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Analytic, symmetric functions that omit zero

Śladkowska Janina

Zeszyty Naukowe Politechniki Rzeszowskiej. Matematyka

2004

z. 27 [212]

69-81

Let Fo] (b), b is an element of R denote the class of functions of the form f{z) = b+b1z+..., analytic in the unit disk U and such that 0 [...] and let Fo(b) be the set of functions of the class Fo(b) such that f(z) €is an element of R iff z (-1,1). The functions from the class ^{b) are closely related to the typically-real functions, namely for each f is an element of Fo(b) there exist two typically-real functions T1 and T2 such that f = bT1/T2. So it is possible to treat Fo (b) as a subclass of the class[...] of all the functions of the form bT1/T2, where T1,T2 are arbitrary typically-real functions. It is easy to obtain the variational formulas in the class [...] (b) by utilizing the known formulas obtained in the class of typically-real functions [1]. Certain extremal problems in the class FoR(b) and also in the class Fo(b), for example estimation from below of the second coefficient, were solved with the aid of these formulas.