Zeros of an infinite series












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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







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    12












    $begingroup$


    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







    share|cite|improve this question









    New contributor




    Yu Feng is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.







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      12












      12








      12


      4



      $begingroup$


      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







      share|cite|improve this question









      New contributor




      Yu Feng is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.







      $endgroup$




      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






      share|cite|improve this question









      New contributor




      Yu Feng is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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      share|cite|improve this question









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      edited 2 days ago









      Peter Mortensen

      1635




      1635






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      asked 2 days ago









      Yu FengYu Feng

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          1 Answer
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          19












          $begingroup$

          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.






          share|cite|improve this answer











          $endgroup$









          • 1




            $begingroup$
            What do you mean by "the is true"?
            $endgroup$
            – Peter Mortensen
            2 days ago












          • $begingroup$
            @Peter Mortensen: Thanks. I corrected the misprint.
            $endgroup$
            – Alexandre Eremenko
            2 days ago











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          1 Answer
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          active

          oldest

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          19












          $begingroup$

          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.






          share|cite|improve this answer











          $endgroup$









          • 1




            $begingroup$
            What do you mean by "the is true"?
            $endgroup$
            – Peter Mortensen
            2 days ago












          • $begingroup$
            @Peter Mortensen: Thanks. I corrected the misprint.
            $endgroup$
            – Alexandre Eremenko
            2 days ago
















          19












          $begingroup$

          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.






          share|cite|improve this answer











          $endgroup$









          • 1




            $begingroup$
            What do you mean by "the is true"?
            $endgroup$
            – Peter Mortensen
            2 days ago












          • $begingroup$
            @Peter Mortensen: Thanks. I corrected the misprint.
            $endgroup$
            – Alexandre Eremenko
            2 days ago














          19












          19








          19





          $begingroup$

          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.






          share|cite|improve this answer











          $endgroup$



          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.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited 2 days ago

























          answered 2 days ago









          Alexandre EremenkoAlexandre Eremenko

          49.8k6138256




          49.8k6138256








          • 1




            $begingroup$
            What do you mean by "the is true"?
            $endgroup$
            – Peter Mortensen
            2 days ago












          • $begingroup$
            @Peter Mortensen: Thanks. I corrected the misprint.
            $endgroup$
            – Alexandre Eremenko
            2 days ago














          • 1




            $begingroup$
            What do you mean by "the is true"?
            $endgroup$
            – Peter Mortensen
            2 days ago












          • $begingroup$
            @Peter Mortensen: Thanks. I corrected the misprint.
            $endgroup$
            – Alexandre Eremenko
            2 days ago








          1




          1




          $begingroup$
          What do you mean by "the is true"?
          $endgroup$
          – Peter Mortensen
          2 days ago






          $begingroup$
          What do you mean by "the is true"?
          $endgroup$
          – Peter Mortensen
          2 days ago














          $begingroup$
          @Peter Mortensen: Thanks. I corrected the misprint.
          $endgroup$
          – Alexandre Eremenko
          2 days ago




          $begingroup$
          @Peter Mortensen: Thanks. I corrected the misprint.
          $endgroup$
          – Alexandre Eremenko
          2 days ago










          Yu Feng is a new contributor. Be nice, and check out our Code of Conduct.










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