Zeros of an infinite series
$begingroup$
Let $sum_{j=1}^{infty}a_{j}$ be a convergent series of positive numbers and ${z_{j}}_{j=1}^infty$ a closed discrete subset of the open unit disc $mathbb{D}$. Then $h(z):=sum_{j=1}^{infty}frac{a_{j}}{z-z_{j}}$ is a meromorphic function on $mathbb{D}$.
If we only consider the case of infinite sum, does $h(z)=sum_{j=1}^{infty}frac{a_{j}}{z-z_{j}}$ always have infinitely many zeros on $mathbb{D}$? Note that $h$ never vanishes outside $mathbb{D}$.
This question comes from the paper Bounded Projective Functions and Hyperbolic Metrics with Isolated Singularities (Example 1.1 and Question 3.3).
cv.complex-variables
edited Jan 20 at 4:04
Peter Mortensen
1635
asked Jan 19 at 14:03
Yu FengYu Feng
634
$endgroup$
add a comment |
$begingroup$
Let $sum_{j=1}^{infty}a_{j}$ be a convergent series of positive numbers and ${z_{j}}_{j=1}^infty$ a closed discrete subset of the open unit disc $mathbb{D}$. Then $h(z):=sum_{j=1}^{infty}frac{a_{j}}{z-z_{j}}$ is a meromorphic function on $mathbb{D}$.
If we only consider the case of infinite sum, does $h(z)=sum_{j=1}^{infty}frac{a_{j}}{z-z_{j}}$ always have infinitely many zeros on $mathbb{D}$? Note that $h$ never vanishes outside $mathbb{D}$.
This question comes from the paper Bounded Projective Functions and Hyperbolic Metrics with Isolated Singularities (Example 1.1 and Question 3.3).
cv.complex-variables
edited Jan 20 at 4:04
Peter Mortensen
1635
asked Jan 19 at 14:03
Yu FengYu Feng
634
$endgroup$
add a comment |
$begingroup$
Let $sum_{j=1}^{infty}a_{j}$ be a convergent series of positive numbers and ${z_{j}}_{j=1}^infty$ a closed discrete subset of the open unit disc $mathbb{D}$. Then $h(z):=sum_{j=1}^{infty}frac{a_{j}}{z-z_{j}}$ is a meromorphic function on $mathbb{D}$.
If we only consider the case of infinite sum, does $h(z)=sum_{j=1}^{infty}frac{a_{j}}{z-z_{j}}$ always have infinitely many zeros on $mathbb{D}$? Note that $h$ never vanishes outside $mathbb{D}$.
This question comes from the paper Bounded Projective Functions and Hyperbolic Metrics with Isolated Singularities (Example 1.1 and Question 3.3).
cv.complex-variables
edited Jan 20 at 4:04
Peter Mortensen
1635
asked Jan 19 at 14:03
Yu FengYu Feng
634
$endgroup$
Let $sum_{j=1}^{infty}a_{j}$ be a convergent series of positive numbers and ${z_{j}}_{j=1}^infty$ a closed discrete subset of the open unit disc $mathbb{D}$. Then $h(z):=sum_{j=1}^{infty}frac{a_{j}}{z-z_{j}}$ is a meromorphic function on $mathbb{D}$.
If we only consider the case of infinite sum, does $h(z)=sum_{j=1}^{infty}frac{a_{j}}{z-z_{j}}$ always have infinitely many zeros on $mathbb{D}$? Note that $h$ never vanishes outside $mathbb{D}$.
This question comes from the paper Bounded Projective Functions and Hyperbolic Metrics with Isolated Singularities (Example 1.1 and Question 3.3).
cv.complex-variables
cv.complex-variables
edited Jan 20 at 4:04
Peter Mortensen
1635
asked Jan 19 at 14:03
Yu FengYu Feng
634
edited Jan 20 at 4:04
Peter Mortensen
1635
asked Jan 19 at 14:03
Yu FengYu Feng
634
edited Jan 20 at 4:04
Peter Mortensen
1635
edited Jan 20 at 4:04
Peter Mortensen
1635
edited Jan 20 at 4:04
Peter Mortensen
1635
1635
asked Jan 19 at 14:03
Yu FengYu Feng
634
asked Jan 19 at 14:03
Yu FengYu Feng
634
asked Jan 19 at 14:03
Yu FengYu Feng
634
634
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
This was conjectured by J. Borcea, and a counterexample was constructed by J. Langley:
MR2317957
Langley, J. K.
Equilibrium points of logarithmic potentials on convex domains,
Proc. Amer. Math. Soc. 135 (2007), no. 9, 2821–2826.
His counterexample has an additional property that $z_k$ tend to a limit on the unit circle.
However, your question has positive answer under some additional conditions imposed on $z_k$; this is proved in the same paper of Langley.
Notice that a similar question in the plane (under the assumptions $z_ktoinfty$, and
$$sum_k a_k/|z_k|<infty,$$
$f$ is meromorphic in the plane) is unsolved, despite a lot of research on this question.
answered Jan 19 at 14:29
Alexandre EremenkoAlexandre Eremenko
49.9k6138256
$endgroup$
1
$begingroup$
What do you mean by "the is true"?
$endgroup$
– Peter Mortensen
Jan 20 at 2:49
$begingroup$
@Peter Mortensen: Thanks. I corrected the misprint.
$endgroup$
– Alexandre Eremenko
Jan 20 at 3:53
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "504"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f321246%2fzeros-of-an-infinite-series%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
This was conjectured by J. Borcea, and a counterexample was constructed by J. Langley:
MR2317957
Langley, J. K.
Equilibrium points of logarithmic potentials on convex domains,
Proc. Amer. Math. Soc. 135 (2007), no. 9, 2821–2826.
His counterexample has an additional property that $z_k$ tend to a limit on the unit circle.
However, your question has positive answer under some additional conditions imposed on $z_k$; this is proved in the same paper of Langley.
Notice that a similar question in the plane (under the assumptions $z_ktoinfty$, and
$$sum_k a_k/|z_k|<infty,$$
$f$ is meromorphic in the plane) is unsolved, despite a lot of research on this question.
answered Jan 19 at 14:29
Alexandre EremenkoAlexandre Eremenko
49.9k6138256
$endgroup$
1
$begingroup$
What do you mean by "the is true"?
$endgroup$
– Peter Mortensen
Jan 20 at 2:49
$begingroup$
@Peter Mortensen: Thanks. I corrected the misprint.
$endgroup$
– Alexandre Eremenko
Jan 20 at 3:53
add a comment |
$begingroup$
This was conjectured by J. Borcea, and a counterexample was constructed by J. Langley:
MR2317957
Langley, J. K.
Equilibrium points of logarithmic potentials on convex domains,
Proc. Amer. Math. Soc. 135 (2007), no. 9, 2821–2826.
His counterexample has an additional property that $z_k$ tend to a limit on the unit circle.
However, your question has positive answer under some additional conditions imposed on $z_k$; this is proved in the same paper of Langley.
Notice that a similar question in the plane (under the assumptions $z_ktoinfty$, and
$$sum_k a_k/|z_k|<infty,$$
$f$ is meromorphic in the plane) is unsolved, despite a lot of research on this question.
answered Jan 19 at 14:29
Alexandre EremenkoAlexandre Eremenko
49.9k6138256
$endgroup$
1
$begingroup$
What do you mean by "the is true"?
$endgroup$
– Peter Mortensen
Jan 20 at 2:49
$begingroup$
@Peter Mortensen: Thanks. I corrected the misprint.
$endgroup$
– Alexandre Eremenko
Jan 20 at 3:53
add a comment |
$begingroup$
This was conjectured by J. Borcea, and a counterexample was constructed by J. Langley:
MR2317957
Langley, J. K.
Equilibrium points of logarithmic potentials on convex domains,
Proc. Amer. Math. Soc. 135 (2007), no. 9, 2821–2826.
His counterexample has an additional property that $z_k$ tend to a limit on the unit circle.
However, your question has positive answer under some additional conditions imposed on $z_k$; this is proved in the same paper of Langley.
Notice that a similar question in the plane (under the assumptions $z_ktoinfty$, and
$$sum_k a_k/|z_k|<infty,$$
$f$ is meromorphic in the plane) is unsolved, despite a lot of research on this question.
answered Jan 19 at 14:29
Alexandre EremenkoAlexandre Eremenko
49.9k6138256
$endgroup$
This was conjectured by J. Borcea, and a counterexample was constructed by J. Langley:
MR2317957
Langley, J. K.
Equilibrium points of logarithmic potentials on convex domains,
Proc. Amer. Math. Soc. 135 (2007), no. 9, 2821–2826.
His counterexample has an additional property that $z_k$ tend to a limit on the unit circle.
However, your question has positive answer under some additional conditions imposed on $z_k$; this is proved in the same paper of Langley.
Notice that a similar question in the plane (under the assumptions $z_ktoinfty$, and
$$sum_k a_k/|z_k|<infty,$$
$f$ is meromorphic in the plane) is unsolved, despite a lot of research on this question.
answered Jan 19 at 14:29
Alexandre EremenkoAlexandre Eremenko
49.9k6138256
edited Jan 20 at 3:59
answered Jan 19 at 14:29
Alexandre EremenkoAlexandre Eremenko
49.9k6138256
answered Jan 19 at 14:29
Alexandre EremenkoAlexandre Eremenko
49.9k6138256
answered Jan 19 at 14:29
Alexandre EremenkoAlexandre Eremenko
49.9k6138256
49.9k6138256
1
$begingroup$
What do you mean by "the is true"?
$endgroup$
– Peter Mortensen
Jan 20 at 2:49
$begingroup$
@Peter Mortensen: Thanks. I corrected the misprint.
$endgroup$
– Alexandre Eremenko
Jan 20 at 3:53
add a comment |
1
$begingroup$
What do you mean by "the is true"?
$endgroup$
– Peter Mortensen
Jan 20 at 2:49
$begingroup$
@Peter Mortensen: Thanks. I corrected the misprint.
$endgroup$
– Alexandre Eremenko
Jan 20 at 3:53
1
1
$begingroup$
What do you mean by "the is true"?
$endgroup$
– Peter Mortensen
Jan 20 at 2:49
$begingroup$
What do you mean by "the is true"?
$endgroup$
– Peter Mortensen
Jan 20 at 2:49
$begingroup$
@Peter Mortensen: Thanks. I corrected the misprint.
$endgroup$
– Alexandre Eremenko
Jan 20 at 3:53
$begingroup$
@Peter Mortensen: Thanks. I corrected the misprint.
$endgroup$
– Alexandre Eremenko
Jan 20 at 3:53
add a comment |
Thanks for contributing an answer to MathOverflow!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f321246%2fzeros-of-an-infinite-series%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
