Estimations of the second coefficient of a univalent, bounded, symmetric and non-vanishing function by means of Loewner's parametric method

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Title

Authors ¹

Institute of Mathematics, Silesian Technical University, Ul. Kaszubska 23, 44-100 Gliwice, Poland

Let

₀^{(R)} (b)

denote the class of functions F(z) = b + A₁z + A₂z² + ...

a n a l y t i c and u n i v a \leq n t \in t h e u n i t d i s k U w h i c h s a t i y t h e c o n d i t i o n s : F (U) \subset U, 0 \notin F (U),

Im F^{(n)}(0) = 0

. U \sin g L o e w \neq r' s p a r a m e t r i c m e t h o d w e o b t a \in l o w e r and u p p e r b o u n d s o f A ₂ \in

₀^{(R)}(b)

and f u n c t i o n s f or w h i c h t h e s e b o u n d s a r e r e a l i z e d . T h e

₀^{(R)}(b)

, \int r o d u c e d \in [6], i s a \subset f t h e

_u

o f b o u n d e d, n o n - v a n i s h \in g u n i v a \leq n t f u n c t i o n s \in t h e u n i t d i s k . T h i s l * c l o s e l y r e l a t e d o \neq s h a v e b e e n s t u d i e d b y v a r i o u s a u t h or s \in [1] - [4] . W e m e n t i o n \in p a r t i c \underset{̲}{a} r t h e p a p e r o f D . V . P r o k h or o v and J . S z y n a l [5], w h e r e a s h a r p u p p e r b o u n d f or t h e \sec o n d c o e f f i c i e n t \in

_u!$! is given.

univalent function, Loewner differential equation

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