# Coefficient problem

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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

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].