Mapping Radius of Regular Function and Center of Convex Region. Duan Wenxi
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1 d Iteatioal Cofeece o Electical Compute Egieeig ad Electoics (ICECEE 5 Mappig adius of egula Fuctio ad Cete of Covex egio Dua Wexi School of Applied Mathematics Beijig Nomal Uivesity Zhuhai Chia @qqcom Keywods: egula fuctio; mappig adius; cofomal tasfomatio; cete of covex egio Abstact Thee is oly a uivalet ad egula fuctio i a simply coected egio It tasfoms the egio ito a uit cicle while satisfies that the fuctio value of a poit is equal to zeo ad the deivative of this poit is geate tha zeo the the ivese of the deivative is amed as the mappig adius of the fuctio at the poit The mappig adius is chagig accodig to the movemet of the poit I fact the mappig adius i the egio is a eal valued fuctio The fuctio is cotiuous ad eaches the maximum value withi the egio If a simply coected egio is the covex egio with the symmetical cete the the mappig adius fuctio of the egio obtais the maximum at the symmetical cete of the egio Itoductio Fist of all we look back o the famous iema Theoem ad his deductio[] iema Theoem: Supposig the bouday poits of the simply coected egio o the plae Z ae ot limited to oe z thee is the oly uivalet ad egula fuctio f( z that will be tasfomed ito the uit cicle: w < o the plaew ad meets f( z ( z > Deductio: Supposig the bouday poits of the simply coected egio o the plae Z ae ot limited to oe z thee is the oly uivalet ad egula fuctio F( z that will be tasfomed to the uit cicle: w < f ( z o the plae W ad meets F F I fact suppose w f is the uivalet ad egula fuctio with the tems of iema f Theoem F f ( z F( z F ( z will be tasfomed to the cicle by the F( zad f w f < ( z ( z Mappig adius Defiitio: Suppose the fuctio f( z meets the tems of iema Theoem ad f ( z is amed as the mappig adius of f at the poit z of We ca see fom the above deductios if will be tasfomed ito the cicle: w < by the F ( F( z F ( z the the mappig adius at the poit z of is the adius of the cicle: w < Suppose is the simply coected egio that the bouday poits ae ot limited to oe o the plae Z ad z Suppose N ( is the family of fuctios which meet the followig coditios ( F( zis the uivalet ad egula fuctio i the 5 The authos - Published by Atlatis Pess 4
2 ( F F The popeties of the family of fuctios ae followig: Popety Suppose M( F sup F the thee is a uivalet ad egula fuctio F( zi N ( z eaches the miimum of M( F Ad the miimum value is equal to the mappig adius at the poit z of Poof: We ae limited to M( F < Suppose w f is the uivalet ad egula fuctio that will oly be tasfomed to the uit cicle: w < ad meets f( z ( z > Its ivese fuctio F( ϕ( w is maked as z ϕ( w Agai makψ ( w the ψ ( w is the uivalet ad egula fuctio o M( F the uit cicle: w < ψ ( ψ ( w < accodig to Schwaz Theoem ψ ( because ( ( ( ( ( F ϕ w ( F ϕ w F ( ϕ( ψ w ϕ w ψ ( M( F M( F M( F M( F so M( F ( z whe ψ ( w w ψ ( M( F f ( z theefoe F( ϕ( w w that is to say M( F w ( z F o f ( z F thus f F f ( z Popety: Whe the aea SF ( F dσ eaches miimum M( F sup F eaches f miimum ad oly whe F f ( z SFis ( the miimum Amog which w f is the uivalet ad egula fuctio that will tasfomed to the uit cicle: w < ad meets f( z ( z > Poof : we ae limited to SF ( is a fiite umbe the ivese fuctio of w f is z ϕ( w i w e the SF ( F dσ F ( ϕ( w ϕ ( w dd lim F ( ϕ( w ϕ ( w dd < w < We will F ( ϕ( w ϕ ( w as powe seies a F ( ϕ( ϕ ( F Pay attetio to π ( m i e m d π m w F ( ϕ( w ϕ ( w aw Amog which m + m+ i( m F ( ϕ( w ϕ ( w dd aw aw dd m w < w < m w < m π i( m m+ + m m e d a a d S( F lim F ( ϕ( w ϕ ( w dd < w + ρ π a lim π a π a ρ + + a the SF ( π a Whe a F a f so F f/ ( z a a e d d z + π a d π a i this case F [ ϕ( w] ϕ ( w a F( ϕ ( w aw
3 Mappig adius fuctios of the cicle Suppose : z a < is the cicle o the plae Z z The fuctio z z f ( z a ( z a will tasfomed the cicle ito the uit cicle: w < o the plae W ad meets f( z ( z > z a The mappig adius at poit z of : z a < is z a ( z accodig to the defiitio If z is looked upo as the movig poit the mappig adius of the cicle : z a < is z a z ( ( Obviously the mappig adius fuctio of the cicle has the followig popeties: ( zis ( the mootoe deceasig fuctio of z a ; ( z ( obtais the maximum at the cete of the cicle ad max z ( Mappig adius fuctio of the simply coected egio Suppose is the simply coected egio o the plae Z which the bouday poits ae ot limited to oe z the uivalet ad egula fuctio f i the will make tasfomed ito the uit cicle: w < o the plaew ad meet f > Fo a the followig fuctio is made: i f f( a gz ( f f( a i [ f f( a] + [ f f( a] f( a g [ f f( a] z i [ f f( a + f f( a f( a f( a] [ f f( a] i [ f( a f( a] [ f f( a] i ( a[ f( a f( a] i f ( a i f ( a g ( a [ f( a f( a] f( a f( a f( a If at fist choose ag ( a thus ( a g ( a > f( a will also tasfomed ito the uit cicle: w < o the plae W by gz ( ad ga ( g ( a > Theefoe the mappig adius of the poit a about egio is f( a g ( a ( a The poit a is cosideed as the movig poit z thus the mappig adius fuctio i is obtaied as f Sice clealy ( z is the cotiuous fuctio i ( 44
4 The maximum of the mappig adius fuctio ad the cete of the covex egio Fo M( F that is itoduced i the fist paagaph the popeties ae followig: Popety : If thus if M( F if M( F (3 F F Poof: The family of the uivalet ad egula fuctios i both ad meet F( z F ( z ae espectively maked as N ( ad N ( N ( N ( theefoe if M( F if M( F F F Defiitio: Suppose is the covex egio of the axial symmety the midpoit of the staight lie segmet of the symmetical axis i is amed as the midpoit of the symmetical axis of the covex egio Popety : Suppose is the covex egio with axial symmety which ouded by a piecewise smooth cuve C The midpoit of the symmetical axis is a max a ζ z ( mi z z ζ C z C ( z thus the mappig adius fuctio ( z of meets (4 Poof: Suppose z z is the cete of the cicle z ( mi z z z C is the adius to the iscibed cicula disc : z ξ z ( ξ C thus ( ( z z accodig to popety ad ( ( z z thus z ( The cicumscibed cicula disc to : ζ a is made with a as the cete of the cicle ad the adius as max a ζ is obtaied accodig to popety Fo ( a ζ C that is Sum up the above statemets Especially whe mi a ζ ζ c the maximum of is maked as ( M thus ( M (5 Theoem : Suppose is the covex egio ouded by a piecewise smooth cuve C ζ C thus lim z ( Poof: Fo fomulatio (4 ( ( z z that is the uivalet ad egula fuctio f that will be chaged ito the uit cicle : w < ad meets ( that is f z f ( z > Fo each f( z cofom to ( z theefoe z ( f( z ( z ( f( z ( z f ( ( ( z f z Whe ζ C lim f thus lim z ( ( z Theoem : If is the covex egio ouded by a piecewise smooth cuve C the mappig adius fuctio ( zof must each the maximum i the egio z ( z Poof: Suppose C is the bouday of the fuctio z C is cotiuous i the closed egio fomed by ad its bouday C has the maximum but the value of o the bouday is zeo The maximum of ( z is at some poit a i the egio that is the maximum of z ( is at the poit a 45
5 The followig thee coclusios [] ae explaied i ode to give aothe theoem: If f is the uivalet ad egula fuctio i the egio the ; If f is the egula fuctio i the egio is also the egula fuctio i the egio ; If is the covex egio with the symmetical axis ouded by a piecewise smooth cuve C ad z ais the midpoit of the symmetical axis z a make M( max f z a z M( is the mootoe iceasig fuctio about accodig to the picipal of the maximum modulus of the egula fuctio Theoem 3: If is the covex egio with the symmetical axis ouded by a piecewise smooth cuve C the mappig adius fuctio of gais the maximum at the midpoit of the symmetical axis of Poof: Suppose f is the uivalet ad egula fuctio which will tasfomed ito the w < o plae W ad meet f > thus is still the egula fuctio i Mak a as the midpoit of the symmetical axis of fo M( max f is the iceasig fuctio of z a thus f ad each the miimum at z a z f the poit a Theefoe the mappig adius fuctio i eaches the maximum at the poit a Deductio : If is the covex egio with the symmetical cete ouded by a piecewise smooth cuve C the mappig adius fuctio ( z of obtais the maximum at the symmetical cete of Deductio : If is the covex egio with the symmetical cete ouded by a piecewise smooth cuve C the poit of the mappig adius fuctio ( z of gais the maximum is i the cete of efeeces [] e Luxi eometic Theoies of Fuctios of a Complex Vaiable [M] Beijig Sciece Pess pp~7 955 [] Zhog Yuqua eometic Theoies of Fuctios of a Complex Vaiable [M] Beijig Highe Educatio Pess pp8 4 46
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