View source for Star-like function

''univalent star-like function''

A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s0872201.png" /> which is regular and univalent in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s0872202.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s0872203.png" />, and maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s0872204.png" /> onto a [[Star-like domain|star-like domain]] with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s0872205.png" />. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s0872206.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s0872207.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s0872208.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s0872209.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722010.png" />, regular in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722011.png" />, is star-like in this disc if and only if it satisfies the condition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722012.png" /></td> </tr></table>

The family of star-like functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722013.png" />, normalized so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722015.png" />, forms the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722017.png" />, which admits a parametric representation by Stieltjes integrals:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722018.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722019.png" /> is a non-decreasing function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722021.png" />.

For the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722022.png" /> the [[Coefficient problem|coefficient problem]] has been solved; sharp estimates have been found for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722026.png" /> (the argument of the function is the branch that vanishes at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722027.png" />). The extremal functions for these estimates are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722028.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722029.png" /> is real. The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722030.png" /> of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722031.png" /> is related to the class of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722034.png" />, that are regular and univalent in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722035.png" /> and map <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722036.png" /> onto a convex domain, by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722037.png" />.

A star-like function such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722038.png" /></td> </tr></table>

is called a star-like function of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722040.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722041.png" />.

Attention has also been given to univalent star-like functions in an annulus (see [[#References|[1]]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722042.png" />-valent star-like functions and weakly star-like functions in a disc (see [[#References|[2]]], [[#References|[4]]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087220/s08722043.png" />-locally star-like functions (see [[#References|[1]]]), and functions which are star-like in the direction of the real axis (see [[#References|[3]]]). For star-like functions of several complex variables, see [[#References|[5]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.M. Goluzin,   "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.A. Hummel,   "Multivalent starlike functions"  ''J. d'Anal. Math.'' , '''18'''  (1967)  pp. 133–160</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.S. Robertson,   "Analytic functions star-like in one direction"  ''Amer. J. Math.'' , '''58''' :  3  (1936)  pp. 465–472</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.W. Goodman,   "Open problems on univalent and mutivalent functions"  ''Bull. Amer. Math. Soc.'' , '''74''' :  6  (1968)  pp. 1035–1050</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  I.I. Bavrin,   "Classes of holomorphic functions of several complex variables and extremal problems for these classes of functions" , Moscow  (1976)  (In Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.W. Goodman,   "Univalent functions" , '''1''' , Mariner  (1983)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.L. Duren,   "Univalent functions" , Springer  (1983)  pp. Sect. 10.11</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  C. Pommerenke,   "Univalent functions" , Vandenhoeck &amp; Ruprecht  (1975)</TD></TR></table>