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A Note on the Maximum Number of Zeros of r(z)-z
An important theorem of Khavinson and Neumann (Proc. Am. Math. Soc. 134: 1077–1085, 2006 ) states that the complex harmonic function r ( z ) - z ¯ , where r is a rational function of degree n ≥ 2 , has at most 5 ( n - 1 ) zeros. In this note, we resolve a slight inaccuracy in their proof and in addi...
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| Published in: | Computational methods and function theory 2015-09, Vol.15 (3), p.439-448 |
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| Format: | Article |
| Language: | English |
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| container_end_page | 448 |
| container_issue | 3 |
| container_start_page | 439 |
| container_title | Computational methods and function theory |
| container_volume | 15 |
| creator | Luce, Robert Sète, Olivier Liesen, Jörg |
| description | An important theorem of Khavinson and Neumann (Proc. Am. Math. Soc. 134: 1077–1085,
2006
) states that the complex harmonic function
r
(
z
)
-
z
¯
, where
r
is a rational function of degree
n
≥
2
, has at most
5
(
n
-
1
)
zeros. In this note, we resolve a slight inaccuracy in their proof and in addition we show that for certain functions of the form
r
(
z
)
-
z
¯
no more than
5
(
n
-
1
)
-
1
zeros can occur. Moreover, we show that
r
(
z
)
-
z
¯
is regular, if it has the maximal number of zeros. |
| doi_str_mv | 10.1007/s40315-015-0110-6 |
| format | article |
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2006
) states that the complex harmonic function
r
(
z
)
-
z
¯
, where
r
is a rational function of degree
n
≥
2
, has at most
5
(
n
-
1
)
zeros. In this note, we resolve a slight inaccuracy in their proof and in addition we show that for certain functions of the form
r
(
z
)
-
z
¯
no more than
5
(
n
-
1
)
-
1
zeros can occur. Moreover, we show that
r
(
z
)
-
z
¯
is regular, if it has the maximal number of zeros.</description><identifier>ISSN: 1617-9447</identifier><identifier>EISSN: 2195-3724</identifier><identifier>DOI: 10.1007/s40315-015-0110-6</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>Analysis ; Computational Mathematics and Numerical Analysis ; Functions of a Complex Variable ; Mathematics ; Mathematics and Statistics</subject><ispartof>Computational methods and function theory, 2015-09, Vol.15 (3), p.439-448</ispartof><rights>Springer-Verlag Berlin Heidelberg 2015</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,780,784,27922,27923</link.rule.ids></links><search><creatorcontrib>Luce, Robert</creatorcontrib><creatorcontrib>Sète, Olivier</creatorcontrib><creatorcontrib>Liesen, Jörg</creatorcontrib><title>A Note on the Maximum Number of Zeros of r(z)-z</title><title>Computational methods and function theory</title><addtitle>Comput. Methods Funct. Theory</addtitle><description>An important theorem of Khavinson and Neumann (Proc. Am. Math. Soc. 134: 1077–1085,
2006
) states that the complex harmonic function
r
(
z
)
-
z
¯
, where
r
is a rational function of degree
n
≥
2
, has at most
5
(
n
-
1
)
zeros. In this note, we resolve a slight inaccuracy in their proof and in addition we show that for certain functions of the form
r
(
z
)
-
z
¯
no more than
5
(
n
-
1
)
-
1
zeros can occur. Moreover, we show that
r
(
z
)
-
z
¯
is regular, if it has the maximal number of zeros.</description><subject>Analysis</subject><subject>Computational Mathematics and Numerical Analysis</subject><subject>Functions of a Complex Variable</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><issn>1617-9447</issn><issn>2195-3724</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2015</creationdate><recordtype>article</recordtype><sourceid/><recordid>eNotTz1PwzAUtBBIhMIPYPMIg-l7tmPjsaqgIJWydGKxHOcFWpEE2Y2E-utJW4bTnW64D8ZuER4QwE6zBoWlgCMQhDljhURXCmWlPmcFGrTCaW0v2VXOW4BSO6UKNp3xVb8j3nd890X8Lfxu2qHlq6GtKPG-4R-U-nwQ6W5_L_bX7KIJ35lu_nnC1s9P6_mLWL4vXuezpciIyghEFylI3Wgrm0BVtBDqaEMYW42rVFlrsk0FJkZ6HLfFkmRlazWurKF2asLkKTb_pE33Sclv-yF1Y6NH8IfH_vTYwxGjadQfUK1HfA</recordid><startdate>20150910</startdate><enddate>20150910</enddate><creator>Luce, Robert</creator><creator>Sète, Olivier</creator><creator>Liesen, Jörg</creator><general>Springer Berlin Heidelberg</general><scope/></search><sort><creationdate>20150910</creationdate><title>A Note on the Maximum Number of Zeros of r(z)-z</title><author>Luce, Robert ; Sète, Olivier ; Liesen, Jörg</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-s1136-119cea24f472faebc70adc7aa49369b35d4e7fb06cce8161c5e2b7d3617d0d93</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2015</creationdate><topic>Analysis</topic><topic>Computational Mathematics and Numerical Analysis</topic><topic>Functions of a Complex Variable</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Luce, Robert</creatorcontrib><creatorcontrib>Sète, Olivier</creatorcontrib><creatorcontrib>Liesen, Jörg</creatorcontrib><jtitle>Computational methods and function theory</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Luce, Robert</au><au>Sète, Olivier</au><au>Liesen, Jörg</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>A Note on the Maximum Number of Zeros of r(z)-z</atitle><jtitle>Computational methods and function theory</jtitle><stitle>Comput. Methods Funct. Theory</stitle><date>2015-09-10</date><risdate>2015</risdate><volume>15</volume><issue>3</issue><spage>439</spage><epage>448</epage><pages>439-448</pages><issn>1617-9447</issn><eissn>2195-3724</eissn><abstract>An important theorem of Khavinson and Neumann (Proc. Am. Math. Soc. 134: 1077–1085,
2006
) states that the complex harmonic function
r
(
z
)
-
z
¯
, where
r
is a rational function of degree
n
≥
2
, has at most
5
(
n
-
1
)
zeros. In this note, we resolve a slight inaccuracy in their proof and in addition we show that for certain functions of the form
r
(
z
)
-
z
¯
no more than
5
(
n
-
1
)
-
1
zeros can occur. Moreover, we show that
r
(
z
)
-
z
¯
is regular, if it has the maximal number of zeros.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s40315-015-0110-6</doi><tpages>10</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 1617-9447 |
| ispartof | Computational methods and function theory, 2015-09, Vol.15 (3), p.439-448 |
| issn | 1617-9447 2195-3724 |
| language | eng |
| recordid | cdi_springer_journals_10_1007_s40315_015_0110_6 |
| source | Springer Link |
| subjects | Analysis Computational Mathematics and Numerical Analysis Functions of a Complex Variable Mathematics Mathematics and Statistics |
| title | A Note on the Maximum Number of Zeros of r(z)-z |
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