# The sharp Remez-type inequality for even trigonometric polynomials on the
period

Tamás Erdélyi  Department of Mathematics, Texas A&M University, College
Station, Texas 77843, College Station, Texas 77843  terdelyimath.tamu.edu

(September 7, 2018 )

We prove that

$\max_{t\in[-\pi,\pi]}{|Q(t)|}\leq
T_{2n}(\sec(s/4))=\frac{1}{2}((\sec(s/4)+\tan(s/4))^{2n}+(\sec(s/4)-\tan(s/4))^{2n})$

for every even trigonometric polynomial $Q$ of degree at most $n$ with complex
coefficients satisfying

$m(\\{t\in[-\pi,\pi]:|Q(t)|\leq 1\\})\geq 2\pi-s\,,\qquad s\in(0,2\pi)\,,$

where $m(A)$ denotes the Lebesgue measure of a measurable set
$A\subset{\mathbb{R}}$ and $T_{2n}$ is the Chebysev polynomial of degree $2n$
on $[-1,1]$ defined by $T_{2n}(\cos t)=\cos(2nt)$ for $t\in{\mathbb{R}}$. This
inequality is sharp. We also prove that

$\max_{t\in[-\pi,\pi]}{|Q(t)|}\leq
T_{2n}(\sec(s/2))=\frac{1}{2}((\sec(s/2)+\tan(s/2))^{2n}+(\sec(s/2)-\tan(s/2))^{2n})$

for every trigonometric polynomial $Q$ of degree at most $n$ with complex
coefficients satisfying

$m(\\{t\in[-\pi,\pi]:|Q(t)|\leq 1\\})\geq 2\pi-s\,,\qquad s\in(0,\pi)\,.$

trigonometric polynomials, Remez-type inequalities, geometry of polynomials

††support: 2010 Mathematics Subject Classifications. 11C08, 41A17, 26C10,
30C15

## 0\. Foreword

I started my Ph.D. program in June, 1987, at The Ohio State University, as a
student of Paul Nevai. In August, 1987, Paul Nevai moved from Columbus, Ohio,
to Columbia, South Carolina, to spend the school year 1987–88 at the
University of South Carolina, and each of his students followed him. This is
how I met Mr. (Yingkang) Hu, who was a student of Ron DeVore at that time. It
was a somewhat short but very exciting time for me at the University of South
Carolina. Not only had I passed each of my necessary exams in my Ph.D. program
but I had the possibility to learn from and interact with some of the very
best researchers in approximation theory. My office neighbors were Paul Nevai,
Ron DeVore, and George Lorenz. George Lorentz happened to be Ron DeVore’s
office neighbor on one hand and my office neighbor on the other hand to
finalize the book entitled “Constructive Approximation” he was writing jointly
with Ron DeVore. It was a privilege for me that the authors of this
fundamental book in approximation theory asked me to read some sections of
their book in preparation and took my remarks seriously. I was also lucky to
have talented, motivated, and helpful class-mates such as Yingkang Hu, George
Kyriazis, Jiaxiang (John) Zhang, and Shing-Whu Jha around. During the school
year 1987–88 Yingkang was a good friend of mine. We spent much time together
as class-mates as well as friends. I liked to eat Chinese food in the company
of Mr. Hu, Jiaxiang Zhang, Shing-Whu Jha, and some of my other Chinese
friends. We all took Paul Nevai’s class on Orthogonal Polynomials. We traveled
quite much together in this school year to attend various conferences, but
Yinkang, Jiaxiang, and me traveled together as tourists as well in the spring
break. In August, 1988, Paul Nevai moved back to Columbus, and I followed him
to write my dissertation at The Ohio State University. I defended my Ph.D.
dissertation at the University of South Carolina in May, 1989. A central part
of my dissertation was a Remez-type inequality for (generalized) algebraic and
trigonometric polynomials and their applications in the proofs of various
other inequalities for generalized polynomials. I was not able to prove the
sharp Remez-type inequality for trigonometric polynomials, but I have managed
to prove the sharp upper bound for $|Q(0)|$ for even trigonometric polynomials
$Q$ satisfying

$m\\{t\in(0,2\pi]:|Q(t)|\leq 1\\})\geq 2\pi-s\,,\qquad s\in(0,2\pi)\,,$

where $m(A)$ denotes the Lebesgue measure of a set $A\subset{\mathbb{R}}$.
After some 30 years I have revisited the topic of my dissertation. While the
sharp Remez-type inequality for trigonometric polynomials remains open, I am
glad to prove the sharp Remez-type inequality at least for even trigonometric
polynomials in this volume dedicated to the memory of Dr. Hu. After defending
my thesis I met Yingkang very rarely, but we were keeping in touch. In March,
2002, Yingkang visited us at Texas A&M University, and gave a talk in our
Center of Approximation Theory Seminar with the title “Equivalence of Moduli
of Smoothness”, a topic on which he published two papers in Journal of
Approximation Theory. During his visit we watched the Academy Award Ceremony
in my apartment as he was very much interested in some of the latest movies.
This may have been the last time I saw him and I was very sad to learn about
his death in March, 2016.

## 1\. Introduction

Let ${\Cal{T}}_{n}$ denote the set of all real trigonometric polynomials of
degree at most $n$. Let ${\Cal{T}}_{n}^{c}$ denote the set of all complex
trigonometric polynomials of degree at most $n$. Let
$K:={\mathbb{R}}\enskip(\text{mod}\,\,2\pi)$. Let $m(A)$ denote the Lebesgue
measure of a measurable set $A\subset{\mathbb{R}}$. For $s\in(0,2\pi)$ let

${\Cal{T}}_{n}(s):=\\{Q\in{\Cal{T}}_{n}:m(\\{t\in K:|Q(t)|\leq 1\\})\geq
2\pi-s\\}$

and

${\Cal{T}}_{n}^{c}(s):=\\{Q\in{\Cal{T}}_{n}^{c}:m(\\{t\in K:|Q(t)|\leq
1\\})\geq 2\pi-s\\}\,.$

Let ${\Cal{P}}_{n}$ denote the set of all algebraic polynomials of degree at
most $n$ with real coefficients. Let ${\Cal{P}}_{n}^{c}$ denote the set of all
algebraic polynomials of degree at most $n$ with complex coefficients. Let
$T_{n}$ be the Chebysev polynomial of degree $n$ on $[-1,1]$ defined by
$T_{n}(\cos t)=\cos(nt)$ for $t\in K$. For real numbers $a<b$ and a Lebesgue
measurable set $A\subset[a,b]$ let

$\mu_{[a,b]}(A):=\int_{A}{\frac{(b-a)/2}{\sqrt{((b-a)/2)^{2}-(x-(a+b)/2)^{2}}}\,dx}\,.$

The classical Remez inequality states that if $P\in{\Cal{P}}_{n}$,
$s\in(0,2)$, and

$m\left(\left\\{x\in[-1,1]:|P(x)|\leq 1\right\\}\right)\geq 2-s\,,$

then

$\max_{x\in[-1,1]}{|P(x)|}\leq T_{n}\left(\frac{2+s}{2-s}\right)\,,$

where $T_{n}$ defined by

$T_{n}(x):=\cos(n\arccos x)\,,\qquad x\in[-1,1]\,,$

is the Chebyshev polynomial of degree $n$. This inequality is sharp and

$T_{n}\left(\frac{2+s}{2-s}\right)\leq\exp(\min\\{5ns^{1/2},2n^{2}s\\})\,,\qquad
s\in(0,1]\,.$

Remez-type inequalities for various classes of functions have been studied by
several authors, and they have turned out to be applicable and connected to
various problems in approximation theory. See [1]–[34], for example.

In [16] we proved that

$|Q(0)|\leq T_{2n}(\sec(s/4))$ $None$

for every even $Q\in{\Cal{T}}_{n}^{c}(s)$ and $s\in(0,2\pi)$. However, $0$ is
a special point in the study of even trigonometric polynomials
$Q\in{\Cal{T}}_{n}^{c}(s)$, so the question whether or not

$\max_{t\in K}{|Q(t)|}\leq T_{2n}(\sec(s/4))$ $None$

holds at least for all even trigonometric polynomials
$Q\in{\Cal{T}}_{n}^{c}(s)$ remained open, while it was speculated that (1.2)
may hold for all $Q\in{\Cal{T}}_{n}^{c}(s)$. In this paper we show that (1.2)
holds for all even $Q\in{\Cal{T}}_{n}^{c}(s)$, while it remains open whether
or not (1.2) holds for all $Q\in{\Cal{T}}_{n}^{c}(s)$.

## 2\. New Results

Our first result is a sharp Remez-type inequality for even trigonometric
polynomials with complex coefficients.

###### Theorem 2.1

We have

$\max_{t\in K}{|Q(t)|}\leq
T_{2n}(\sec(s/4))=\frac{1}{2}((\sec(s/4)+\tan(s/4))^{2n}+(\sec(s/4)-\tan(s/4))^{2n})$

for every even $Q\in{\Cal{T}}_{n}^{c}(s)$ and $s\in(0,2\pi)$. Equality holds
if and only if

$\\{t\in K:|Q(t)|\leq 1\\}=[s/2,2\pi-s/2]$

and $Q\in{\Cal{T}}_{n}(s)$ is of the form

$Q(t)=T_{2n}\left(\frac{\cos(t/2)}{\cos(s/4)}\right)\,.$

Our next result is a Remez-type inequality for odd trigonometric polynomials
with complex coefficients.

###### Theorem 2.2

We have

$\max_{t\in
K}{|Q(t)|}\leq\frac{1}{2}((\sec(s/4)+\tan(s/4))^{2n}+(\sec(s/4)-\tan(s/4))^{2n})+\frac{1}{\sqrt{2}}$

for every odd $Q\in{\Cal{T}}_{n}^{c}(s)$ and $s\in(0,2\pi)$.

Theorem 2.1 implies he following result for all trigonometric polynomials with
complex coefficients.

###### Theorem 2.3

We have

$\max_{t\in K}{|R(t)|}\leq
T_{2n}(\sec(s/2))=\frac{1}{2}((\sec(s/2)+\tan(s/2))^{2n}+(\sec(s/2)-\tan(s/2))^{2n})$

for every $R\in{\Cal{T}}_{n}^{c}(s)$ and $s\in(0,\pi)$.

Note that Theorem 2.2 was proved in [16] only for all $R\in{\Cal{T}}_{n}(s)$
(rather than $R\in{\Cal{T}}_{n}^{c}(s)$) and $s\in(0,\pi)$. It remains open
whether or not Theorem 2.1 can be extended to all $Q\in{\Cal{T}}_{n}^{c}(s)$
and $s\in(0,2\pi)$.

## 3\. Lemmas

Our first lemma is the conclusion (13) of Section 7 in [16]. This deep result
plays a central role in the proof of Theorem 2.1.

###### Lemma 3.1

We have

$|U(0)|\leq T_{2n}(\sec(s/4))$

for every even $U\in{\Cal{T}}_{n}(s)$ and $s\in(0,2\pi)$. Equality holds if
and only if

$\\{t\in K:|U(t)|\leq 1\\}=[s/2,2\pi-s/2]$

and

$U(t)=T_{2n}\left(\frac{\cos(t/2)}{\cos(s/4)}\right)\,.$

Our next lemma extends Lemma 3.1 to all even trigonometric polynomials with
complex coefficients.

###### Lemma 3.2

We have

$|Q(0)|\leq T_{2n}(\sec(s/4))$

for every even $Q\in{\Cal{T}}_{n}^{c}(s)$ and $s\in(0,2\pi)$. Equality holds
if and only if

$\\{t\in K:|Q(t)|\leq 1\\}=[s/2,2\pi-s/2]$

and

$Q(t)=T_{2n}\left(\frac{\cos(t/2)}{\cos(s/4)}\right)\,.$

###### Demonstration Proof

Let $Q\in{\Cal{T}}_{n}^{c}(s)$. Choose $c\in{\mathbb{C}}$ with $|c|=1$ so that
$cQ(0)$ is real. Define $U\in{\Cal{T}}_{n}(s)$ by

$U(t):=\text{\rm Re}(cQ(t))\,,\qquad t\in K\,.$

Applying Lemma 3.1 to $U\in{\Cal{T}}_{n}(s)$ we obtain

$|Q(0)|=|S(0)|\leq T_{n}(\sec(s/4))\,,$

and equality holds if and only if

$\\{t\in K:|Q(t)|\leq 1\\}=[-\pi+s/2,\pi-s/2]$

and

$Q(t)=T_{2n}\left(\frac{\cos(t/2)}{\cos(s/4)}\right)\,.$

∎

Observe that every even $U\in{\Cal{T}}_{n}^{c}$ can be written as $U(t)=P(\cos
t)$, where $P\in{\Cal{P}}_{n}^{c}$. In terms of algebraic polynomials Lemma
3.1 can be formulated as follows.

###### Lemma 3.3

We have

$|P(1)|\leq T_{2n}(\sec(s/4))$

for every $P\in{\Cal{P}}_{n}^{c}$ such that

$\mu_{[-1,1]}(\\{x\in[-1,1]:|P(x)|\leq 1\\})\geq\pi-s/2\,.$

Equality holds if and only if

$\\{x\in[-1,1]:|P(x)|\leq 1\\}=[-1,\cos(s/2)]$

and

$P(\cos t)=T_{2n}\left(\frac{\cos(t/2)}{\cos(s/4)}\right)\,.$

Associated with $P\in{\Cal{P}}_{n}^{c}$ let
${\widetilde{P}}\in{\Cal{P}}_{n}^{c}$ be defined by
${\widetilde{P}}(x)=P(-x)$. Observing that

$\mu_{[-1,1]}(\\{x\in[-1,1]:|P(x)|\leq
1\\})=\mu_{[-1,1]}(\\{x\in[-1,1]:|{\widetilde{P}}(x)|\leq 1\\})$

we obtain the following.

###### Lemma 3.3*

We have

$|P(-1)|\leq T_{2n}(\sec(s/4))$

for every $P\in{\Cal{P}}_{n}^{c}$ such that

$\mu_{[-1,1]}(\\{x\in[-1,1]:|P(x)|\leq 1\\})\geq\pi-s/2\,.$

Equality holds if and only if

$\\{x\in[-1,1]:|P(x)|\leq 1\\}=[-\cos(s/2),1]$

and

$P(-\cos t)=T_{2n}\left(\frac{\cos(t/2)}{\cos(s/4)}\right)\,.$

We need to transform the above two lemmas linearly from the interval $[-1,1]$
to the interval $[a,b]$.

###### Lemma 3.4

We have

$|P(b)|\leq T_{2n}(\sec(s/4))$

for every $P\in{\Cal{P}}_{n}^{c}$ such that

$\mu_{[a,b]}(\\{x\in[a,b]:|P(x)|\leq 1\\})\geq\frac{b-a}{2}\,(\pi-s/2)\,.$

Equality holds if and only if

$\\{x\in[a,b]:|P(x)|\leq 1\\}=[a,(a+b)/2+((b-a)/2)\cos(s/2)]$

and

$P(((b-a)/2)\cos t+(a+b)/2)=T_{2n}\left(\frac{\cos(t/2)}{\cos(s/4)}\right)\,.$

###### Lemma 3.4*

We have

$|P(a)|\leq T_{2n}(\sec(s/4))$

for every $P\in{\Cal{P}}_{n}^{c}$ such that

$\mu_{[a,b]}(\\{x\in[a,b]:|P(x)|\leq 1\\})\geq\frac{b-a}{2}\,(\pi-s/2)\,.$

Equality holds if and only if

$\\{x\in[a,b]:|P(x)|\leq 1\\}=[(a+b)/2-((b-a)/2)\cos(s/2),b]$

and

$P((-(b-a)/2)\cos
t+(a+b)/2)=T_{2n}\left(\frac{\cos(t/2)}{\cos(s/4)}\right)\,.$

Observing that $P\in{\Cal{P}}_{n}^{c}$ implies
${\widetilde{P}}\in{\Cal{P}}_{n}^{c}$, where ${\widetilde{P}}(x)=P(-x)$ we
obtain the following.

###### Lemma 3.5

We have

$\mu_{[1-2r,1]}(A)>r^{1/2}\mu_{[-1,1]}(A)$

for every $r\in(0,1)$ and for every Lebesgue measurable set $A\subset[1-2r,1]$
with $m(A)>0$.

###### Demonstration Proof

Let $r\in(0,1)$ and let $A\subset[1-2r,1]$ be a Lebesgue measurable set with
$m(A)>0$. We have

$\begin{split}\frac{\mu_{[1-2r,1]}(A)}{\mu_{[-1,1]}(A)}&=\frac{\displaystyle{\int_{A}{\frac{r}{\sqrt{r^{2}-(x-(1-r))^{2}}}\,dx}}}{\displaystyle{\int_{A}{\frac{1}{\sqrt{(1-x^{2}}}\,dx}}}=\frac{\displaystyle{\int_{A}{\frac{r}{\sqrt{(1-x^{2})}}\,\frac{\sqrt{(1-x^{2}}}{\sqrt{r^{2}-(x-(1-r))^{2}}}}\,dx}}{\displaystyle{\int_{A}{\frac{1}{\sqrt{1-x^{2}}}}\,dx}}\cr&>\frac{\displaystyle{\int_{A}{\frac{r}{\sqrt{1-x^{2}}}\,dx}}\,\min_{y\in[1-2r,1]}{\displaystyle{\frac{\sqrt{1-y^{2}}}{\sqrt{r^{2}-(y-(1-r))^{2}}}}}}{\displaystyle{\int_{A}{\frac{1}{\sqrt{1-x^{2}}}\,dx}}}\cr&\geq
rr^{-1/2}=r^{1/2}\cr\end{split}$

as

$\min_{y\in[1-2r,1]}{\frac{\sqrt{1-y^{2}}}{\sqrt{r^{2}-(y-(1-r))^{2}}}}=\min_{y\in[1-2r,1]}{\frac{\sqrt{1+y}}{\sqrt{y-(1-2r)}}}=\frac{\sqrt{2}}{\sqrt{2r}}=r^{-1/2}\,.$

∎

###### Lemma 3.6

We have

$\mu_{[-1,1-2r]}(A)>(1-r)^{1/2}\mu_{[-1,1]}(A)$

for every $r\in(0,1)$ and for every Lebesgue measurable set
$A\subset[-1,-1+2r]$ with $m(A)>0$.

###### Demonstration Proof

Let $r\in(0,1)$ and let $A\subset[-1,1-2r]$ be a Lebesgue measurable set with
$m(A)>0$.. Applying Lemma 3.5 with $A$ replaced by $-A:=\\{-x:x\in
A\\}\subset[1-2(1-r),1]$ and $r\in(0,1)$ replaced by $1-r\in(0,1)$, we get

$\mu_{[-1,-1+2r]}(A)=\mu_{[1-2(1-r),1]}(-A)>(1-r)^{1/2}\mu_{[-1,1]}(-A)=(1-r)^{1/2}\mu_{[-1,1]}(A)\,.$

∎

###### Lemma 3.7

We have

$\arccos(1-2r)>\pi r^{1/2}\qquad and\qquad\pi-\arccos(1-2r)>\pi(1-r)^{1/2}$

for all $r\in(0,1)$.

###### Demonstration Proof

The first inequality is equivalent to $\sin t>(2/\pi)t$ for all
$t=(\pi/2)r^{1/2}\in(0,\pi/2)$. The second inequality follows from the first
one as

$\pi-\arccos(1-2r)=\arccos(-1+2r)=\arccos(1-2(1-r))>\pi(1-r)^{1/2}$

for all $r\in(0,1)$. ∎

## 4\. Proof of Theorem 2.1

###### Demonstration Proof of Theorem 2.1

Let $Q\in{\Cal{T}}_{n}^{c}(s)$ be even with $s\in(0,2\pi)$. Then $Q$ is of the
form $Q(t)=P(\cos t)$, where $P\in{\Cal{P}}_{n}^{c}$ satisfies

$\mu_{[-1,1]}(\\{x\in[-1,1]:|P(x)|\leq 1\\})\geq\pi-s/2\,.$ $None$

We want to prove that

$|P(\alpha)|\leq T_{2n}(\sec(s/4))$ $None$

for all $\alpha\in[-1,1]$.

If $\alpha=1$ then (4.2) holds by Lemma 3.3. If $\alpha=-1$ then (4.2) holds
by Lemma 3.3*. Assume now that $\alpha=1-2r\in(-1,1)$, that is, $\alpha=1-2r$
with $r\in(0,1)$. It follows from (4.1) and the definition of the measure
$\mu_{[-1,1]}$ that

$m(\\{t\in[0,\pi]:|P(\cos t)|\leq 1)\\})\geq\pi-s/2\,,$

and hence we have either

$m(\\{t\in[0,\arccos(1-2r)]:|P(\cos t)|\leq
1)\\})\geq\frac{\pi-\arccos(1-2r)}{\pi}(\pi-s/2)$ $None$

or

$m(\\{t\in[\arccos(1-2r),\pi]:|P(\cos t)|\leq
1)\\})\geq\frac{\arccos(1-2r)}{\pi}(\pi-s/2)\,.$ $None$

For the sake of brevity let

$A_{1}:=\\{x\in[-1,1-2r]:|P(x)|\leq 1\\}\quad\text{\rm and}\quad
A_{2}:=\\{x\in[1-2r,1]:|P(x)|\leq 1\\}\,.$ $None$

By (4.3), (4.4), and (4.5) we have either

$\mu_{[-1,1]}(A_{1})\geq\frac{\pi-\arccos(1-2r)}{\pi}(\pi-s/2)$ $None$

or

$\mu_{[-1,1]}(A_{2})\geq\frac{\arccos(1-2r)}{\pi}(\pi-s/2)$ $None$

Suppose that (4.6) holds. Then, combining Lemmas 3.6 and 3.7 we obtain

$\begin{split}\mu_{[-1,1-2r]}(A_{1})>&(1-r)^{1/2}\mu_{[-1,1]}(A_{1})\geq(1-r)^{1/2}\frac{\pi-\arccos(1-2r)}{\pi}(\pi-s/2)\cr>&(1-r)^{1/2}(1-r)^{1/2}(\pi-s/2)\geq(1-r)(\pi-s/2)\,,\cr\end{split}$

and

$|P(\alpha)|=|P(1-2r)|<T_{2n}(\sec(s/4))$

follows from Lemma 3.4.

Suppose that (4.7) holds. Then, combining Lemmas 3.5 and 3.7 we obtain

$\begin{split}\mu_{[1-2r,1]}(A_{2})>&\,r^{1/2}\mu_{[-1,1]}(A_{2})\geq
r^{1/2}\frac{\arccos(1-2r)}{\pi}(\pi-s/2)\cr>\,&r^{1/2}r^{1/2}(\pi-s/2)\geq
r(\pi-s/2)\,,\cr\end{split}$

and

$|P(\alpha)|=|P(1-2r)|<T_{2n}(\sec(s/4))$

follows from Lemma 3.4*. ∎

###### Demonstration Proof of Theorem 2.2

Let $Q\in{\Cal{T}}_{n}^{c}(s)$ be odd. Then $R$ defined by
$R(t)=2|Q(t)|^{2}-1$ is even and $R\in{\Cal{T}}_{2n}^{c}(s)$. Applying Theorem
2.1 to $R$ we obtain

$\max_{t\in K}{|2|Q(t)|^{2}-1|}=\max_{t\in
K}{|R(t)|}\leq\frac{1}{2}((\sec(s/4)+\tan(s/4))^{4n}+(\sec(s/4)-\tan(s/4))^{4n})\,,$

and hence

$\begin{split}\max_{t\in
K}{|Q(t)|}\leq&\left(\frac{1}{4}((\sec(s/4)+\tan(s/4))^{4n}+(\sec(s/4)-\tan(s/4))^{4n})+\frac{1}{2}\right)^{1/2}\cr\leq&\frac{1}{2}((\sec(s/4)+\tan(s/4))^{2n}+(\sec(s/4)-\tan(s/4))^{2n})+\frac{1}{\sqrt{2}}\cr\end{split}$

∎

###### Demonstration Proof of Theorem 2.3

Observe that if $R\in{\Cal{T}}_{n}^{c}(s)$, $s\in(0,\pi)$, and $Q$ is defined
by $Q(t)=\frac{1}{2}(R(t)+R(-t))$, then $R\in{\Cal{T}}_{n}^{c}(2s)$. Hence
Theorem 2.1 implies that

$|R(0)|=|Q(0)|\leq T_{2n}(\sec(s/2))$ $None$

holds for every even $R\in{\Cal{T}}_{n}^{c}(s)$ and $s\in(0,\pi)$. The theorem
now follows by a simple shift. Namely, observe that $R\in{\Cal{T}}_{n}^{c}(s)$
implies that $R_{a}\in{\Cal{T}}_{n}^{c}(s)$ where $R_{a}$ is defined by
$R_{a}(t):=R(t+a)$ for $t\in K$ and $a\in K$. Hence (4.8) implies that

$|R(a)|=|R_{a}(0)|\leq T_{2n}(\sec(s/2))$

holds for every even $R\in{\Cal{T}}_{n}^{c}(s)$, $s\in(0,\pi)$, and $a\in K$.
∎

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