Analogue of the Mean value theorem for holomorphic functions












2












$begingroup$


Let $f:mathbb{C}rightarrow mathbb{C}$ be entire. Let $w_1,w_2$ be any two distinct complex numbers. Must there exist $cin overline{B_{|w_2-w_1|}(w_1)}$ such that $f'(c)=frac{f(w_2)-f(w_1)}{w_2-w_1}$ ?



Note:



I have shown the folowing: Let $[w_1, w_2]$ be the line segment with endpoints $w_1,w_2$. Then there exists $c_1,c_2in [w_1,w_2]$ such that:
$$Re(f'(c_1))+Im(f'(c_2))i=frac{f(w_2)-f(w_1)}{w_2-w_1} ...(1)$$



I was wondering whether allowing $c_1,c_2$ to lie in $overline{B_{|w_2-w_1|}(w_1)}$ (instead of just the line segment joining $w_1,w_2$) enables us to choose $c_1,c_2$ such that $(1)$ holds and $c_1=c_2$.



Thank you






complex-analysis examples-counterexamples







share|cite|improve this question











$endgroup$








  • 2




    $begingroup$
    No, $f(z) = e^z$, $w_2 = w_1 + 2pi i$ is the classical counterexample. You have the estimate, $lvert f(w_2) - f(w_1)rvert leqslant lvert w_2-w_1rvert cdot max { lvert f'(z)rvert : z in [w_1,w_2]}$, and the two point-form for real and imaginary part. That's it.
    $endgroup$
    – Daniel Fischer♦
    Jan 10 '14 at 18:38










  • $begingroup$
    @DanielFischer I don't understand the part : "and the two point-form ..."
    $endgroup$
    – Amr
    Jan 10 '14 at 18:43










  • $begingroup$
    Your equation $(1)$ in the Note:.
    $endgroup$
    – Daniel Fischer♦
    Jan 10 '14 at 18:44










  • $begingroup$
    @DanielFischer Ahhhhh :) Thanks a lot.
    $endgroup$
    – Amr
    Jan 10 '14 at 18:49
















2












$begingroup$


Let $f:mathbb{C}rightarrow mathbb{C}$ be entire. Let $w_1,w_2$ be any two distinct complex numbers. Must there exist $cin overline{B_{|w_2-w_1|}(w_1)}$ such that $f'(c)=frac{f(w_2)-f(w_1)}{w_2-w_1}$ ?



Note:



I have shown the folowing: Let $[w_1, w_2]$ be the line segment with endpoints $w_1,w_2$. Then there exists $c_1,c_2in [w_1,w_2]$ such that:
$$Re(f'(c_1))+Im(f'(c_2))i=frac{f(w_2)-f(w_1)}{w_2-w_1} ...(1)$$



I was wondering whether allowing $c_1,c_2$ to lie in $overline{B_{|w_2-w_1|}(w_1)}$ (instead of just the line segment joining $w_1,w_2$) enables us to choose $c_1,c_2$ such that $(1)$ holds and $c_1=c_2$.



Thank you






complex-analysis examples-counterexamples







share|cite|improve this question











$endgroup$








  • 2




    $begingroup$
    No, $f(z) = e^z$, $w_2 = w_1 + 2pi i$ is the classical counterexample. You have the estimate, $lvert f(w_2) - f(w_1)rvert leqslant lvert w_2-w_1rvert cdot max { lvert f'(z)rvert : z in [w_1,w_2]}$, and the two point-form for real and imaginary part. That's it.
    $endgroup$
    – Daniel Fischer♦
    Jan 10 '14 at 18:38










  • $begingroup$
    @DanielFischer I don't understand the part : "and the two point-form ..."
    $endgroup$
    – Amr
    Jan 10 '14 at 18:43










  • $begingroup$
    Your equation $(1)$ in the Note:.
    $endgroup$
    – Daniel Fischer♦
    Jan 10 '14 at 18:44










  • $begingroup$
    @DanielFischer Ahhhhh :) Thanks a lot.
    $endgroup$
    – Amr
    Jan 10 '14 at 18:49














2












2








2


2



$begingroup$


Let $f:mathbb{C}rightarrow mathbb{C}$ be entire. Let $w_1,w_2$ be any two distinct complex numbers. Must there exist $cin overline{B_{|w_2-w_1|}(w_1)}$ such that $f'(c)=frac{f(w_2)-f(w_1)}{w_2-w_1}$ ?



Note:



I have shown the folowing: Let $[w_1, w_2]$ be the line segment with endpoints $w_1,w_2$. Then there exists $c_1,c_2in [w_1,w_2]$ such that:
$$Re(f'(c_1))+Im(f'(c_2))i=frac{f(w_2)-f(w_1)}{w_2-w_1} ...(1)$$



I was wondering whether allowing $c_1,c_2$ to lie in $overline{B_{|w_2-w_1|}(w_1)}$ (instead of just the line segment joining $w_1,w_2$) enables us to choose $c_1,c_2$ such that $(1)$ holds and $c_1=c_2$.



Thank you






complex-analysis examples-counterexamples







share|cite|improve this question











$endgroup$




Let $f:mathbb{C}rightarrow mathbb{C}$ be entire. Let $w_1,w_2$ be any two distinct complex numbers. Must there exist $cin overline{B_{|w_2-w_1|}(w_1)}$ such that $f'(c)=frac{f(w_2)-f(w_1)}{w_2-w_1}$ ?



Note:



I have shown the folowing: Let $[w_1, w_2]$ be the line segment with endpoints $w_1,w_2$. Then there exists $c_1,c_2in [w_1,w_2]$ such that:
$$Re(f'(c_1))+Im(f'(c_2))i=frac{f(w_2)-f(w_1)}{w_2-w_1} ...(1)$$



I was wondering whether allowing $c_1,c_2$ to lie in $overline{B_{|w_2-w_1|}(w_1)}$ (instead of just the line segment joining $w_1,w_2$) enables us to choose $c_1,c_2$ such that $(1)$ holds and $c_1=c_2$.



Thank you






complex-analysis examples-counterexamples




complex-analysis examples-counterexamples






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 12 '18 at 11:26









Brahadeesh

6,46942363




6,46942363










asked Jan 10 '14 at 18:32









AmrAmr

14.4k43292




14.4k43292








  • 2




    $begingroup$
    No, $f(z) = e^z$, $w_2 = w_1 + 2pi i$ is the classical counterexample. You have the estimate, $lvert f(w_2) - f(w_1)rvert leqslant lvert w_2-w_1rvert cdot max { lvert f'(z)rvert : z in [w_1,w_2]}$, and the two point-form for real and imaginary part. That's it.
    $endgroup$
    – Daniel Fischer♦
    Jan 10 '14 at 18:38










  • $begingroup$
    @DanielFischer I don't understand the part : "and the two point-form ..."
    $endgroup$
    – Amr
    Jan 10 '14 at 18:43










  • $begingroup$
    Your equation $(1)$ in the Note:.
    $endgroup$
    – Daniel Fischer♦
    Jan 10 '14 at 18:44










  • $begingroup$
    @DanielFischer Ahhhhh :) Thanks a lot.
    $endgroup$
    – Amr
    Jan 10 '14 at 18:49














  • 2




    $begingroup$
    No, $f(z) = e^z$, $w_2 = w_1 + 2pi i$ is the classical counterexample. You have the estimate, $lvert f(w_2) - f(w_1)rvert leqslant lvert w_2-w_1rvert cdot max { lvert f'(z)rvert : z in [w_1,w_2]}$, and the two point-form for real and imaginary part. That's it.
    $endgroup$
    – Daniel Fischer♦
    Jan 10 '14 at 18:38










  • $begingroup$
    @DanielFischer I don't understand the part : "and the two point-form ..."
    $endgroup$
    – Amr
    Jan 10 '14 at 18:43










  • $begingroup$
    Your equation $(1)$ in the Note:.
    $endgroup$
    – Daniel Fischer♦
    Jan 10 '14 at 18:44










  • $begingroup$
    @DanielFischer Ahhhhh :) Thanks a lot.
    $endgroup$
    – Amr
    Jan 10 '14 at 18:49








2




2




$begingroup$
No, $f(z) = e^z$, $w_2 = w_1 + 2pi i$ is the classical counterexample. You have the estimate, $lvert f(w_2) - f(w_1)rvert leqslant lvert w_2-w_1rvert cdot max { lvert f'(z)rvert : z in [w_1,w_2]}$, and the two point-form for real and imaginary part. That's it.
$endgroup$
– Daniel Fischer♦
Jan 10 '14 at 18:38




$begingroup$
No, $f(z) = e^z$, $w_2 = w_1 + 2pi i$ is the classical counterexample. You have the estimate, $lvert f(w_2) - f(w_1)rvert leqslant lvert w_2-w_1rvert cdot max { lvert f'(z)rvert : z in [w_1,w_2]}$, and the two point-form for real and imaginary part. That's it.
$endgroup$
– Daniel Fischer♦
Jan 10 '14 at 18:38












$begingroup$
@DanielFischer I don't understand the part : "and the two point-form ..."
$endgroup$
– Amr
Jan 10 '14 at 18:43




$begingroup$
@DanielFischer I don't understand the part : "and the two point-form ..."
$endgroup$
– Amr
Jan 10 '14 at 18:43












$begingroup$
Your equation $(1)$ in the Note:.
$endgroup$
– Daniel Fischer♦
Jan 10 '14 at 18:44




$begingroup$
Your equation $(1)$ in the Note:.
$endgroup$
– Daniel Fischer♦
Jan 10 '14 at 18:44












$begingroup$
@DanielFischer Ahhhhh :) Thanks a lot.
$endgroup$
– Amr
Jan 10 '14 at 18:49




$begingroup$
@DanielFischer Ahhhhh :) Thanks a lot.
$endgroup$
– Amr
Jan 10 '14 at 18:49










1 Answer
1






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$begingroup$

No, in general such a $c$ does not exist. The classical counterexample is $f(z) = e^z$, with $w_2 = w_1 + 2pi i$. Then



$$frac{f(w_2) - f(w_1)}{w_2-w_1} = 0,$$



but $f'(c) neq 0$ for all $cinmathbb{C}$.



What you have in terms of the mean value theorem for holomorphic functions is the estimate



$$lvert f(w_2) - f(w_1)rvert leqslant lvert w_2-w_1rvertcdot max leftlbrace lvert f'(zeta)rvert : zeta in [w_1,w_2]rightrbrace,$$



and your equation $(1)$ with generally different points for the real and imaginary part.






share|cite|improve this answer











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    1 Answer
    1






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    active

    oldest

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    active

    oldest

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    5












    $begingroup$

    No, in general such a $c$ does not exist. The classical counterexample is $f(z) = e^z$, with $w_2 = w_1 + 2pi i$. Then



    $$frac{f(w_2) - f(w_1)}{w_2-w_1} = 0,$$



    but $f'(c) neq 0$ for all $cinmathbb{C}$.



    What you have in terms of the mean value theorem for holomorphic functions is the estimate



    $$lvert f(w_2) - f(w_1)rvert leqslant lvert w_2-w_1rvertcdot max leftlbrace lvert f'(zeta)rvert : zeta in [w_1,w_2]rightrbrace,$$



    and your equation $(1)$ with generally different points for the real and imaginary part.






    share|cite|improve this answer











    $endgroup$


















      5












      $begingroup$

      No, in general such a $c$ does not exist. The classical counterexample is $f(z) = e^z$, with $w_2 = w_1 + 2pi i$. Then



      $$frac{f(w_2) - f(w_1)}{w_2-w_1} = 0,$$



      but $f'(c) neq 0$ for all $cinmathbb{C}$.



      What you have in terms of the mean value theorem for holomorphic functions is the estimate



      $$lvert f(w_2) - f(w_1)rvert leqslant lvert w_2-w_1rvertcdot max leftlbrace lvert f'(zeta)rvert : zeta in [w_1,w_2]rightrbrace,$$



      and your equation $(1)$ with generally different points for the real and imaginary part.






      share|cite|improve this answer











      $endgroup$
















        5












        5








        5





        $begingroup$

        No, in general such a $c$ does not exist. The classical counterexample is $f(z) = e^z$, with $w_2 = w_1 + 2pi i$. Then



        $$frac{f(w_2) - f(w_1)}{w_2-w_1} = 0,$$



        but $f'(c) neq 0$ for all $cinmathbb{C}$.



        What you have in terms of the mean value theorem for holomorphic functions is the estimate



        $$lvert f(w_2) - f(w_1)rvert leqslant lvert w_2-w_1rvertcdot max leftlbrace lvert f'(zeta)rvert : zeta in [w_1,w_2]rightrbrace,$$



        and your equation $(1)$ with generally different points for the real and imaginary part.






        share|cite|improve this answer











        $endgroup$



        No, in general such a $c$ does not exist. The classical counterexample is $f(z) = e^z$, with $w_2 = w_1 + 2pi i$. Then



        $$frac{f(w_2) - f(w_1)}{w_2-w_1} = 0,$$



        but $f'(c) neq 0$ for all $cinmathbb{C}$.



        What you have in terms of the mean value theorem for holomorphic functions is the estimate



        $$lvert f(w_2) - f(w_1)rvert leqslant lvert w_2-w_1rvertcdot max leftlbrace lvert f'(zeta)rvert : zeta in [w_1,w_2]rightrbrace,$$



        and your equation $(1)$ with generally different points for the real and imaginary part.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Jan 10 '14 at 21:36


























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