# polynomials and maximum principle

Let $P$ be a polynomial of degree $n$. For every $r>0$, let $M(r):=\max \{|P(z)| :|z|=r\}$. I want to show that the function $F(r)=\frac{M(r)}{r}$ is monotonically decreasing in $(0, +\infty)$. Second question is: if $F(r_1)=F(r_2)$ for $r_1\neq r_2$, what can be said on polynomial $P$?

For the first question, how can i use maximum modulus principle? for the second, i have no idea....any help?

-
If $P(z) = z^n$, then $F(r) = r^{n-1}$ is not decreasing. Did you make a mistake in the definition of $F$? –  Greg Martin Nov 5 '12 at 22:47
Possible duplicate of Domination of complex-value polynomial by highest power. –  WimC Nov 6 '12 at 17:29
yes, there's an error. Put "increasing" in place of "decreasing" –  bateman Nov 6 '12 at 18:29

my solution: put $P(z)=z^nQ(\frac{1}{z})$. Then Q(z) is entire on $\mathbb C-{0}$. By maximum modulus principle, the function $M^Q(r)=max \{|Q(\frac{1}{z})|:|z|=r\}$ is increasing for $r>0$.
Now, $M(r)=r^nM^Q(r)$ and so $F(r)=r^{n-1}M^Q(r)$ is also increasing. Did i make any mistake?
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9674160480499268, "perplexity": 396.0249746759355}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500823169.67/warc/CC-MAIN-20140820021343-00259-ip-10-180-136-8.ec2.internal.warc.gz"}