Some remarks for a certain class of holomorphic functions at the boundary of the unit disc

Abstract

We consider a boundary version of the Schwarz Lemma on a certain class which is

denoted by K(alfa) . For the function  f(z)=z+c₂z²+....... which is defined in the unit disc E

such that the function f (z) belongs to the class K(alfa) , we estimate from below the modulus of the

angular derivative of the function ( )

( )

zf ' z

f z

at the boundary point b with ( ) = 1

( ) 1

bf ' b

f b 

. Moreover,

we get the Schwarz Lemma for the class K(alfa) . We also investigate some inequalities obtained in

terms of sharpness.

Keywords

• Schwarz Lemma,Holomorphic function,Jack’s Lemma,Angular derivative

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