Coefficient problem

for the class $ S $

A problem for the class of functions

$$ f ( z) = \ z + \sum _ {n = 2 } ^ \infty c _ {n} z ^ {n} $$

which are regular and univalent in the disc $ | z | < 1 $. It consists of determining for every $ n $, $ n \geq 2 $, the region of values $ V _ {n} $ for the system of coefficients $ \{ c _ {2} \dots c _ {n} \} $ of the functions of this class and, in particular, to find sharp bounds for $ | c _ {n} | $, $ n \geq 2 $, in the class $ S $( see Bieberbach conjecture). The coefficient problem for a class $ R $ of functions regular in $ | z | < 1 $ consists in determining in $ R $, for every $ n $, $ n \geq 1 $, the region of values of the first $ n $ coefficients in the series expansions of the functions of $ R $ in powers of $ z $ and, in particular, in obtaining sharp bounds for these coefficients in the class $ R $. The coefficient problem has been solved for the Carathéodory class, for the class of univalent star-like functions, and for the class of functions regular and bounded in $ | z | < 1 $.

It is known that $ V _ {2} $ is a disc: $ | c _ {2} | \leq 2 $. Profound qualitative results with regard to the coefficient problem have been obtained for the class $ S $( see [7]). The set $ V _ {n} $ is a bounded closed domain; the point $ c _ {2} = 0 \dots c _ {n} = 0 $ is an interior point of $ V _ {n} $; $ V _ {n} $ is homeomorphic to a closed $ ( 2n - 2) $- dimensional ball; the boundary of $ V _ {n} $ is a union of finitely many parts $ \Pi _ {1} \dots \Pi _ {N} $; the coordinates of a point $ ( c _ {2} \dots c _ {n} ) $ on any one of these parts are functions of a finite number ( $ \leq 2n - 3 $) of parameters. To every boundary point of $ V _ {n} $ there corresponds a unique function of the class $ S $. The boundary of $ V _ {3} $ is a union of two hyperplanes $ \Pi _ {1} $ and $ \Pi _ {2} $ of dimension 3 and their intersections: planes $ \Pi _ {3} $ and $ \Pi _ {4} $ and a curve $ \Pi _ {5} $. Parametric formulas have been derived for $ \Pi _ {1} $ and $ \Pi _ {2} $ in terms of elementary functions. The intersection of $ V _ {3} $ with the plane $ \mathop{\rm Im} c _ {2} = 0 $ is symmetric about the planes $ \mathop{\rm Re} c _ {2} = 0 $ and $ \mathop{\rm Im} c _ {3} = 0 $. The intersection of $ V _ {3} $ with the plane $ \mathop{\rm Im} c _ {3} = 0 $ is symmetric about the planes $ \mathop{\rm Im} c _ {2} = 0 $ and $ \mathop{\rm Re} c _ {2} = 0 $. A function $ w = f ( z) $ corresponding to a point on $ \Pi _ {1} $ maps $ | z | < 1 $ onto the $ w $- plane cut by an analytic curve going to infinity. A function $ w = f ( z) $ corresponding to a point on $ \Pi _ {2} $ maps $ | z | < 1 $ onto the $ w $- plane cut by three analytic arcs, issuing from a finite point and inclined to one another at angles $ 2 \pi /3 $; one of these arcs lies on a straight line $ \mathop{\rm arg} w = \textrm{ const } $ and goes to infinity.

Among the other special regions that have been investigated are the following: the region of values $ \{ c _ {2} , c _ {3} \} $ in the subclass of $ S $ consisting of functions with real $ c _ {2} $ and $ c _ {3} $; the region of values $ \{ | c _ {k + 1 } |, | c _ {2k + 1 } | \} $ and $ \{ c _ {k + 1 } , c _ {2k + 1 } \} $, if $ \mathop{\rm Im} c _ {k + 1 } = \mathop{\rm Im} c _ {2k + 1 } = 0 $, on the subclass of bounded functions in $ S $ representable as

$$ f ( z) = z + \sum _ {n = 1 } ^ \infty c _ {nk + 1 } z ^ {nk + 1 } ; $$

the region of values $ \{ c _ {2} , c _ {3} \} $ on the subclass of bounded functions in $ S $; the region of values $ \{ c _ {2} , c _ {3} , c _ {4} \} $ on the subclass of functions in $ S $ with real $ c _ {2} , c _ {3} $ and $ c _ {4} $.

Sharp bounds for the coefficients, of the type $ | c _ {n} | \leq A _ {n} $, $ n \geq 2 $, have been obtained in the subclass of convex functions in $ S $ with $ A _ {n} = 1 $( cf. Convex function (of a complex variable)), in the subclass of star-like functions in $ S $ with $ A _ {n} = n $, in the subclass of odd star-like functions in $ S $ with $ A _ {n} = 1 $, $ n = 3, 5 \dots $ in the class of univalent functions having real coefficients with $ A _ {n} = n $, in the subclass of close-to-convex functions in $ S $ with $ A _ {n} = n $, and in the class $ S $ itself with $ A _ {n} = n $( cf. Bieberbach conjecture, [8]). In the class of functions

$$ f ( z) = z + \sum _ {n = 2 } ^ \infty c _ {n} z ^ {n} $$

which are regular and typically real in $ | z | < 1 $ one has the sharp bound $ | c _ {n} | \leq n $, $ n \geq 2 $, and in the class of Bieberbach–Eilenberg functions $ f ( z) = a _ {1} z + a _ {2} z ^ {2} + \dots $ one has the sharp bound $ | a _ {n} | \leq 1 $, $ n \geq 1 $.

Sharp bounds are known for the class $ \Sigma $ of functions

$$ F ( \zeta ) = \ \zeta + \sum _ {n = 0 } ^ \infty \frac{b _ {n} }{\zeta ^ {n} } $$

which are meromorphic and univalent in $ | \zeta | > 1 $; these are

$$ | b _ {1} | \leq 1,\ \ | b _ {2} | \leq { \frac{2}{3} } ,\ \ | b _ {3} | \leq { \frac{1}{2} } + e ^ {-} 6 . $$

For the subclass of star-like functions in $ \Sigma $, one has the sharp bound

$$ | b _ {n} | \leq \ \frac{2}{n + 1 } ,\ \ n \geq 1. $$

Sharp bounds are also known for other subclasses of $ S $ and $ \Sigma $( see [1]–[4]), and also for some classes of $ p $- valent functions and in classes of functions which are $ p $- valent in the mean (see [5]).

References

[1]	G.M. Goluzin, "Geometric methods in the theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)
[2]	I.E. Bazilevich, , Mathematics in the USSR during 40 years: 1917–1957 , 1 , Moscow (1959) pp. 444–472 (In Russian)
[3]	J.A. Jenkins, "Univalent functions and conformal mapping" , Springer (1958)
[4]	W.K. Hayman, "Coefficient problems for univalent functions and related function classes" J. London Math. Soc. , 40 : 3 (1965) pp. 385–406
[5]	A.W. Goodman, "Open problems on univalent and multivalent functions" Bull. Amer. Math. Soc. , 74 : 6 (1968) pp. 1035–1050
[6]	D. Phelps, "On a coefficient problem in univalent functions" Trans. Amer. Math. Soc. , 143 (1969) pp. 475–485
[7]	A.C. Schaeffer, D.C. Spencer, "Coefficient regions for schlicht functions" , Amer. Math. Soc. Coll. Publ. , 35 , Amer. Math. Soc. (1950)
[8]	L. de Branges, "A proof of the Bieberbach conjecture" Acta Math. , 154 : 1–2 (1985) pp. 137–152

Comments

For functions in the class $ \Sigma $, the estimates for $ | b _ {2} | $ and $ | b _ {3} | $ mentioned above are due to M. Schiffer [a1] and P.R. Garabedian and Schiffer [a2], respectively. The sharp bound for star-like functions in $ \Sigma $ is due to J. Clunie [a3] and C. Pommerenke [a4]. Standard references in English include [a5]–[a7].

References

[a1]	M. Schiffer, "Sur un problème d'extrémum de la répresentation conforme" Bull. Soc. Math. France , 66 (1938) pp. 48–55
[a2]	P.R. Garabedian, M. Schiffer, "A coefficient inequality for schlicht functions" Ann. of Math. , 61 (1955) pp. 116–136
[a3]	J. Clunie, "On meromorphic schlicht functions" J. London Math. Soc. , 34 (1959) pp. 215–216
[a4]	C. Pommerenke, "On meromorphic starlike functions" Pacific J. Math. , 13 (1963) pp. 221–235
[a5]	W.K. Hayman, "Multivalent functions" , Cambridge Univ. Press (1967)
[a6]	C. Pommerenke, "Univalent functions" , Vandenhoeck & Ruprecht (1975)
[a7]	P.L. Duren, "Univalent functions" , Springer (1983) pp. 258
[a8]	O. Tammi, "Extremum problems for bounded univalent functions II" , Lect. notes in math. , 913 , Springer (1982)