Difference between revisions of "Level lines"
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+ | $#C+1 = 71 : ~/encyclopedia/old_files/data/L058/L.0508210 Level lines | ||
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''of a Green function'' | ''of a Green function'' | ||
The point sets | The point sets | ||
− | + | $$ | |
+ | L _ \lambda = \{ {z \in D } : { | ||
+ | G ( z , z _ {0} ) = \lambda = \textrm{ const } } \} | ||
+ | ,\ | ||
+ | 0 \leq \lambda < \infty , | ||
+ | $$ | ||
− | where | + | where $ G ( z , z _ {0} ) $ |
+ | is the [[Green function|Green function]] for the domain $ D $ | ||
+ | in the complex plane with pole at the point $ z _ {0} \in D $. | ||
+ | If $ D $ | ||
+ | is simply connected, then the structure of this set is easily determined by conformally mapping $ D $ | ||
+ | onto the disc $ | \zeta | < 1 $, | ||
+ | taking the point $ z _ {0} $ | ||
+ | to $ \zeta = 0 $. | ||
+ | The Green function is invariant under this transformation, while the level lines of the Green function for the disc $ | \zeta | < 1 $ | ||
+ | with pole at $ \zeta = 0 $, | ||
+ | i.e. $ - \mathop{\rm log} | \zeta | $, | ||
+ | are the circles $ | \zeta | = \textrm{ const } $. | ||
+ | So, in the case of a simply-connected domain, the level line $ G ( z , z _ {0} ) = \lambda $ | ||
+ | is a simple closed curve, coinciding for $ \lambda = 0 $ | ||
+ | with the boundary of $ D $ | ||
+ | and tending to $ z _ {0} $ | ||
+ | as $ \lambda \rightarrow + \infty $. | ||
+ | If the domain $ D $ | ||
+ | is $ m $- | ||
+ | connected and its boundary consists of Jordan curves $ C _ \nu $, | ||
+ | $ \nu = 1 \dots m $, | ||
+ | then: if $ \lambda > 0 $ | ||
+ | is sufficiently large, the level line is a Jordan curve; for $ \lambda \rightarrow + \infty $ | ||
+ | the corresponding level line tends to the point $ z _ {0} $, | ||
+ | while for decreasing $ \lambda $ | ||
+ | it moves away from $ z _ {0} $; | ||
+ | if $ m > 1 $, | ||
+ | then for certain values of $ \lambda $ | ||
+ | the level line has self-intersection, and decomposes into non-intersecting simple closed curves; for sufficiently small $ \lambda $ | ||
+ | the level line consists of $ m $ | ||
+ | Jordan curves and for $ \lambda \rightarrow 0 $ | ||
+ | each of these curves tends to one of the boundary curves of $ D $. | ||
− | In questions of the approximation of functions by polynomials on a closed bounded set | + | In questions of the approximation of functions by polynomials on a closed bounded set $ B $ |
+ | with a simply-connected complement, an important role is played by estimates for the distance between boundary points of $ B $ | ||
+ | and level lines of the complement of $ B $( | ||
+ | cf. [[#References|[4]]], [[#References|[5]]]). | ||
− | For univalent conformal mappings of the disc < | + | For univalent conformal mappings of the disc $ | z | < 1 $ |
+ | by functions of the class $ S = \{ {f } : {f ( z) = z + \dots, f \textrm{ regular and univalent in } | z | < 1 } \} $( | ||
+ | cf. [[Univalent function|Univalent function]]), the behaviour of the level line $ L ( f , r ) $( | ||
+ | the image of the circle $ | z | = r < 1 $) | ||
+ | intuitively gives the degree of distortion. Any function of class $ S $ | ||
+ | maps the disc $ | z | < r $, | ||
+ | $ 0 < r < 2 - \sqrt 3 $, | ||
+ | onto a [[Convex domain|convex domain]], while the disc $ | z | < r $, | ||
+ | $ 0 < r < \mathop{\rm tanh} \pi / 4 $, | ||
+ | is mapped onto a [[Star-like domain|star-like domain]]. The level line $ L ( f , r ) $, | ||
+ | $ f \in S $, | ||
+ | $ 0 < r < 1 $, | ||
+ | belongs to the annulus | ||
− | + | $$ | |
+ | K _ {r} = \{ {w } : {r | ||
+ | ( 1 + r ) ^ {-} 2 \leq | w | \leq r ( 1 - r ) ^ {-} 2 } \} | ||
+ | $$ | ||
and bounds a simply-connected domain comprising the coordinate origin. | and bounds a simply-connected domain comprising the coordinate origin. | ||
− | For the curvature | + | For the curvature $ K ( f , r ) $ |
+ | of the level line $ L ( f , r ) $ | ||
+ | in the class $ S $ | ||
+ | one has the following sharp estimate: | ||
+ | |||
+ | $$ | ||
+ | K ( f , r ) \geq \ | ||
+ | |||
+ | \frac{1 - 4 r + r ^ {2} }{r} | ||
+ | |||
+ | \left ( 1+ | ||
+ | \frac{r}{1-} | ||
+ | r \right ) ^ {2} , | ||
+ | $$ | ||
+ | |||
+ | and equality holds only for the function $ f ( z) = z / ( 1 + z ) ^ {2} $ | ||
+ | at the point $ z = r $. | ||
+ | The exact upper bound for $ K ( f , r ) $ | ||
+ | in the class $ S $ | ||
+ | is at present (1984) not known. The exact upper bound for $ K ( f , r ) $ | ||
+ | in the subclass of star-like functions in $ S $( | ||
+ | cf. [[Star-like function|Star-like function]]) has the form | ||
− | + | $$ | |
+ | K ( f , r ) \leq \ | ||
− | + | \frac{1 + 4 r + r ^ {2} }{r} | |
− | + | \left ( 1- | |
+ | \frac{r}{1+} | ||
+ | r \right ) ^ {2} , | ||
+ | $$ | ||
− | and equality holds only for the function | + | and equality holds only for the function $ f ( z) = z / ( 1 - z ) ^ {2} $ |
+ | at $ z = r $. | ||
− | For mappings of the disc < | + | For mappings of the disc $ | z | < 1 $ |
+ | by functions of the class $ S $ | ||
+ | the number of points of inflection of the level line $ L ( f , r ) $ | ||
+ | and the number of points violating the star-likeness condition (i.e. points of the level line at which the direction of rotation of the radius vector changes when $ z $ | ||
+ | runs over the circle $ | z | = r $ | ||
+ | in a given direction) may change non-monotonically for increasing $ r $, | ||
+ | i.e. if $ r _ {1} < r _ {2} $, | ||
+ | one can show that the level line $ L ( f , r _ {1} ) $ | ||
+ | may have more points of inflection and more points violating the star-likeness condition than $ L ( f , r _ {2} ) $. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> S. Stoilov, "The theory of functions of a complex variable" , '''1''' , Moscow (1962) (In Russian; translated from Rumanian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> G.M. Goluzin, "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc. (1969) pp. Appendix (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> I.A. Aleksandrov, "Parametric extensions in the theory of univalent functions" , Moscow (1976) (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> V.K. Dzyadyk, "On a problem of S.M. Nikol'skii in a complex region" ''Izv. Akad. Nauk SSSR Mat.'' , '''23''' : 5 (1959) pp. 697–763 (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> N.A. Lebedev, N.A. Shirokov, "The uniform approximation of functions on closed sets with a finite number of angular points with non-zero exterior angles" ''Izv. Akad. Nauk Armen. SSR Ser. Mat.'' , '''6''' : 4 (1971) pp. 311–341 (In Russian)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> S. Stoilov, "The theory of functions of a complex variable" , '''1''' , Moscow (1962) (In Russian; translated from Rumanian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> G.M. Goluzin, "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc. (1969) pp. Appendix (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> I.A. Aleksandrov, "Parametric extensions in the theory of univalent functions" , Moscow (1976) (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> V.K. Dzyadyk, "On a problem of S.M. Nikol'skii in a complex region" ''Izv. Akad. Nauk SSSR Mat.'' , '''23''' : 5 (1959) pp. 697–763 (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> N.A. Lebedev, N.A. Shirokov, "The uniform approximation of functions on closed sets with a finite number of angular points with non-zero exterior angles" ''Izv. Akad. Nauk Armen. SSR Ser. Mat.'' , '''6''' : 4 (1971) pp. 311–341 (In Russian)</TD></TR></table> | ||
− | |||
− | |||
====Comments==== | ====Comments==== |
Revision as of 22:16, 5 June 2020
of a Green function
The point sets
$$ L _ \lambda = \{ {z \in D } : { G ( z , z _ {0} ) = \lambda = \textrm{ const } } \} ,\ 0 \leq \lambda < \infty , $$
where $ G ( z , z _ {0} ) $ is the Green function for the domain $ D $ in the complex plane with pole at the point $ z _ {0} \in D $. If $ D $ is simply connected, then the structure of this set is easily determined by conformally mapping $ D $ onto the disc $ | \zeta | < 1 $, taking the point $ z _ {0} $ to $ \zeta = 0 $. The Green function is invariant under this transformation, while the level lines of the Green function for the disc $ | \zeta | < 1 $ with pole at $ \zeta = 0 $, i.e. $ - \mathop{\rm log} | \zeta | $, are the circles $ | \zeta | = \textrm{ const } $. So, in the case of a simply-connected domain, the level line $ G ( z , z _ {0} ) = \lambda $ is a simple closed curve, coinciding for $ \lambda = 0 $ with the boundary of $ D $ and tending to $ z _ {0} $ as $ \lambda \rightarrow + \infty $. If the domain $ D $ is $ m $- connected and its boundary consists of Jordan curves $ C _ \nu $, $ \nu = 1 \dots m $, then: if $ \lambda > 0 $ is sufficiently large, the level line is a Jordan curve; for $ \lambda \rightarrow + \infty $ the corresponding level line tends to the point $ z _ {0} $, while for decreasing $ \lambda $ it moves away from $ z _ {0} $; if $ m > 1 $, then for certain values of $ \lambda $ the level line has self-intersection, and decomposes into non-intersecting simple closed curves; for sufficiently small $ \lambda $ the level line consists of $ m $ Jordan curves and for $ \lambda \rightarrow 0 $ each of these curves tends to one of the boundary curves of $ D $.
In questions of the approximation of functions by polynomials on a closed bounded set $ B $ with a simply-connected complement, an important role is played by estimates for the distance between boundary points of $ B $ and level lines of the complement of $ B $( cf. [4], [5]).
For univalent conformal mappings of the disc $ | z | < 1 $ by functions of the class $ S = \{ {f } : {f ( z) = z + \dots, f \textrm{ regular and univalent in } | z | < 1 } \} $( cf. Univalent function), the behaviour of the level line $ L ( f , r ) $( the image of the circle $ | z | = r < 1 $) intuitively gives the degree of distortion. Any function of class $ S $ maps the disc $ | z | < r $, $ 0 < r < 2 - \sqrt 3 $, onto a convex domain, while the disc $ | z | < r $, $ 0 < r < \mathop{\rm tanh} \pi / 4 $, is mapped onto a star-like domain. The level line $ L ( f , r ) $, $ f \in S $, $ 0 < r < 1 $, belongs to the annulus
$$ K _ {r} = \{ {w } : {r ( 1 + r ) ^ {-} 2 \leq | w | \leq r ( 1 - r ) ^ {-} 2 } \} $$
and bounds a simply-connected domain comprising the coordinate origin.
For the curvature $ K ( f , r ) $ of the level line $ L ( f , r ) $ in the class $ S $ one has the following sharp estimate:
$$ K ( f , r ) \geq \ \frac{1 - 4 r + r ^ {2} }{r} \left ( 1+ \frac{r}{1-} r \right ) ^ {2} , $$
and equality holds only for the function $ f ( z) = z / ( 1 + z ) ^ {2} $ at the point $ z = r $. The exact upper bound for $ K ( f , r ) $ in the class $ S $ is at present (1984) not known. The exact upper bound for $ K ( f , r ) $ in the subclass of star-like functions in $ S $( cf. Star-like function) has the form
$$ K ( f , r ) \leq \ \frac{1 + 4 r + r ^ {2} }{r} \left ( 1- \frac{r}{1+} r \right ) ^ {2} , $$
and equality holds only for the function $ f ( z) = z / ( 1 - z ) ^ {2} $ at $ z = r $.
For mappings of the disc $ | z | < 1 $ by functions of the class $ S $ the number of points of inflection of the level line $ L ( f , r ) $ and the number of points violating the star-likeness condition (i.e. points of the level line at which the direction of rotation of the radius vector changes when $ z $ runs over the circle $ | z | = r $ in a given direction) may change non-monotonically for increasing $ r $, i.e. if $ r _ {1} < r _ {2} $, one can show that the level line $ L ( f , r _ {1} ) $ may have more points of inflection and more points violating the star-likeness condition than $ L ( f , r _ {2} ) $.
References
[1] | S. Stoilov, "The theory of functions of a complex variable" , 1 , Moscow (1962) (In Russian; translated from Rumanian) |
[2] | G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) pp. Appendix (Translated from Russian) |
[3] | I.A. Aleksandrov, "Parametric extensions in the theory of univalent functions" , Moscow (1976) (In Russian) |
[4] | V.K. Dzyadyk, "On a problem of S.M. Nikol'skii in a complex region" Izv. Akad. Nauk SSSR Mat. , 23 : 5 (1959) pp. 697–763 (In Russian) |
[5] | N.A. Lebedev, N.A. Shirokov, "The uniform approximation of functions on closed sets with a finite number of angular points with non-zero exterior angles" Izv. Akad. Nauk Armen. SSR Ser. Mat. , 6 : 4 (1971) pp. 311–341 (In Russian) |
Comments
Some non-Soviet references for the approximation questions mentioned are [a1] and [a2], in which other references can be found. See also Approximation of functions of a complex variable.
References
[a1] | L. Bijvoets, W. Hogeveen, J. Korevaar, "Inverse approximation theorems of Lebedev and Tamrazov" P.L. Butzer (ed.) , Functional analysis and approximation (Oberwolfach 1980) , Birkhäuser (1981) pp. 265–281 |
[a2] | D. Gaier, "Vorlesungen über Approximation im Komplexen" , Birkhäuser (1980) pp. Chapt. 1, §6 |
Level lines. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Level_lines&oldid=13582