{tmm}. 
They proved   that the Hyers-Ulam stability holds
for the Banach space valued differential equation
$y'(x) = \lambda y(x)$ (see also \cite{mjt,motn,popa}).

Using the conventional power series method, the author
investigated the general solution of the inhomogeneous linear
first-order differential equation
$$
y'(x) - \lambda y(x) = \sum_{m=0}^\infty a_m (x-c)^m,
$$
where $\lambda$ is a complex number and the convergence radius
of the power series is positive.
This result was applied for proving an approximation property
of exponential functions in a neighborhood of $c$
(see \cite{115} and  \cite{jung4,jung3}).

Throughout this article, we assume that $\rho_1$ is a positive
real number or infinity.
In Section 2, using an idea from \cite{115}, we 
investigate the general solution of the inhomogeneous Gegenbauer
differential equation
\begin{equation}
\big( 1-x^2 \big) y''(x) + n(n-1)y(x) = \sum_{m=0}^\infty a_m x^m,
\label{eq:1.1}
\end{equation}
where the power series has a radius of convergence greater than or equal to
 $\rho_1$.
Moreover, we prove the Hyers-Ulam stability of the Gegenbauer
differential equation \eqref{eq:2.1} in a certain class of 
analytic functions.

\section{General solution of \eqref{eq:1.1}}

For an integer $n \geq 2$, the second-order ordinary
differential equation
\begin{equation}
\big( 1-x^2 \big) y''(x) + n(n-1)y(x) = 0
\label{eq:2.1}
\end{equation}
is a kind of the ultraspherical or Gegenbauer differential
equation and has a general solution of the form
$y(x) = C_1 J_n(x) + C_2 H_n(x)$, where we denote by
$J_n(x)$ and $H_n(x)$ the Gegenbauer functions which are
expressed by using the Legendre functions of the first and
second kind as follows:
$$
J_n(x) = \frac{P_{n-2}(x) - P_n(x)}{2n-1},\quad 
H_n(x) = \frac{Q_{n-2}(x) - Q_n(x)}{2n-1}.
$$
The Gegenbauer differential equation \eqref{eq:2.1} is
encountered in hydrodynamics when describing axially symmetric
Stokes flows  \cite{polyanin}.
We recall that $\rho_1$ is a positive real number or 
infinity.

\begin{theorem}\label{thm:2.1}
Let $n$ be an integer greater than $1$ and let $\rho_1$ be
the radius of convergence of power series
$\sum_{m=0}^\infty a_m x^m$.
Define $\rho := \min \{ 1, \rho_1 \}$.
Then every solution $y : (-\rho, \rho) \to \mathbb{C}$ of the
inhomogeneous Gegenbauer differential equation \eqref{eq:1.1}
can be expressed as
\begin{equation}
y(x) = y_h(x) + \sum_{m=2}^\infty c_m x^m, \label{eq:2.2}
\end{equation}
where the coefficients $c_m$'s are given by
\begin{gather*}
c_{2m}   = \sum_{k=0}^{m-1} \frac{(2k)! a_{2k}}{(2m)!}
            \prod_{i=k+1}^{m-1} (2i-n)(2i+n-1), \\
c_{2m+1} = \sum_{k=0}^{m-1} \frac{(2k+1)! a_{2k+1}}{(2m+1)!}
            \prod_{i=k+1}^{m-1} (2i-n+1)(2i+n) 
\end{gather*}
for each $m \in \mathbb{N}$ and $y_h(x)$ is a solution of the
Gegenbauer differential equation \eqref{eq:2.1}.
\end{theorem}


\begin{proof}
Since each solution of \eqref{eq:1.1} can be expressed as a
power series in $x$, we put
$y(x) = \sum_{m=0}^\infty c_m x^m$ in \eqref{eq:1.1} to obtain
\begin{align*}
&\big( 1-x^2 \big)y''(x) + n(n-1)y(x) \\
&= \sum_{m=0}^\infty
     \big[ (m+2)(m+1) c_{m+2} - (m-n)(m+n-1) c_m \big] x^m \\
&= \sum_{m=0}^\infty a_m x^m,
\end{align*}
from which we obtain the  recurrence formula
\begin{align}
(m+2)(m+1) c_{m+2} - (m-n)(m+n-1) c_m = a_m \label{eq:rec}
\end{align}
for all $m \in \mathbb{N}_0$.

Now we prove that the formula
\begin{equation}
\begin{aligned}
c_{2m}  &= \sum_{k=0}^{m-1} \frac{(2k)! a_{2k}}{(2m)!}
            \prod_{i=k+1}^{m-1} (2i-n)(2i+n-1) \\
       &\quad  + \frac{c_0}{(2m)!} \prod_{i=0}^{m-1} (2i-n)(2i+n-1)
\end{aligned}\label{eq:2.3a}
\end{equation}
holds for any $m \in \mathbb{N}$:
If we set $m = 1$ in \eqref{eq:2.3a}, then we obtain
$2c_2 + n(n-1)c_0 = a_0$ which coincides with \eqref{eq:rec}
when $m = 0$.
We assume that the formula \eqref{eq:2.3a} is true for some
$m \in \mathbb{N}$.
Then, it follows from \eqref{eq:rec} and the induction hypothesis
that
\begin{align*}
c_{2m+2}
  & = \frac{a_{2m}}{(2m+2)(2m+1)} +
      \frac{(2m-n)(2m+n-1)}{(2m+2)(2m+1)} c_{2m} \\
  & = \frac{a_{2m}}{(2m+2)(2m+1)} +
      \sum_{k=0}^{m-1} \frac{(2k)! a_{2k}}{(2m+2)!}
      \prod_{i=k+1}^m (2i-n)(2i+n-1) \\
  &\quad  + \frac{c_0}{(2m+2)!} \prod_{i=0}^m (2i-n)(2i+n-1) \\
  &= \sum_{k=0}^m \frac{(2k)! a_{2k}}{(2m+2)!}
      \prod_{i=k+1}^m (2i-n)(2i+n-1) \\
  &\quad  + \frac{c_0}{(2m+2)!} \prod_{i=0}^m (2i-n)(2i+n-1),
\end{align*}
which can be obtained provided we replace $m$ in \eqref{eq:2.3a}
with $m+1$.
Hence, we conclude that the formula \eqref{eq:2.3a} is true for
all $m \in \mathbb{N}$.
Similarly, we can prove the validity of the formula
\begin{equation}
\begin{aligned}
c_{2m+1} &= \sum_{k=0}^{m-1}
            \frac{(2k+1)! a_{2k+1}}{(2m+1)!}
            \prod_{i=k+1}^{m-1} (2i-n+1)(2i+n) \\
      &\quad + \frac{c_1}{(2m+1)!}
             \prod_{i=0}^{m-1} (2i-n+1)(2i+n)
\end{aligned} \label{eq:2.3b}
\end{equation}
for all $m \in \mathbb{N}$.

Indeed, we can set $c_0 = c_1 = 0$ in \eqref{eq:2.3a} and
\eqref{eq:2.3b}.
Under this assumption, we have
\begin{align*}
c_{2m} &= \sum_{k=0}^{m-1} \frac{(2k)! a_{2k}}{(2m)!}
          \prod_{i=k+1}^{m-1} (2i-n)(2i+n-1) \\
       &= \sum_{k=0}^{[n/2]-1} \frac{(2k)! a_{2k}}{(2m)!}
          \prod_{i=k+1}^{m-1} (2i-n)(2i+n-1) \\
       &\quad + \sum_{k=[n/2]}^{m-1} \frac{(2k)! a_{2k}}{(2m)!}
          \prod_{i=k+1}^{m-1} (2i-n)(2i+n-1) \\
       &= \sum_{k=0}^{[n/2]-1} \frac{(2k)! a_{2k}}{(2m)!}
          \Big( \prod_{i=k+1}^{[n/2]} (2i-n)(2i+n-1) \Big)\!
          \Big( \prod_{i=[n/2]+1}^{m-1} (2i-n)(2i+n-1) \Big) \\
         &+ \sum_{k=[n/2]}^{m-1} \frac{(2k)! a_{2k}}{(2m)!}
          \prod_{i=k+1}^{m-1} (2i-n)(2i+n-1).
\end{align*}
Hence, since $| 2i-n | | 2i+n-1 | < 2i(2i-1)$ for $i > [n/2]$,
we obtain
\begin{align*}
| c_{2m} |
&\leq \sum_{k=0}^{[n/2]-1} \frac{(2k)! | a_{2k} |}{(2m)!}
      \Big( \prod_{i=k+1}^{[n/2]} | 2i-n | | 2i+n-1 | \Big)
      \Big( \prod_{i=[n/2]+1}^{m-1} (2i)(2i-1) \Big) \\
     &+ \sum_{k=[n/2]}^{m-1} \frac{(2k)! | a_{2k} |}{(2m)!}
      \prod_{i=k+1}^{m-1} (2i)(2i-1) \\
   &= \sum_{k=0}^{[n/2]-1} \frac{(2k)! | a_{2k} |}{(2m)!}
      \alpha_n(k) \prod_{i=[n/2]+1}^{m-1} (2i)(2i-1) \\
     &\quad + \sum_{k=[n/2]}^{m-1} \frac{(2k)! | a_{2k} |}{(2m)!}
      \prod_{i=k+1}^{m-1} (2i)(2i-1),
\end{align*}
where $\alpha_n(k) := \prod_{i=k+1}^{[n/2]} | 2i-n | | 2i+n-1 |$
for $k \in \{ 0, 1, \ldots, [n/2]-1 \}$.
Moreover, taking into account that
$\prod_{i=k+1}^{m-1} (2i)(2i-1) = (2m-2)!/(2k)!$, we have
\begin{equation}
| c_{2m} |
\leq \sum_{k=0}^{[n/2]-1}
     \frac{\alpha_n(k) | a_{2k} |}{2m(2m-1)} +
     \sum_{k=[n/2]}^{m-1} \frac{| a_{2k} |}{2m(2m-1)}
\leq \frac{1}{m} \sum_{k=0}^{m-1}
     \frac{\alpha_n | a_{2k} |}{2(2m-1)},
\label{eq:20130507-1}
\end{equation}
for all $m \in \mathbb{N}$, where
$\alpha_n := \max \{ \alpha_n(0), \alpha_n(1), \ldots,  \alpha_n([n/2]-1), 1 \}$.
Similarly, we obtain
\begin{equation}
| c_{2m+1} | \leq \frac{1}{m} \sum_{k=0}^{m-1}
                  \frac{\beta_n | a_{2k+1} |}{2(2m+1)}
\label{eq:20130507-2}
\end{equation}
for any $m \in \mathbb{N}$, where
$\beta_n := \max \{ \beta_n(0), \beta_n(1), \ldots,
 \beta_n([n/2]-1), 1 \}$ and
$\beta_n(k) := \prod_{i=k+1}^{[n/2]} | 2i-n+1 | | 2i+n |$ for
$k \in \{ 0, 1, \ldots, [n/2]-1 \}$.

It  follows from \eqref{eq:20130507-1}, \eqref{eq:20130507-2},
and \cite[Problem 8.8.1 (p)]{kosmala} that
\[
\limsup_{m \to \infty} | c_{2m} |
\leq \limsup_{m \to \infty} \frac{1}{m}
     \sum_{k=0}^{m-1} \frac{\alpha_n | a_{2k} |}{2(2m-1)}
\leq \limsup_{m \to \infty}
     \frac{\alpha_n | a_{2m-2} |}{2(2m-1)}
\leq \limsup_{m \to \infty} | a_{2m-2} |
\]
and
\[
\limsup_{m \to \infty} | c_{2m+1} |
\leq \limsup_{m \to \infty} \frac{1}{m}
     \sum_{k=0}^{m-1} \frac{\beta_n | a_{2k+1} |}{2(2m+1)}
\leq \limsup_{m \to \infty} \frac{\beta_n | a_{2m-1} |}{2(2m+1)}
\leq \limsup_{m \to \infty} | a_{2m-1} |
\]
which imply that the radius $\rho_2$ of convergence of the
power series $\sum_{m=2}^\infty c_m x^m$ is not less than the
radius $\rho_1$ of the power series $\sum_{m=0}^\infty a_m x^m$.

If we define $\rho_3 := \min \{ \rho_0, \rho_1, \rho_2 \}$,
where $\rho_0 = 1$ is the radius of convergence of the general
solution to \eqref{eq:2.1}, then $\rho = \rho_3$.
According to \cite[Theorem 2.1]{jungsevli} and our assumption
that $c_0 = c_1 = 0$, every solution
$y : (-\rho_3, \rho_3) \to \mathbb{C}$ of the inhomogeneous
Gegenbauer differential equation \eqref{eq:1.1} can be
expressed by \eqref{eq:2.2}.
\end{proof}

\section{Hyers-Ulam stability for \eqref{eq:2.1}}


Let $n$ be an integer larger than $1$ and let $\rho_1$ be a
positive real number larger than $1$ or  infinity.
We denote by $\tilde{C}$ the set of all functions
$f : (-1, 1) \to \mathbb{C}$ with the following properties:
\begin{itemize}
\item[(a)] $f(x)$ is expressible by a power series
                 $\sum_{m=0}^\infty b_m x^m$ whose radius of
                 convergence is at least $\rho_1$;
\item[(b)] There exists a constant $K \geq 0$ such that
                 $\sum_{m=0}^\infty | a_m x^m | \leq K
                  | \sum_{m=0}^\infty a_m x^m |$ for all
                 $x \in (-\rho_1, \rho_1)$, where
                 $a_m = (m+2)(m+1)b_{m+2} - (m-n)(m+n-1)b_m$
                 for all $m \in \mathbb{N}_0$.
\end{itemize}
If we define
$$
( y_1 + y_2 )(x) = y_1(x) + y_2(x) \quad\text{and}\quad
( \lambda y_1 )(x) = \lambda y_1(x)
$$
for all $y_1, y_2 \in \tilde{C}$ and $\lambda \in \mathbb{C}$,
then $\tilde{C}$ is a vector space over the complex numbers.
We remark that the set $\tilde{C}$ is  a vector space.

In the following theorem, we investigate the Hyers-Ulam
stability of the Gegenbauer differential equation \eqref{eq:2.1}
for functions in $\tilde{C}$.

\begin{theorem}\label{thm:3.1}
If a function $y \in \tilde{C}$ satisfies the differential
inequality
\begin{equation}
\big| \big( 1 - x^2 \big) y''(x) + n(n-1) y(x) \big|
\leq \varepsilon
\label{eq:3.1}
\end{equation}
for all $x \in (-1, 1)$ and for some $\varepsilon \geq 0$,
then there exist constants $C_1, C_2 > 0$ and a solution
$y_h : (-1, 1) \to \mathbb{C}$ of the Gegenbauer differential
equation \eqref{eq:2.1} such that
$$
| y(x) - y_h(x) |
\leq C_1 | x | \ln \frac{1 + | x |}{1 - | x |} +
     C_2 \Big( \ln \frac{1 + | x |}{1 - | x |} - 2 | x | \Big)
$$
for any $x \in (-1, 1)$.
\end{theorem}


\begin{proof}
According to (a), $y(x)$ can be expressed as
$y(x) = \sum_{m=0}^\infty b_m x^m$ and it follows from (a) and
(b) that
\begin{equation}
\begin{aligned}
&\big( 1 - x^2 \big) y''(x) + n(n-1) y(x) \\
&= \sum_{m=0}^\infty
   \big[ (m+2)(m+1) b_{m+2} - (m-n)(m+n-1) b_m \big] x^m
   \\
&= \sum_{m=0}^\infty a_m x^m
\end{aligned}\label{eq:3.3}
\end{equation}
for all $x \in (-1, 1)$.
By considering \eqref{eq:3.1} and \eqref{eq:3.3}, we have
$$
\Big| \sum_{m=0}^\infty a_m x^m \Big| \leq \varepsilon
$$
for any $x \in (-1, 1)$.
This inequality, together with (b), yields that
\begin{equation}
\sum_{m=0}^\infty \big| a_m x^m \big|
\leq K \Big| \sum_{m=0}^\infty a_m x^m \Big|
\leq K \varepsilon
\label{eq:condition1}
\end{equation}
for all $x \in (-1, 1)$.

Now, it follows from Theorem \ref{thm:2.1}, \eqref{eq:3.3},
and \eqref{eq:condition1} that there exists a solution
$y_h : (-1, 1) \to \mathbb{C}$ of the Gegenbauer differential
equation \eqref{eq:2.1} such that
\begin{align*}
| y(x) - y_h(x) |
\leq \Big| \sum_{m=2}^\infty c_m x^m \Big|
\leq \sum_{m=1}^\infty | c_{2m} | | x |^{2m} +
     \sum_{m=1}^\infty | c_{2m+1} | | x |^{2m+1}
\end{align*}
for all $x \in (-1, 1)$.
By \eqref{eq:20130507-1} and \eqref{eq:20130507-2}, we moreover
have
\begin{equation}
\begin{aligned}
&| y(x) - y_h(x) | \\
& \leq \alpha_n \sum_{m=1}^\infty \frac{|x|^{2m}}{2(2m-1)}
       \frac{1}{m} \sum_{k=0}^{m-1} | a_{2k} | +
       \beta_n \sum_{m=1}^\infty \frac{|x|^{2m+1}}{2(2m+1)}
       \frac{1}{m} \sum_{k=0}^{m-1} | a_{2k+1} |
\end{aligned}\label{eq:20130604-1}
\end{equation}
for all $x \in (-1, 1)$.
(See the proof of Theorem \ref{thm:2.1} for the definitions
of $\alpha_n$ and $\beta_n$).

In view of (a) and (b), the radius of convergence of the power
series $\sum_{m=0}^\infty a_m x^m$ is $\rho_1$ which is larger
than $1$.
This fact implies that
$$
\sum_{m=0}^\infty | a_m |
= \sum_{k=0}^\infty | a_{2k} | + \sum_{k=0}^\infty | a_{2k+1} |
< \infty,
$$
which again implies that
$$
\lim_{k \to \infty} | a_{2k} | = 0, \quad
\lim_{k \to \infty} | a_{2k+1} | = 0.
$$

According to \cite[Theorem 2.8.6]{kosmala}, the sequences
$\big\{ | a_{2k} | \big\}$ and $\big\{ | a_{2k+1} | \big\}$
are $(C, 1)$ summable to $0$; i.e.,
$$
\lim_{m \to \infty} \frac{1}{m} \sum_{k=0}^{m-1} | a_{2k} | = 0, \quad
\lim_{m \to \infty} \frac{1}{m} \sum_{k=0}^{m-1} | a_{2k+1} |
= 0.
$$
Thus, there exists a constant $C > 0$ such that
$$
\frac{1}{m} \sum_{k=0}^{m-1} | a_{2k} | \leq C, \quad
\frac{1}{m} \sum_{k=0}^{m-1} | a_{2k+1} | \leq C
$$
for any $m \in \mathbb{N}$.

Hence,  from \eqref{eq:20130604-1} it follows that
\begin{equation}
| y(x) - y_h(x) |
\leq \frac{\alpha_n C}{2}
     \sum_{m=1}^\infty \frac{|x|^{2m}}{2m-1} +
     \frac{\beta_n C}{2}
     \sum_{m=1}^\infty \frac{|x|^{2m+1}}{2m+1}
\label{eq:20130606-1}
\end{equation}
for all $x \in (-1, 1)$.
Since
$$
\frac{1}{2} \ln \frac{1 + | x |}{1 - | x |}
= \sum_{m=1}^\infty \frac{| x |^{2m-1}}{2m-1}
= \sum_{m=0}^\infty \frac{| x |^{2m+1}}{2m+1}
$$
for $x \in (-1, 1)$, it holds that
$$
| y(x) - y_h(x) |
\leq C_1 | x | \ln \frac{1 + | x |}{1 - | x |} +
     C_2 \Big( \ln \frac{1 + | x |}{1 - | x |} - 2 | x | \Big)
$$
for any $x \in (-1, 1)$, where we set
$$
C_1 = \frac{\alpha_n C}{4}, \quad C_2 = \frac{\beta_n C}{4},
$$
which completes the proof.
\end{proof}

According to the previous theorem, each approximate solution
of the Gegenbauer differential equation \eqref{eq:2.1} can be
well approximated by an exact solution of the Gegenbauer
differential equation in a (small) neighborhood of $0$.

\begin{corollary}\label{cor:3.2}
If a function $y \in \tilde{C}$ satisfies the differential
inequality \eqref{eq:3.1} for all $x \in (-1, 1)$ and for
some $\varepsilon \geq 0$, then there exists a solution
$y_h : (-1, 1) \to \mathbb{C}$ of the Gegenbauer differential
equation \eqref{eq:2.1} such that
$$
| y(x) - y_h(x) | = O\big( x^2 \big)
$$
as $x \to 0$, where $O(\cdot)$ denotes the Landau symbol
$($big-$O)$.
\end{corollary}


\begin{proof}
According to Theorem \ref{thm:3.1} and \eqref{eq:20130606-1},
there exists a solution $y_h : (-1, 1) \to \mathbb{C}$ of the
Gegenbauer differential equation \eqref{eq:2.1} such that
$$
| y(x) - y_h(x) |
\leq \frac{\alpha_n C}{2} | x |^2
     \sum_{m=1}^\infty \frac{|x|^{2m-2}}{2m-1} +
     \frac{\beta_n C}{2} | x |^3
     \sum_{m=1}^\infty \frac{|x|^{2m-2}}{2m+1}
$$
for any $x \in (-1, 1)$, where we see the proof of Theorem
\ref{thm:3.1} for the definition of $C$, which completes our
proof.
\end{proof}


\subsection*{Acknowledgments}
This work was supported by the 2013 Hongik University
Research Fund.


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\end{document}