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)

References

[a1]	P.L. Duren, "Univalent functions" , Springer (1983) pp. Sect. 10.11
[a2]	A.W. Goodman, "Univalent functions" , 1 , Mariner (1983)