If $f(z)=z+a_2z^2+…+a_nz^n$ is injective on the unit disk, then $|a_2|≤frac12(n-1)$
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Suppose $f(z)=z+a_2z^2+...+a_nz^n$ is injective in $D:={z:|z|<1}$. Show that $|a_2|≤(n-1)/2$ and $|a_n|≤1/n$.
Partial Solution
Assume without loss of generality that $a_nne0$. If $f$ is injective, then $f'(z)≠0$ in $D$. Now, $$f'(z)=1+2a_2z+...+na_nz^{n-1}$$ By the fundamental theorem of algebra, $$f'(z)=na_n(z-r_1)...(z-r_{n-1})$$ where $r_1,...,r_{n-1}$ are the roots of $f'(z)$, and $$f'(0)=1=na_n(-1)^nr_1r_2...r_{n-1}$$ Note that $|r_i|geq1$ for each $i=1,...,n-1$, hence $$|r_1r_2...r_{n-1}|geq1$$ and, finally, $$|a_n|=frac1{n,|r_1r_2...r_{n-1}|}leqfrac{1}{n}$$
What I cannot figure out, though, is how to prove that $|a_2|≤(n-1)/2$. I have tried similar arguments, but cannot seem to get it. Any hints would be greatly appreciated.
complex-analysis polynomials
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$begingroup$
Suppose $f(z)=z+a_2z^2+...+a_nz^n$ is injective in $D:={z:|z|<1}$. Show that $|a_2|≤(n-1)/2$ and $|a_n|≤1/n$.
Partial Solution
Assume without loss of generality that $a_nne0$. If $f$ is injective, then $f'(z)≠0$ in $D$. Now, $$f'(z)=1+2a_2z+...+na_nz^{n-1}$$ By the fundamental theorem of algebra, $$f'(z)=na_n(z-r_1)...(z-r_{n-1})$$ where $r_1,...,r_{n-1}$ are the roots of $f'(z)$, and $$f'(0)=1=na_n(-1)^nr_1r_2...r_{n-1}$$ Note that $|r_i|geq1$ for each $i=1,...,n-1$, hence $$|r_1r_2...r_{n-1}|geq1$$ and, finally, $$|a_n|=frac1{n,|r_1r_2...r_{n-1}|}leqfrac{1}{n}$$
What I cannot figure out, though, is how to prove that $|a_2|≤(n-1)/2$. I have tried similar arguments, but cannot seem to get it. Any hints would be greatly appreciated.
complex-analysis polynomials
$endgroup$
add a comment |
$begingroup$
Suppose $f(z)=z+a_2z^2+...+a_nz^n$ is injective in $D:={z:|z|<1}$. Show that $|a_2|≤(n-1)/2$ and $|a_n|≤1/n$.
Partial Solution
Assume without loss of generality that $a_nne0$. If $f$ is injective, then $f'(z)≠0$ in $D$. Now, $$f'(z)=1+2a_2z+...+na_nz^{n-1}$$ By the fundamental theorem of algebra, $$f'(z)=na_n(z-r_1)...(z-r_{n-1})$$ where $r_1,...,r_{n-1}$ are the roots of $f'(z)$, and $$f'(0)=1=na_n(-1)^nr_1r_2...r_{n-1}$$ Note that $|r_i|geq1$ for each $i=1,...,n-1$, hence $$|r_1r_2...r_{n-1}|geq1$$ and, finally, $$|a_n|=frac1{n,|r_1r_2...r_{n-1}|}leqfrac{1}{n}$$
What I cannot figure out, though, is how to prove that $|a_2|≤(n-1)/2$. I have tried similar arguments, but cannot seem to get it. Any hints would be greatly appreciated.
complex-analysis polynomials
$endgroup$
Suppose $f(z)=z+a_2z^2+...+a_nz^n$ is injective in $D:={z:|z|<1}$. Show that $|a_2|≤(n-1)/2$ and $|a_n|≤1/n$.
Partial Solution
Assume without loss of generality that $a_nne0$. If $f$ is injective, then $f'(z)≠0$ in $D$. Now, $$f'(z)=1+2a_2z+...+na_nz^{n-1}$$ By the fundamental theorem of algebra, $$f'(z)=na_n(z-r_1)...(z-r_{n-1})$$ where $r_1,...,r_{n-1}$ are the roots of $f'(z)$, and $$f'(0)=1=na_n(-1)^nr_1r_2...r_{n-1}$$ Note that $|r_i|geq1$ for each $i=1,...,n-1$, hence $$|r_1r_2...r_{n-1}|geq1$$ and, finally, $$|a_n|=frac1{n,|r_1r_2...r_{n-1}|}leqfrac{1}{n}$$
What I cannot figure out, though, is how to prove that $|a_2|≤(n-1)/2$. I have tried similar arguments, but cannot seem to get it. Any hints would be greatly appreciated.
complex-analysis polynomials
complex-analysis polynomials
edited Dec 7 '18 at 19:04
Did
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247k23222457
asked Dec 6 '18 at 23:54
greeliousgreelious
19410
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Figured it out. Just to answer my own question: $$f'(z)=1+2a_2z+ ... +na_nz^{n-1}=na_n(z-r_1)...(z-r_{n-1})$$
$$implies2a_2=na_n(r_2r_3...r_{n-1}(-1)^{n-2}+r_1r_3...r_{n-1}(-1)^{n-2}+ ... +r_1r_2...r_{n-2}(-1)^{n-2})$$
$$begin{align}
implies|2a_2| & = n|a_n||r_2r_3...r_{n-1}(-1)^{n-2}+r_1r_3...r_{n-1}(-1)^{n-2}+ ... +r_1r_2...r_{n-2}(-1)^{n-2}|\
& leq nbigl(frac{1}{n}bigr)bigl(|r_2r_3...r_{n-1}(-1)^{n-2}|+|r_1r_3...r_{n-1}(-1)^{n-2}|+ ... +|r_1r_2...r_{n-2}(-1)^{n-2}|bigr)\
end{align}$$
$$implies |2a_2|leq1+1+ ... +1=n-1$$
$$implies |a_2|leqfrac{n-1}{2}$$
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More simply, $$2a_2=-sum_{k=1}^{n-1}frac1{r_k}$$
$endgroup$
– Did
Dec 7 '18 at 14:40
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1 Answer
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$begingroup$
Figured it out. Just to answer my own question: $$f'(z)=1+2a_2z+ ... +na_nz^{n-1}=na_n(z-r_1)...(z-r_{n-1})$$
$$implies2a_2=na_n(r_2r_3...r_{n-1}(-1)^{n-2}+r_1r_3...r_{n-1}(-1)^{n-2}+ ... +r_1r_2...r_{n-2}(-1)^{n-2})$$
$$begin{align}
implies|2a_2| & = n|a_n||r_2r_3...r_{n-1}(-1)^{n-2}+r_1r_3...r_{n-1}(-1)^{n-2}+ ... +r_1r_2...r_{n-2}(-1)^{n-2}|\
& leq nbigl(frac{1}{n}bigr)bigl(|r_2r_3...r_{n-1}(-1)^{n-2}|+|r_1r_3...r_{n-1}(-1)^{n-2}|+ ... +|r_1r_2...r_{n-2}(-1)^{n-2}|bigr)\
end{align}$$
$$implies |2a_2|leq1+1+ ... +1=n-1$$
$$implies |a_2|leqfrac{n-1}{2}$$
$endgroup$
$begingroup$
More simply, $$2a_2=-sum_{k=1}^{n-1}frac1{r_k}$$
$endgroup$
– Did
Dec 7 '18 at 14:40
add a comment |
$begingroup$
Figured it out. Just to answer my own question: $$f'(z)=1+2a_2z+ ... +na_nz^{n-1}=na_n(z-r_1)...(z-r_{n-1})$$
$$implies2a_2=na_n(r_2r_3...r_{n-1}(-1)^{n-2}+r_1r_3...r_{n-1}(-1)^{n-2}+ ... +r_1r_2...r_{n-2}(-1)^{n-2})$$
$$begin{align}
implies|2a_2| & = n|a_n||r_2r_3...r_{n-1}(-1)^{n-2}+r_1r_3...r_{n-1}(-1)^{n-2}+ ... +r_1r_2...r_{n-2}(-1)^{n-2}|\
& leq nbigl(frac{1}{n}bigr)bigl(|r_2r_3...r_{n-1}(-1)^{n-2}|+|r_1r_3...r_{n-1}(-1)^{n-2}|+ ... +|r_1r_2...r_{n-2}(-1)^{n-2}|bigr)\
end{align}$$
$$implies |2a_2|leq1+1+ ... +1=n-1$$
$$implies |a_2|leqfrac{n-1}{2}$$
$endgroup$
$begingroup$
More simply, $$2a_2=-sum_{k=1}^{n-1}frac1{r_k}$$
$endgroup$
– Did
Dec 7 '18 at 14:40
add a comment |
$begingroup$
Figured it out. Just to answer my own question: $$f'(z)=1+2a_2z+ ... +na_nz^{n-1}=na_n(z-r_1)...(z-r_{n-1})$$
$$implies2a_2=na_n(r_2r_3...r_{n-1}(-1)^{n-2}+r_1r_3...r_{n-1}(-1)^{n-2}+ ... +r_1r_2...r_{n-2}(-1)^{n-2})$$
$$begin{align}
implies|2a_2| & = n|a_n||r_2r_3...r_{n-1}(-1)^{n-2}+r_1r_3...r_{n-1}(-1)^{n-2}+ ... +r_1r_2...r_{n-2}(-1)^{n-2}|\
& leq nbigl(frac{1}{n}bigr)bigl(|r_2r_3...r_{n-1}(-1)^{n-2}|+|r_1r_3...r_{n-1}(-1)^{n-2}|+ ... +|r_1r_2...r_{n-2}(-1)^{n-2}|bigr)\
end{align}$$
$$implies |2a_2|leq1+1+ ... +1=n-1$$
$$implies |a_2|leqfrac{n-1}{2}$$
$endgroup$
Figured it out. Just to answer my own question: $$f'(z)=1+2a_2z+ ... +na_nz^{n-1}=na_n(z-r_1)...(z-r_{n-1})$$
$$implies2a_2=na_n(r_2r_3...r_{n-1}(-1)^{n-2}+r_1r_3...r_{n-1}(-1)^{n-2}+ ... +r_1r_2...r_{n-2}(-1)^{n-2})$$
$$begin{align}
implies|2a_2| & = n|a_n||r_2r_3...r_{n-1}(-1)^{n-2}+r_1r_3...r_{n-1}(-1)^{n-2}+ ... +r_1r_2...r_{n-2}(-1)^{n-2}|\
& leq nbigl(frac{1}{n}bigr)bigl(|r_2r_3...r_{n-1}(-1)^{n-2}|+|r_1r_3...r_{n-1}(-1)^{n-2}|+ ... +|r_1r_2...r_{n-2}(-1)^{n-2}|bigr)\
end{align}$$
$$implies |2a_2|leq1+1+ ... +1=n-1$$
$$implies |a_2|leqfrac{n-1}{2}$$
edited Dec 7 '18 at 14:07
answered Dec 7 '18 at 0:14
greeliousgreelious
19410
19410
$begingroup$
More simply, $$2a_2=-sum_{k=1}^{n-1}frac1{r_k}$$
$endgroup$
– Did
Dec 7 '18 at 14:40
add a comment |
$begingroup$
More simply, $$2a_2=-sum_{k=1}^{n-1}frac1{r_k}$$
$endgroup$
– Did
Dec 7 '18 at 14:40
$begingroup$
More simply, $$2a_2=-sum_{k=1}^{n-1}frac1{r_k}$$
$endgroup$
– Did
Dec 7 '18 at 14:40
$begingroup$
More simply, $$2a_2=-sum_{k=1}^{n-1}frac1{r_k}$$
$endgroup$
– Did
Dec 7 '18 at 14:40
add a comment |
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