The maximum number of zeros of $r(z) - \overline{z}$ revisited

TL;DR

This paper establishes bounds on the maximum number of zeros for rational harmonic functions of a specific form, depending on polynomial degrees, and proves the regularity of functions attaining these bounds.

Contribution

It generalizes previous results by deriving degree-dependent bounds and proving regularity for extremal functions.

Findings

Derived bounds on zeros depending on polynomial degrees

Proved regularity of functions reaching these bounds

Extended previous literature on rational harmonic functions

Abstract

Generalizing several previous results in the literature on rational harmonic functions, we derive bounds on the maximum number of zeros of functions $f(z) = \frac{p(z)}{q(z)} - \overline{z}$, which depend on both $\mathrm{deg}(p)$ and $\mathrm{deg}(q)$. Furthermore, we prove that any function that attains one of these upper bounds is regular.

Equations

$$f(z)=r(z)-\overline{z}\text{mbox}withr(z)=\frac{p(z)}{q(z)},$$

$$n:=\mathrm{deg}(r)=\max \{n_p,n_q\}\ge 2.$$

$$N(f_c)\le \{\begin{array}{ll}2n_p+3n_q-3,& \text{ if }{n}_p<n_q,\\ 5n_p-5,& \text{ if }{n}_p=n_q,\\ 3n_p+2n_q-2,& \text{ if }{n}_p>n_q+1.\end{array}$$

$$f_c(z)=r_c(z)-\overline{z},\text{mbox}where{r}_c(z):=\frac{p(z)-cq(z)}{q(z)},$$

$$J_f(z)=|{\partial }_{z}f(z){|}^2-|{\partial }_{\bar{z}}f(z){|}^2=|r^{\prime }(z){|}^2-1.$$

$$V(g;\Gamma ):=\frac{1}{2\pi }{\Delta }_{\Gamma }\arg g(z).$$

$$V(f;\Gamma )=N_{+}(f;\mathrm{int}(\Gamma ))-N_{-}(f;\mathrm{int}(\Gamma ))-P(f;\mathrm{int}(\Gamma )),$$

$$N_{+}(f_c)-N_{-}(f_c)=\{\begin{array}{ll}{n}_q-1,& \text{ if }{n}_p\le n_q,\\ n_p,& \text{ if }{n}_p>n_q+1.\end{array}$$

$$|f_c(z)+\overline{z}|=|r(z)-c|<|z|\le |f_c(z)|+|\overline{z}|\text{ for all }z\in \partial B_M(0).$$

$$-1$$

$$=N_{+}(f_c;B_M(0))-N_{-}(f_c;B_M(0))-P(f_c;B_M(0))$$

$$=N_{+}(f_c)-N_{-}(f_c)-n_q,$$

$$|f_c(z)-\tilde{p}(z)|=|\tilde{r}(z)-\overline{z}-c|<|\tilde{p}(z)|\le |f_c(z)|+|\tilde{p}(z)|\text{ for all }z\in \partial B_M(0).$$

$$n_p-n_q$$

$$=N_{+}(f_c;B_M(0))-N_{-}(f_c;B_M(0))-P(f_c;B_M(0))$$

$$=N_{+}(f_c)-N_{-}(f_c)-n_q,$$

$$N_0(f_c)+N_{-}(f_c)\le n_p+n_q-1.$$

$$R(z_0)=z_0,\text{mbox}whereR(z):=(\overline{r}\circ r)(z).$$

$$R^{\prime }(z)=(\overline{r}\circ r{)}^{\prime }(z)={\overline{r}}^{\prime }(r(z))r^{\prime }(z).$$

$$|R^{\prime }(z_0)|=|{\overline{r}}^{\prime }(r(z_0))r^{\prime }(z_0)|=|{\overline{r}}^{\prime }({\overline{z}}_0)r^{\prime }(z_0)|=|\overline{r^{\prime }(z_0)}{r}^{\prime }(z_0)|=|r^{\prime }(z_0){|}^2\le 1,$$

$$R^{\prime }(z_c)={\overline{r}}^{\prime }(r(z_c))r^{\prime }(z_c)=0,$$

$$\underset{k\to \infty }{\lim }{R}^k(z_c)=z_0.$$

$$\underset{k\to \infty }{\lim }{R}^k({\overline{w}}_c)$$

$$\displaystyle=\overline{r}\Big{(}\lim_{k\rightarrow\infty}(r\circ\overline{r})^{k}(\overline{z}_{c})\Big{)}=\overline{r}\Big{(}\overline{\lim_{k\rightarrow\infty}(\overline{r}\circ r)^{k}(z_{c})}\Big{)}$$

$$\displaystyle=\overline{r}\Big{(}\overline{\lim_{k\rightarrow\infty}R^{k}(z_{c})}\Big{)}=\overline{r}(\overline{z}_{0})=\overline{r(z_{0})}=z_{0}.$$

$$r^{\prime }(z)=\frac{p^{\prime }(z)q(z)-p(z)q^{\prime }(z)}{q(z{)}^2},$$

$$\delta:=\min\big{\{}|f(z)|:z\in\cup_{j=1}^{k}\,\partial B_{\varepsilon}(z_{j})\big{\}}>0.$$

$$|f(z)-f_c(z)|=|c|<\delta \le |f(z)|\le |f(z)|+|f_c(z)|.$$

$$1=N_{+}(f;B_{\epsilon }(z_j))=V(f;\partial B_{\epsilon }(z_j))=V(f_c;\partial B_{\epsilon }(z_j))=N_{+}(f_c;B_{\epsilon }(z_j)),$$

$$N(f_c)$$

$$\le N_{+}(f_{c_k})-N_{-}(f_{c_k})+N_{-}(f_{c_k})+N_0(f_c)+N_{-}(f_c)$$

$$\le (n_q-1)+(n_p+n_q-1)+(n_p+n_q-1)$$

$$=2n_p+3n_q-3,$$

$$N(f_c)$$

$$\le n_p+(n_p+n_q-1)+(n_p+n_q-1)$$

Full text

Keywords:

Zeros of rational harmonic functions; Rational harmonic functions; Harmonic polynomials; Complex valued harmonic functions

AMS Subject Classification (2010):

30D05, 31A05, 37F10

1 Introduction

We study the zeros of rational harmonic functions of the form

[TABLE]

where $p$ and $q$ are coprime polynomials of respective degrees $n_{p}$ and $n_{q}$, and

[TABLE]

If $z_{0}\in\mathbb{C}$ is a zero of $f$, i.e., $r(z_{0})-\overline{z}_{0}=0$, then also $z_{0}=\overline{r(z_{0})}$. Inserting this into the first equation and taking complex conjugates gives $\overline{r}(r(z_{0}))-z_{0}=0$. This can be transformed into a polynomial equation (of degree $n^{2}+1$), which shows that $f$ has finitely many zeros. It is also important to note that because of the term $\overline{z}$, the zeros of $f$ can not be “factored out”, and hence they do not have a multiplicity in the usual sense. “Numbers of zeros” in this context therefore refer to numbers of distinct complex points.

Several authors have studied upper bounds on $N(f)$, the number of zeros of a function $f$ as in (1). In particular, Khavinson and Neumann [4] showed that in general $N(f)\leq 5n-5$, and Khavinson and Świa̧tek [6] showed that if $n_{q}=0$, i.e., $f$ is a harmonic polynomial, then $N(f)\leq 3n-2$. Results of Rhie [11] and Geyer [3], respectively, show that these two upper bounds are sharp for each $n\geq 2$.

The main purpose of this note is to prove the following theorem, which takes the individual degrees $n_{p}$ and $n_{q}$ into account, and which generalizes (almost) all previously known bounds on the maximal number of zeros of $f$.

Theorem 1.1

Let $f$ be as in (1). Then for every $c\in\mathbb{C}$, the number of zeros of $f_{c}(z):=f(z)-c$ satisfies

[TABLE]

The only case “missing” in this theorem is $n_{p}=n_{q}+1$. For this case we know from [9] that $N(f_{c})\leq 5n_{p}-6$.

Note that for each $c\in\mathbb{C}$ we can write

[TABLE]

which is again a rational harmonic function of the form (1). The degree of the numerator polynomial of $r_{c}$ is potentially different from $n_{p}$, and this allows for some flexibility in applications of Theorem 1.1. A second reason why we have formulated the result for the function $f_{c}$ (rather than just $f$) is the application of rational harmonic functions in the context of gravitational lensing; see [5] for a survey. In that application a constant shift represents the position of the light source of the lens, and the change of the number of zeros under movements of the light source is of great interest; see, e.g., [8] for more details.

Our note is organized as follows. In Section 2 we briefly recall the mathematical background, in particular the argument principle for continuous functions and a helpful result from complex dynamics. In Section 3 we prove Theorem 1.1. We also prove that a function that attains the bound in Theorem 1.1 is regular, which generalizes a result from [10]. In Section 4 we explain why the special case $n_{p}=n_{q}+1$ cannot be completely resolved by our method of proof, and we discuss the relation of Theorem 1.1 to all previously published bounds that we are aware of.

2 Mathematical background

Let $f$ be as in (1). Using the Wirtinger derivatives $\partial_{z}$ and $\partial_{\bar{z}}$ we can write the Jacobian of $f$ as

[TABLE]

If $z_{0}\in\mathbb{C}$ is a zero of $f$, i.e., $f(z_{0})=0$, then $z_{0}$ is called a sense-preserving, sense-reversing, or singular zero of $f$, if $|r^{\prime}(z_{0})|>1$, $|r^{\prime}(z_{0})|<1$, or $|r^{\prime}(z_{0})|=1$, respectively. The sense-preserving and sense-reversing zeros of $f$ are called the regular zeros. If $f$ has only such zeros, then $f$ is called regular, and otherwise $f$ is called singular. We denote the number of sense-preserving, sense-reversing, and singular zeros of $f$ in a set $S\subseteq\mathbb{C}$ by $N_{+}(f;S)$, $N_{-}(f;S)$, and $N_{0}(f;S)$, respectively. For $S=\mathbb{C}$ we simply write $N_{+}(f)$, $N_{-}(f)$, and $N_{0}(f)$. In our proofs we will use the following result on regular functions; see [4, Lemma].

Lemma 2.1

If $f$ is as in (1), then the set of complex numbers $c$ for which $f_{c}(z)=f(z)-c$ is regular, is open and dense in $\mathbb{C}$.

This lemma can be easily shown when using the fact that the function $f_{c}$ is singular if and only if $c$ is a caustic point of $f$; see [8, Proposition 2.2].

Let $\Gamma$ be a closed Jordan curve, and let $g$ be any function that is continuous and nonzero on $\Gamma$. Then the winding of $g$ on $\Gamma$ is defined as the change in the argument of $g(z)$ as $z$ travels once around $\Gamma$ in the positive direction, divided by $2\pi$, i.e.,

[TABLE]

The following result holds for the winding of the functions of our interest.

Theorem 2.2

Let $f$ be as in (1). If $f$ is nonzero and finite on a closed Jordan curve $\Gamma$ and has no singular zero in $\operatorname{int}(\Gamma)$, then

[TABLE]

where $P(f;\operatorname{int}(\Gamma))$ denotes the number of poles with multiplicities of $r$, and hence of $f$, in $\operatorname{int}(\Gamma)$.

We will frequently use the following version of Rouché’s theorem.

Theorem 2.3

Let $\Gamma$ be a closed Jordan curve and suppose that $f,g:\Gamma\rightarrow\mathbb{C}$ are continuous. If $|f(z)-g(z)|<|f(z)|+|g(z)|$ holds for all $z\in\Gamma$, then $V(f;\Gamma)=V(g;\Gamma)$.

For more details on the mathematical background described above we refer to [4], [8] and [13].

In addition, we will need a result on fixed points from complex dynamics. Let $z_{f}\in\mathbb{C}$ be a fixed point of a rational function $r$, i.e., $r(z_{f})=z_{f}$. Then $z_{f}$ is called attracting, repelling, or rationally neutral, if respectively $|r^{\prime}(z_{f})|<1$, $|r^{\prime}(z_{f})|>1$, or $|r^{\prime}(z_{f})|=1$. The following is a combination of [2, Chapter III, Theorem 2.2 and 2.3].

Theorem 2.4

If $z_{f}$ is an attracting or rationally neutral fixed point of a rational function $R$ with $\deg(R)\geq 2$, then exists a critical point $z_{c}$ of $R$, i.e., $R^{\prime}(z_{c})=0$, with $\lim_{k\rightarrow\infty}R^{k}(z_{c})=z_{f}$, where $R^{k}:=R\circ\cdots\circ R$ ($k$ times).

3 Main results

Our strategy to prove Theorem 1.1 is the following: First we determine $N_{+}(f_{c})-N_{-}(f_{c})$ for regular functions $f$ as in (1) and $f_{c}(z)=f(z)-c$ with respect to $n_{p}$ and $n_{q}$ using Theorem 2.2 and 2.3. Then we bound $N_{0}(f)+N_{-}(f)$ for general $f$ by the number of zeros of $r^{\prime}$ using Theorem 2.4. Finally, we combine both results with Lemma 2.1 in order to obtain the proof also for non-regular $f$.

We denote by $B_{M}(z_{0})$ the open disk of radius $M>0$ around $z_{0}\in\mathbb{C}$.

Lemma 3.1

Let $f$ be as in (1) and suppose that $f$ is regular, i.e., $N_{0}(f)=0$. Then for every $c\in\mathbb{C}$ the function $f_{c}(z)=f(z)-c$ satisfies

[TABLE]

{proof}

For each $c\in\mathbb{C}$ we can write $f_{c}$ as in (2). Obviously, $r$ and $r_{c}$ have the same poles, and the argument given in the Introduction shows that $f_{c}$ has finitely many zeros. Thus, if $\widetilde{M}>0$ is sufficiently large, we have $P(f_{c};B_{\widetilde{M}}(0))=n_{q}$ and $N_{+/-}(f_{c};B_{\widetilde{M}}(0))=N_{+/-}(f_{c})$.

First we assume $n_{p}\leq n_{q}$. Then $\lim_{|z|\rightarrow\infty}|r(z)|$ is finite (possibly zero), and for a sufficiently large $M\geq\widetilde{M}>0$,

[TABLE]

Using Theorem 2.2 and 2.3,

[TABLE]

which gives $N_{+}(f_{c})-N_{-}(f_{c})=n_{q}-1$.

Next, we assume $n_{p}>n_{q}+1$. Then $r$ can be written as $r(z)=\widetilde{p}(z)+\widetilde{r}(z)$, where $\widetilde{p}$ is a polynomial of exact degree $n_{p}-n_{q}\geq 2$, and $\widetilde{r}$ is a rational function with $\lim_{|z|\rightarrow\infty}|\widetilde{r}(z)|=0$. For a sufficiently large $M\geq\widetilde{M}>0$,

[TABLE]

Using again Theorem 2.2 and 2.3,

[TABLE]

and hence $n_{p}=N_{+}(f_{c})-N_{-}(f_{c})$.\eop

Much of the proof of the next result is based on the proof of [1, Theorem C.3]. We nevertheless include all steps for clarity and completeness of our presentation.

Lemma 3.2

Let $f$ be as in (1). Then for every $c\in\mathbb{C}$ the function $f_{c}(z)=f(z)-c$ satisfies

[TABLE]

{proof}

Let $z_{0}\in\mathbb{C}$ be a non-sense-preserving zero of $f$, i.e., $r(z_{0})=\overline{z}_{0}$ with $|r^{\prime}(z_{0})|\leq 1$. Then also $\overline{r}(r(z_{0}))=z_{0}$, or

[TABLE]

The derivative of the rational function $R$ is given by

[TABLE]

Thus,

[TABLE]

which shows that $z_{0}$ is an attracting or rationally neutral fixed point of $R$. By Theorem 2.4, there exits a critical point $z_{c}$ of $R$, i.e.,

[TABLE]

such that

[TABLE]

From (3) we obtain $r^{\prime}(z_{c})=0$ or $r^{\prime}(\overline{w}_{c})=0$, where $w_{c}:=r(z_{c})$. In the second case we use (4) and the continuity of $\overline{r}$ to obtain

[TABLE]

In summary, we have shown that if $z_{0}$ is a non-sense-preserving zero of $f$, and hence an attracting or rationally neutral fixed point of $R$, then there exists a critical point $\widetilde{z}_{c}$ of $r$ (in the first case $\widetilde{z}_{c}=z_{c}$, and in the second case $\widetilde{z}_{c}=\overline{w}_{c}$) with $\lim_{k\rightarrow\infty}R^{k}(\widetilde{z}_{c})=z_{0}$. Clearly, different fixed points of $R$ attract disjoint sets of (critical) points, and therefore $N_{0}(f)+N_{-}(f)$ is less than or equal to the number of zeros of

[TABLE]

which is at most $n_{p}+n_{q}-1$. Finally, for every $c\in\mathbb{C}$ we can write $f_{c}$ in the form (2). Now, by the same argument as above, $N_{0}(f_{c})+N_{-}(f_{c})$ is less than or equal to the number of zeros of $r_{c}^{\prime}=r^{\prime}$, which is at most $n_{p}+n_{q}-1$. \eop

In order to control the behavior of singular functions we will also need the fact that a small constant perturbation does not reduce the number of sense-preserving zeros of $f$.

Lemma 3.3

Let $f$ be as in (1) and let $z_{1},\dots,z_{k}$ be the sense-preserving zeros of $f$. Then $N_{+}(f)\leq N_{+}(f_{c})$ for all $c\in\mathbb{C}$ with sufficiently small $|c|$.

{proof}

Since $f$ has finitely many zeros we can always find an $\varepsilon>0$, such that $f$ is sense-preserving on $B_{\varepsilon}(z_{j})$ for $j=1,\dots,k$, and $B_{\varepsilon}(z_{j})\cap B_{\varepsilon}(z_{\ell})=\varnothing$ for $j\neq\ell$, and

[TABLE]

In particular, the condition $\delta>0$ just means that none of the boundaries $\partial B_{\varepsilon}(z_{j})$ contains a zero of $f$. By construction, for all $z\in\cup_{j=1}^{k}\,\partial B_{\varepsilon}(z_{j})$ we have for $|c|<\delta$

[TABLE]

With Theorem 2.3 we get, for each $j=1,\dots,k$,

[TABLE]

where in the third equality we used that a constant shift preserves the orientation of $f$ on $\mathbb{C}$. \eop

Proof of Theorem 1.1:

Let $f$ be as in (1) and let $c\in\mathbb{C}$ be arbitrary. Due to Lemma 2.1, there exists a sequence $\{c_{k}\}_{k\in\mathbb{N}}\subset\mathbb{C}$, such that the functions $f_{c_{k}}(z):=f(z)-c_{k}$ are regular, and $c_{k}\rightarrow c$. If $f_{c}$ is regular, we can chose $c_{k}=c$ for all $k\in\mathbb{N}$.

If $n_{p}<n_{q}$, then for sufficiently small $|c_{k}-c|$,

[TABLE]

where we have used Lemma 3.1–3.3. Analogously, if $n_{p}>n_{q}+1$, then

[TABLE]

Finally, if $n_{p}=n_{q}$, then the rational function $r$ in (1) can be written as

[TABLE]

where $\alpha\neq 0$, ${\rm deg}(\widetilde{p})<n_{p}=n$ and ${\rm deg}(\widetilde{q})<n_{q}=n$. Now the numerator degree is

[TABLE]

and applying the bound from the first case to the function

[TABLE]

gives

[TABLE]

which completes the proof.

We will now show that any function $f$ as in (1) that attains one of the bounds of Theorem 1.1 is regular, which generalizes [10, Theorem 3.1].

Lemma 3.4

Let $f$ be as in (1) and suppose that $f$ is singular. Then there exists a constant $c\in\mathbb{C}$ such that $N_{+}(f)<N_{+}(f_{c})$.

{proof}

Note that $f$ has at least one sense-preserving zero due to Lemma 3.1. Let $z_{1},\dots,z_{k}$ be the sense-preserving zeros of $f$ with the corresponding disks $B_{\varepsilon}(z_{1}),\dots,B_{\varepsilon}(z_{k})$ as well as $\delta>0$ as in the proof of Lemma 3.3.

Let $z_{0}$ be a singular zero of $f$. We then have $|f(z)|<\delta$ in $B_{\widetilde{\varepsilon}}(z_{0})$ and

[TABLE]

if $\widetilde{\varepsilon}>0$ is small enough. Let $\widetilde{z}\in B_{\widetilde{\varepsilon}}(z_{0})$ be arbitrary with $|r^{\prime}(\widetilde{z})|>1$. Then $|f(\widetilde{z})|<\delta$, and the proof of Lemma 3.3 shows that the function $\widetilde{f}:=f-f(\widetilde{z})$ has one sense-preserving zero in each of the $k$ disks $B_{\varepsilon}(z_{1}),\dots,B_{\varepsilon}(z_{k})$. Moreover, $\widetilde{f}$ has an additional sense-preserving zero at $\widetilde{z}$, which means that $N_{+}(f)<N_{+}(\widetilde{f})$.\eop

Note that the bounds of Lemma 3.3 and 3.4 also hold for $N_{-}$.

Theorem 3.5

Let $f$ as in (1) and $c\in\mathbb{C}$. If $f_{c}$ attains one of the bounds of Theorem 1.1, then $f_{c}$ is regular.

{proof}

Let $f$ be as in (1) with $n_{p}\leq n_{q}$, and let $c\in\mathbb{C}$ be arbitrary. Due to Lemma 2.1 we can choose a sequence $\{c_{k}\}_{k\in\mathbb{N}}$, such that the functions $f_{c_{k}}$ are regular and $c_{k}\rightarrow c$. Using Lemma 3.1–3.3 we obtain

[TABLE]

Now suppose that $f_{c}$ attains the bound of Theorem 1.1. Then by Lemma 3.2 we get

[TABLE]

and with (5) we obtain $N_{+}(f_{c})=n_{p}+2n_{p}-2$. If $f_{c}$ would be singular, we could choose a constant $\widetilde{c}\in\mathbb{C}$ such that $N_{+}(f_{\widetilde{c}})>N_{+}(f_{c})$, but this is in contradiction to the upper bound (5), which holds for an arbitrary constant.

The proof for the case $n_{p}>n_{q}+1$ is analogous. \eop

4 Discussion of Theorem 1.1

The reason why the special case $n_{p}=n_{q}+1$ is “missing” in Theorem 1.1 is that this case is not covered in Lemma 3.1. Note that in this case we can write $r(z)=\alpha z+\widetilde{r}(z)$ for some $\alpha\neq 0$ and with $\lim_{|z|\rightarrow\infty}|\widetilde{r}(z)|$ finite (possibly zero). If $|\alpha|>1$, then for a sufficiently large $M\geq\widetilde{M}>0$ we obtain (cf. the proof of Lemma 3.1)

[TABLE]

for all $z\in\partial B_{M}(0)$, leading to $1=n_{p}-n_{q}=N_{+}(f_{c})-N_{-}(f_{c})-n_{q}$, or $N_{+}(f_{c})-N_{-}(f_{c})=n_{q}+1=n_{p}$. This gives the bound $N(f_{c})\leq 3n_{p}+2n_{q}-2=5n_{p}-4$, but we know that $N(f_{c})\leq 5n_{p}-6$ from [9]. For the case $|\alpha|=1$ the method of proof used for Lemma 3.1 would give no result, and for $|\alpha|<1$ we would indeed obtain $N(f_{c})\leq 5n_{p}-6$, since then

[TABLE]

for all $z\in\partial B_{M}(0)$, giving $N_{+}(f_{c})-N_{-}(f_{c})=n_{q}-1=n_{p}-2$.

Apart from the special case $n_{p}=n_{q}+1$, Theorem 1.1 covers all possible choices of $n_{p}$ and $n_{p}$ with $n=\max\{n_{p},n_{q}\}\geq 2$, and all previous results in this area that we are aware of. In particular:

1. (i)

For any choices of $n_{p}$ and $n_{p}$ in Theorem 1.1 (except $n_{p}=n_{p}+1$) we get $N(f_{c})\leq 5n-5$, which is the general bound from [4]. 2. (i)

For $n=n_{p}\geq 2$ and $n_{q}=0$, Theorem 1.1 gives the bound for harmonic polynomials from [6]. 3. (iii)

For $n=n_{p}>n_{q}+1$, Theorem 1.1 gives the same bound as in [7]. 4. (iv)

For $n=n_{p}=n_{q}+j$ with $j\geq 2$ we get $N(f)\leq 5n-2(j+1)$. For $j=2$ this is the same bound as in [10], and for $j>2$ our new bound is smaller than the bound in [10] ($5n-6$). 5. (v)

For $n=n_{q}=n_{p}+j$ with $j\geq 1$ we get $N(f)\leq 5n-(2j+3)$. For $j=1$ this is the same bound as in [4], and for $j>1$ our new bound is smaller than the previous one.

The following upper bounds on the maximal number of zeros of $f$ as in (1) have been shown to be sharp:

[TABLE]

Let $f$ be the Rhie function from [11] (see also [12, 13]), which has a rational function $r$ of the type $(n-1,n)$, and which has $5n-5$ zeros. The results in [8] imply that for sufficiently small $|c|>0$ the function $f_{c}$ has the same number of zeros as $f$. The corresponding rational function $r_{c}$ then is of the type $(n,n)$, so the bound $N(f)\leq 5n-5$ is sharp also in the case $(n,n)$. More generally, the sharpness of the bounds in Theorem 1.1 for $n_{p}\geq n_{q}$ is discussed in [7, Theorem C], while the case $n_{p}<n_{q}+1$ remains a subject of future research.

Acknowledgements

We thank an anonymous referee for several helpful suggestions, and in particular for pointing out the technical report [7].