# Typically-real function

in a domain

A function $f ( z)$, analytic in some domain $B$ in the $z$- plane containing segments of the real axis, which is real on these segments and for which $( \mathop{\rm Im} f ( z)) ( \mathop{\rm Im} z) > 0$ whenever $\mathop{\rm Im} z \neq 0$. A fundamental class of typically-real functions is the class $T$ of functions

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

that are regular and typically real in the disc $| z | < 1$( cf. [1]). It follows from the definition of the class $T$ that $c _ {n}$ is real for $n \geq 2$. The class $T$ contains the class $S _ {r}$ of functions

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

with real coefficients $c _ {n}$, that are regular and univalent in $| z | < 1$( cf. Univalent function). If $f \in T$, then

$$\phi ( z) = \ { \frac{1 - z ^ {2} }{z} } f ( z) \in C _ {r} ,$$

and, conversely, if $\phi \in C _ {r}$, then

$$f ( z) = \ { \frac{z}{1 - z ^ {2} } } \phi ( z) \in T,$$

where $C _ {r}$ is the class of functions

$$\phi ( z) = 1 + \sum _ {n = 1 } ^ \infty \alpha _ {n} z ^ {n}$$

that are regular in $| z | < 1$ with $\mathop{\rm Re} \phi ( z) > 0$ in $| z | < 1$ and such that $\alpha _ {n}$ is real for $n \geq 1$.

Let $M _ {1}$ be the class of non-decreasing functions $\alpha ( t)$ on $[- 1, 1]$ for which $\alpha ( 1) - \alpha (- 1) = 1$. Functions of class $T$ can be represented in $| z | < 1$ by Stieltjes integrals (cf. [2]):

$$\tag{1 } f ( z) = \ \int\limits _ { - } 1 ^ { 1 } s ( z, t) \ d \alpha ( t),$$

$$s ( z, t) = z ( 1 - 2tz + z ^ {2} ) ^ {-} 1 ,\ \alpha ( t) \in M _ {1} ,$$

in the sense that for each $f \in T$ there exists an $\alpha \in M _ {1}$ such that (1) holds and, conversely, for any $\alpha \in M _ {1}$ formula (1) defines some function $f \in T$. One has $s ( z, t) \in S _ {r}$ for any fixed $t \in [- 1, 1]$. The largest domain in which every function in $T$ is univalent is $\{ | z + i | < \sqrt 2 \} \cap \{ | z - i | < \sqrt 2 \}$. From the representation (1) for the class $T$, a number of rotation and distortion theorems have been obtained (cf. Distortion theorems; Rotation theorems). The following hold in the class $T$:

$$\tag{2 } - n \leq c _ {n} \leq n \ \ \textrm{ if } n \textrm{ is } \textrm{ even } ,$$

$$\tag{3 } k _ {n} = \min _ {0 \leq \theta \leq \pi } \ \frac{\sin n \theta }{\sin \theta } \leq c _ {n} \leq n \ \textrm{ if } n \textrm{ is } \textrm{ odd }$$

$$\left ( k _ {n} \sim \frac{- 2n }{3 \pi } \right ) ,$$

with equality on the left in (2) only for $s ( z, - 1)$ and on the right only for $s ( z, 1)$, on the left in (3) only for functions $f ( z) = \lambda s ( z, t _ {n} ) + ( 1 - \lambda ) s ( z, - t _ {n} )$ for some $t _ {n} \in [- 1, 1]$, and on the right only for $f ( z) = \lambda s ( z, 1) + ( 1 - \lambda ) s ( z, - 1)$, $0 \leq \lambda \leq 1$.

For $T$, the coefficient regions for the systems $\{ c _ {2} \dots c _ {n} \}$, $\{ f ( z) \}$, $\{ f ( z), c _ {2} \dots c _ {n} \}$, $n \geq 2$, have been found (cf. [3]).

#### References

 [1] W. Rogosinski, "Ueber positive harmonische Entwicklungen und typische-reelle Potenzreihen" Math. Z. , 35 (1932) pp. 93–121 [2] G.M. Goluzin, "On typically real functions" Mat. Sb. , 27 : 2 (1950) pp. 201–218 (In Russian) [3] G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)