Lower bounds on the $L_p$ discrepancy of digital NUT sequences

TL;DR

This paper establishes lower bounds on the $L_p$ discrepancy for digital NUT sequences, a significant subclass of digital sequences, contributing to the understanding of their uniformity properties.

Contribution

It provides the first known lower bounds for the $L_p$ discrepancy of specific subclasses of digital NUT sequences, advancing discrepancy theory.

Findings

Lower bounds for $L_p$ discrepancy of digital NUT sequences

Identification of subclasses with provable discrepancy limits

Enhancement of theoretical understanding of digital sequence uniformity

Abstract

We study the $L_p$ discrepancy of digital NUT sequences which are an important sub-class of digital $(0,1)$-sequences in the sense of Niederreiter. The main result is a lower bound for certain sub-classes of digital NUT sequences.

Equations

$$L_p(\mathcal{P})=\Vert {\Delta }_{\mathcal{P}}{\Vert }_{L_p([0,1])}={\left({\int }_0^1|{\Delta }_{\mathcal{P}}(t){|}^p\mathrm{d}t\right)}^{\frac{1}{p}}$$

$$\Delta_{\mathcal{P}}(t)=\sum_{n=0}^{N-1}\boldsymbol{1}_{[0,t)}(x_{n})-Nt\ \ \ \mbox{for $t\in[0,1]$}$$

$$L_{p}(X_{N})\gtrsim\sqrt{\log N}\ \ \ \mbox{$\infty$-often}$$

$$L_{\infty}(X_{N})\gtrsim\log N\ \ \ \mbox{$\infty$-often.}$$

$$C\vec{n}=:(y_1^{(n)},y_2^{(n)},y_3^{(n)},\dots {)}^{\top }\in {\mathbb{Z}}_2^{\infty }$$

$$x_n:=\frac{y_1^{(n)}}{2}+\frac{y_2^{(n)}}{2^2}+\frac{y_3^{(n)}}{2^3}+\cdots .$$

$$C=\left(\begin{array}{cccc}1& c_{1,2}& c_{1,3}& \cdots \\ 0& 1& c_{2,3}& \cdots \\ 0& 0& 1& \cdots \\ ⋮& ⋮& ⋮& \ddots \end{array}\right).$$

$$L_2(X_N^C)\le L_2(X_N^I)\le {\left({\left(\frac{\log N}{6\log 2}\right)}^2+O(\log N)\right)}^{1/2}$$

$$L_p(X_N^C)\le L_{\infty }(X_N^C)\le L_{\infty }(X_N^I)\le \frac{\log N}{3\log 2}+1.$$

$$\underset{N\to \infty }{\mathrm{limsup}}\frac{L_p(X_N^I)}{\log N}=\frac{1}{6\log 2}$$

$$U=\left(\begin{array}{cccc}1& 1& 1& \cdots \\ 0& 1& 1& \cdots \\ 0& 0& 1& \cdots \\ ⋮& ⋮& ⋮& \ddots \end{array}\right)$$

$$\displaystyle C=\left(\begin{array}[]{ccc}\boldsymbol{a}_{1}&&\\ 0&\boldsymbol{a}_{2}&\\ 0&0&\boldsymbol{a}_{3}\\ \lx@intercol\dotfill\hfil\hfil\lx@intercol\end{array}\right)$$

$${\mathbf{a}}_i=(1,0,0,\dots )\text{mbox}or{\mathbf{a}}_i=(1,1,1,\dots )\text{mbox}fori\in \mathbb{N}.$$

$$L_{p}(X_{N}^{C})\gtrsim\log N\ \ \ \mbox{$\infty$-often.}$$

$$L_p(X_N^C)\le L_{\infty }(X_N^C)\le L_{\infty }(X_N^I)\le s_2(N),$$

$$L_{p}(X_{N}^{C})\leq c_{p}s_{2}(N)\ \ \ \mbox{for all $N\in\mathbb{N}$,}$$

$$c_{p}=\left\{\begin{array}[]{ll}1/\sqrt{3}=0.5773\ldots&\mbox{ if $p\in[1,2]$,}\\ 1&\mbox{ if $p\in(2,\infty]$.}\end{array}\right.$$

$$L_{p}(X_{N}^{C})\leq 1\ \ \ \mbox{for all $N$ of the form $N=2^{m}$}$$

$$L_{p}(X_{N}^{C})\lesssim\sqrt{\log N}\ \ \ \mbox{for all $N$ of the form $N=2^{m}+2^{\lfloor\sqrt{m}\rfloor-1}-1$}$$

$$L_{p}(X_{N}^{C})\lesssim\log N\ \ \ \mbox{ for all $N\geq 2$.}$$

$$c_{i,j}=\left\{\begin{array}[]{ll}1&\mbox{if $i\leq j<i+\alpha$,}\\ 0&\mbox{in all other cases.}\end{array}\right.$$

$$L_{p}(X_{N}^{C(\alpha)})\geq\frac{2^{\alpha-1}}{2^{2\alpha}-1}\frac{\log N}{2\alpha\log 2}+O_{\alpha}(1)\ \ \ \mbox{$\infty$-often.}$$

$$N=\sum_{\ell=1}^{r}2^{2\alpha(r-\ell)}=\frac{2^{2\alpha r}-1}{2^{2\alpha}-1}\ \ \ \mbox{for arbitrary $r\in\mathbb{N}$.}$$

$$L_{p}(X_{N}^{I})\geq\frac{\log N}{6\log 2}+O(1)\ \ \ \mbox{$\infty$-often.}$$

$$C(\mathbf{a})=\left(\begin{array}{ccccc}1& a_1& a_2& a_3& \cdots \\ 0& 1& a_2& a_3& \cdots \\ 0& 0& 1& a_3& \cdots \\ 0& 0& 0& 1& \cdots \\ ⋮& ⋮& ⋮& ⋮& \ddots \end{array}\right),$$

$$d_{\ell }(m):=\underset{1\le n\le m-1}{\min }\left\{\exists i\in \{1,\dots ,m-n\}:a_i=a_{i+n}=\ell ,a_{i+1}=\cdots =a_{i+n-1}\ne \ell \right\}.$$

$$L_p(X_{N_{\mathbf{a}}}^{C(\mathbf{a})})\ge \frac{1}{3}{l}_0(m)+O(1)$$

$$L_p(X_{N_{\mathbf{a}}}^{C(\mathbf{a})})\ge \frac{1}{3}{l}_1(m)+O(1)$$

$$L_{p}(X_{N}^{C(\boldsymbol{a})})\geq c\log{N}\ \ \ \mbox{for infinitely many $N$}$$

Full text

Keywords: $L_{p}$-discrepancy, van der Corput sequence, digital $(0,1)$ sequence

MSC 2000: 11K38, 11K31

1 Introduction

For a set $\mathcal{P}=\{x_{0},\dots,x_{N-1}\}$ of $N$ points in $[0,1)$ the (non-normalized) $L_{p}$* discrepancy* for $p\in[1,\infty]$ is defined as

[TABLE]

(with the usual modification if $p=\infty$), where

[TABLE]

is the (non-normalized) discrepancy function of $\mathcal{P}$.

We denote by $\mathbb{N}$ the set of positive integers and define $\mathbb{N}_{0}=\mathbb{N}\cup\{0\}$. Let $X=(x_{n})_{n\geq 0}$ be an infinite sequence in $[0,1)$ and, for $N\in\mathbb{N}$, let $X_{N}=\{x_{0},x_{1},\ldots,x_{N-1}\}$ denote the set consisting of the first $N$ elements of $X$. It is well known that for all $p\in[1,\infty)$ we have

[TABLE]

and

[TABLE]

(For functions $f,g:\mathbb{N}\rightarrow\mathbb{R}^{+}$, we write $g(N)\lesssim f(N)$ or $g(N)\gtrsim f(N)$, if there exists a positive constant $C$ that is independent of $N$ such that $g(N)\leq Cf(N)$ or $g(N)\geq Cf(N)$, respectively.) The lower estimate for finite $p$ was first shown by Proĭnov [13] (see also [3]) based on famous results of Roth [14] and Schmidt [16] for finite point sets in dimension two. Using the method of Proĭnov in conjunction with a result of Halász [7] for finite point sets in dimension two the lower bound follows also for the $L_{1}$-discrepancy. The estimate for $p=\infty$ was first shown by Schmidt [15] in 1972 (see also [1, 9, 17]).

In this paper we investigate the $L_{p}$ discrepancy of digital $(0,1)$-sequences. Since we only deal with digital sequences over $\mathbb{Z}_{2}$ and in dimension 1 we restrict the necessary definitions to this case. For the general setting we refer to [2, 10, 11].

Let $\mathbb{Z}_{2}$ be the finite field of order 2, which we identify with the set $\{0,1\}$ equipped with arithmetic operations modulo 2. For the generation of a digital $(0,1)$ sequence $(x_{n})_{n\geq 0}$ over $\mathbb{Z}_{2}$ we require an infinite matrix $C=(c_{i,j})_{i,j\geq 1}$ over $\mathbb{Z}_{2}$ with the following property111A further technical condition which is sometimes required, see [11, p.72, (S6)], is that for each $j\geq 1$ the sequence $(c_{i,j})_{i\geq 1}$ becomes eventually zero. Otherwise it could happen that one or more elements of the digital $(0,1)$-sequence are 1 and therefore do not belong to $[0,1)$.: for every $n\in\mathbb{N}$ the left upper $n\times n$ submatrix $(c_{i,j})_{i,j=1}^{n}$ has full rank. In order to construct the $n^{{\rm th}}$ element $x_{n}$ for $n\in\mathbb{N}_{0}$ compute the base 2 expansion $n=n_{0}+n_{1}2+n_{2}2^{2}+\cdots$ (which is actually finite), set $\vec{n}:=(n_{0},n_{1},n_{2},\ldots)^{\top}\in\mathbb{Z}_{2}^{\infty}$ and compute the matrix vector product

[TABLE]

over $\mathbb{Z}_{2}$. Finally, set

[TABLE]

We denote the digital $(0,1)$-sequence222In the general notation, the $1$ refers to the dimension and the [math] refers to the full rank condition of the generator matrix $C$. constructed in this way by $X^{C}$.

An important sub-class of digital $(0,1)$-sequences which is studied in many papers (initiated by Faure [5]) are so-called digital NUT sequences whose generator matrices are of non-singular upper triangular (NUT) form

[TABLE]

For example, if $C=I$ is the identity matrix, then the corresponding digital NUT sequence is the van der Corput sequence in base 2. For information about digital NUT sequences and the van der Corput sequence see the survey [6] and the references therein.

For digital NUT sequences $X^{C}$ it is known (see [4, Theorem 1]), that

[TABLE]

and for general $p\geq 1$ it is known (see [12, Theorem 2]) that

[TABLE]

Note that according to the lower bound of Schmidt the upper bound for the $L_{\infty}$ discrepancy in (2) is optimal in the order of magnitude in $N$. This is not the case for finite $p$.

Concerning lower bounds on the $L_{p}$ discrepancy of digital NUT sequences very few is known and only for very special cases. For the van der Corput sequence we have for all $p\in[1,\infty)$

[TABLE]

and hence $L_{p}(X_{N}^{I})\gtrsim\log N$ for infinitely many $N\in\mathbb{N}$; see [12, Corollary 1].

For the so-called upper-$\boldsymbol{1}$-sequence $X^{U}$, which is generated by the matrix

[TABLE]

it is known that for every $p\geq 1$ we have $L_{p}(X_{N}^{U})\geq\frac{\log N}{20\log 2}+O(1)$ infinitely often; see [4].

In [4] the authors study the $L_{p}$ discrepancy of $X^{C}$ for special types of NUT matrices $C$ of the form

[TABLE]

with

[TABLE]

Note that these NUT sequences comprise the van der Corput sequence and the upper-$\boldsymbol{1}$-sequence $X^{U}$ as special cases. For $m\in\mathbb{N}$ let $h(m)$ denote the number of $(1,0,0,\ldots)$ rows among the first $m$ rows of $C$. For example, $h(m)=m$ in case of the van der Corput sequence and $h(m)=0$ in case of the sequence $X^{U}$. Then it follows from [4, Lemma 4] that for every $m\in\mathbb{N}$ there exists an integer $N\in[2^{m},2^{m+1})$ such that $L_{1}(X_{N}^{C})\gtrsim(m+h(m)^{2})^{1/2}$. This implies that if $h(m)\gtrsim m$ we have for every $p\geq 1$

[TABLE]

In general, however, it is a very difficult task to give precise lower bounds on the $L_{p}$ discrepancy of digital NUT sequences. We strongly conjecture the following:

Conjecture 1

For every digital NUT sequence $X^{C}$ we have

[TABLE]

Note that for every digital NUT sequence and for every $p\geq 1$ we have

[TABLE]

where $s_{2}:\mathbb{N}\rightarrow\mathbb{N}$ denotes the binary sum-of-digits function which is defined as $s_{2}(N)=N_{0}+N_{1}+\cdots+N_{m}$ whenever $N$ has binary expansion $N=N_{0}+N_{1}2+\cdots+N_{m}2^{m}$. The very last estimate in (11) follows from the proof of [8, Theorem 3.5 in Chapter 2].

Remark 1

The result in (11) can be generalized and improved in the following sense: For every $p\in[1,\infty]$ and for every digital $(0,1)$-sequence $X^{C}$ we have

[TABLE]

where

[TABLE]

We omit the proof.

The sum-of-digits function is very fluctuating. For example we have $s_{2}(2^{m})=1$, but $s_{2}(2^{m}-1)=m$. In any case we have $s_{2}(N)\leq\frac{\log N}{\log 2}+1$.

Remark 2

The inequalities in (11) shows that having only very few non-zero binary digits is a sufficient condition on $N\in\mathbb{N}$ which guarantees that $X_{N}^{C}$ has very low $L_{p}$ discrepancy. For example we have

[TABLE]

or

[TABLE]

or

[TABLE]

See Figure 1 for a comparison for the van der Corput sequence.

However, the condition on $N$ of having very few non-zero binary digits is not a necessary one for low discrepancy. For example, consider $N$ of the form $N=2^{m}-1$. Then we have $s_{2}(N)=m=\lfloor\frac{\log N}{\log 2}+1\rfloor$ but: since the discrepancy of $X_{N}^{C}$ and of $X_{N+1}^{C}$ differ at most by 1 and since $L_{p}(X_{N+1}^{C})=L_{p}(X_{2^{m}}^{C})\leq 1$ we obtain $L_{p}(X_{N}^{C})\lesssim 1$. Hence, while $s_{2}(N)$ is very large, the discrepancy $L_{p}(X_{N}^{C})$ is low.

But in any case: the only possible candidates of $N$ that satisfy (10) are required to have $s_{2}(N)\gtrsim\log N$.

In Section 2 we provide a lower bound for $L_{p}(X_{N}^{C})$ for special types of NUT matrices.

2 Lower bound on $L_{p}(X_{N}^{C})$

We study two sub-classes of NUT matrices. The first class has a certain band structure. More detailled, the considered matrices are of the form $C(\alpha)=(c_{i,j})_{i,j\geq 1}$ where, for fixed $\alpha\in\mathbb{N}$,

[TABLE]

For example, if $\alpha=1$, we obtain the identity matrix, i.e., $C(1)=I$.

Theorem 1

For all $\alpha\in\mathbb{N}$ and $p\in[1,\infty]$ we have

[TABLE]

The bound above is satisfied for $N$ of the form

[TABLE]

Remark 3

Following all the details in the proof the constant hidden in $O_{\alpha}(1)$ can be computed exactly. 2. 2.

For $\alpha=1$ we have $C(1)=I$ and hence the resulting NUT sequence is the van der Corput sequence. Theorem 1 gives

[TABLE]

This matches the corresponding value in (3).

We also study NUT matrices which have the same entries in each column above the diagonal; i.e. we deal with matrices of the form

[TABLE]

where $\boldsymbol{a}=(a_{1},a_{2},\dots)\in\mathbb{Z}_{2}^{\mathbb{N}}$ is chosen arbitrarily. We set $l_{0}(m):=\#\{i\in\{1,\dots,m\}:a_{i}=0\}$ and $l_{1}(m):=\#\{i\in\{1,\dots,m\}:a_{i}=1\}$. For $m\geq 2$ let further $d_{0}(m)$ be the minimal distance of consecutive zeroes and $d_{1}(m)$ be the minimal distance of consecutive ones in the string $(a_{1},\dots,a_{m})$, i.e. for $\ell\in\{0,1\}$ we define

[TABLE]

Theorem 2

Let $m\geq 2$. For all $p\in[1,\infty]$ and for N_{\boldsymbol{a}}=1+\sum_{i=1}^{m-1}2^{i}(1-a_{i})+2^{m}\ \ \ \mbox{for arbitrary m\in\mathbb{N}} we have

[TABLE]

if $d_{0}(m)\geq 2$, and

[TABLE]

if $d_{1}(m)\geq 2$.

Corollary 1

The first $N$ elements of a NUT-sequence generated by a matrix of the form $C(\boldsymbol{a})$ satisfy

[TABLE]

for some constant $c>0$ if $l_{1}(m)\geq c_{1}m$ for some $c_{1}>0$ and $d_{1}(m)\geq 2$ for all $m\geq 2$ or if $l_{0}(m)\geq c_{2}m$ for some $c_{2}>0$ and $d_{0}(m)\geq 2$ for all $m\geq 2$.

One example for a generator matrix satisfying the hypotheses of Corollary (1) is

[TABLE]

3 The proofs

The following auxiliary result will be the main tool of our proofs.

Lemma 1

For every NUT digital sequence $X^{C}$ and every $N\in\mathbb{N}$ of the form $N=2^{n_{1}}+2^{n_{2}}+\cdots+2^{n_{r}}$ with $n_{1}>n_{2}>\ldots>n_{r}$ and $r\in\mathbb{N}$ we have

[TABLE]

where the $\sigma_{r,j}$ are given by the following matrix-vector product over $\mathbb{Z}_{2}$:

[TABLE]

where the digits 1 in the latter vector are placed at positions $n_{l}-n_{r}+1$ for $l\in\{1,\dots,r-1\}$.

• Proof.

Let $X^{C}=(x_{n})_{n\geq 0}$ be the NUT digital sequence which is generated by the $\mathbb{N}\times\mathbb{N}$ matrix $C$. Let $N\in\mathbb{N}$ be of the form

[TABLE]

where $n_{1}>\ldots>n_{r}$. For $i=1,\ldots,r$ consider

[TABLE]

where for $i=1$ we define $2^{n_{1}}+\cdots+2^{n_{i-1}}=0$. Every

[TABLE]

can be written as

[TABLE]

where $a\in\{0,1,\ldots,2^{n_{i}}-1\}$ and

[TABLE]

For fixed $i=1,\ldots,r$ we decompose the matrix $C$ in the form

[TABLE]

where $C^{(n_{i}\times n_{i})}$ is the left upper $n_{i}\times n_{i}$ sub-matrix of $C$. To $n$ in (16) we associate

[TABLE]

where $a_{0},\ldots,a_{n_{i}-1}$ are the binary digits of $a$ and $\ell_{0},\ell_{1},\ell_{2},\ldots$ are the binary digits of $l_{i}$. With this notation for $n$ in the range (16) we have

[TABLE]

This shows that the point set $\mathcal{P}_{i}$ is a digitally shifted digital net with generating matrix $C^{(n_{i}\times n_{i})}$ and with digital shift vector

[TABLE]

Since $F^{(\mathbb{N}\times\mathbb{N})}$ is also a NUT matrix we find that the shift is of the form

[TABLE]

Note that the matrix $C^{(n_{i}\times n_{i})}$ has full rank, as $X^{C}$ is a NUT digital sequence. Hence the shifted digital net $\mathcal{P}_{i}$ can be written as the set of points

[TABLE]

where $b_{k}=c_{k,1}a_{0}\oplus\dots\oplus c_{k,n_{i}}a_{n_{i}-1}\oplus\sigma_{i,k}$ for $1\leq k\leq n_{i}$. Here and in the following $\oplus$ denotes addition in $\mathbb{Z}_{2}$.

We emphasize that $\sigma_{i,1},\dots,\sigma_{i,n_{i}}$ do not depend on the $a_{i}$’s, whereas the components $\sigma_{i,j}$ for $j\geq n_{i}+1$ may do so. Therefore we can also write

[TABLE]

where $\delta_{i}=\sum_{j=1}^{n_{1}-n_{i}+1}\frac{\sigma_{i,n_{i}+j}}{2^{n_{i}+j}}$ for $i>1$ and $\delta_{1}=0$.

We have the following decomposition of $X_{N}^{C}$:

[TABLE]

Therefore and from the fact that

[TABLE]

we obtain

[TABLE]

where

[TABLE]

and, for $k\geq 1$,

[TABLE]

where $a_{n_{\ell}}=1$ for $\ell=1,\ldots,i-1$ and all other $a_{r}$’s are zero. Note that $\sigma_{i,n_{i}+k}\in\mathbb{Z}_{2}$.

We have

[TABLE]

Observe that

[TABLE]

and

[TABLE]

Hence

[TABLE]

This shows that we have

[TABLE]

From this we obtain for all $i\in\{2,3,\ldots,r\}$ that

[TABLE]

Hence

[TABLE]

For the very last double sum we have

[TABLE]

Hence

[TABLE]

This gives

[TABLE]

$\hfill\Box$

Now we give the proof of Theorem 1.

• Proof.

In order to simplify the notation we will write $C$ instead of $C(\alpha)$ in the following. For every $N$ and $p$ we have

[TABLE]

Now choose $n_{i}=2\alpha(r-i)$ for $i\in\{1,2,\ldots,r\}$, i.e.

[TABLE]

We have

[TABLE]

Therefore

[TABLE]

Hence, using Lemma 1, we get

[TABLE]

Now we have to determine the numbers $\sigma_{r,j}$. Observe that

[TABLE]

where the digits 1 in the latter vector are placed at positions $l\alpha+1$ for $l\in\{2,\dots,2r-2\}$. From the structure of the matrix we find that

[TABLE]

and therefore

[TABLE]

Furthermore

[TABLE]

Putting all together we obtain

[TABLE]

Hence, using (18), we get

[TABLE]

$\hfill\Box$

In the following, we give the proof of Theorem 2.

• Proof.

Note that in the case $n_{r}=0$ the numbers $\sigma_{r,j}$ appearing in Lemma 1 can also be understood in the following way: Let $N=2^{n_{1}}+\sum_{i=0}^{n_{1}-1}N_{i}2^{i}=\sum_{i=1}^{r}2^{n_{i}}$ with $n_{1}=m\in\mathbb{N}$, $N_{i}\in\mathbb{Z}_{2}$ for $i\in\{0,\dots,n_{1}-1\}$ and $r=s_{2}(N)$. Let

[TABLE]

Then we have for $j\in\{1,\dots,n_{1}+1\}$

[TABLE]

Now consider a matrix of the form $C(\boldsymbol{a})$ and set $N_{\boldsymbol{a}}=2^{n_{1}}+\sum_{i=1}^{n_{1}-1}(1-a_{i})2^{i}+1=\sum_{i=1}^{r}2^{n_{i}}$, where $r=l_{0}(m)+2$. Then we have

[TABLE]

We observe that for $N_{\boldsymbol{a}}$ and $j\in\{1,\dots,n_{1}\}$ we have $\sigma_{r,j}=a_{n_{1}}\oplus 1$ if and only if $j=n_{k}+1$ for some $k\in\{2,\dots,r\}$, and $\sigma_{r,j}=a_{n_{1}}$ otherwise. Hence with Lemma 1 we find

[TABLE]

The fact that $d_{0}(m)\geq 2$ implies $n_{i}-n_{k}\leq 2(k-i)$ and further

[TABLE]

This completes the proof of the first claim (13). To derive (14) from (13), we show that changing the tuple $\boldsymbol{a}$ which defines the matrix to $\tilde{\boldsymbol{a}}=(1-a_{i})_{i\geq 1}$ does not change the integral of $\Delta_{X_{N_{\boldsymbol{a}}}^{C(\boldsymbol{a})}}$ much. We use the following argument: Let for $2^{n_{1}}\leq N\leq 2^{n_{1}+1}-1$ with $N=2^{n_{1}}+\sum_{i=0}^{n_{1}-1}N_{i}2^{i}=\sum_{i=1}^{r}2^{n_{i}}$

[TABLE]

It is not hard to show that $S(N)=\frac{1}{2}\sum_{\ell=0}^{n_{1}-1}\left\|\frac{N}{2^{\ell+1}}\right\|+O(1),$ where $\|x\|$ denotes the distance of a real number $x$ to its nearest integer. For $N$ as defined above we define the integer $N^{\prime}:=2^{n_{1}}+\sum_{i=0}^{n_{1}-1}(1-N_{i})2^{i}$ and prove $S(N^{\prime})=S(N)+O(1)$. This is the case, since

[TABLE]

and therefore

[TABLE]

This implies inequality (14). $\hfill\Box$