INEQUALITIES FOR THE ANGULAR DERIVATIVES OF CERTAIN CLASSES OF HOLOMORPHIC FUNCTIONS IN THE UNIT DISC

Abstract

In this paper, a boundary version of the Schwarz lemma is investigated. We take into consideration a function <TEX>$f(z)=z+c_{p+1}z^{p+1}+c_{p+2}z^{p+2}+{\cdots}$</TEX> holomorphic in the unit disc and <TEX>$\|\frac{f(z)}{{\lambda}f(z)+(1-{\lambda})z}-{\alpha}\|$</TEX> < <TEX>${\alpha}$</TEX> for <TEX>${\mid}z{\mid}$</TEX> < 1, where <TEX>$\frac{1}{2}$</TEX> < <TEX>${\alpha}$</TEX> <TEX>${\leq}{\frac{1}{1+{\lambda}}}$</TEX>, <TEX>$0{\leq}{\lambda}$</TEX> < 1. If we know the second and the third coefficient in the expansion of the function <TEX>$f(z)=z+c_{p+1}z^{p+1}+c_{p+2}z^{p+2}+{\cdots}$</TEX>, then we can obtain more general results on the angular derivatives of certain holomorphic function on the unit disc at boundary by taking into account <TEX>$c_{p+1}$</TEX>, <TEX>$c_{p+2}$</TEX> and zeros of f(z) - z. We obtain a sharp lower bound of <TEX>${\mid}f^{\prime}(b){\mid}$</TEX> at the point b, where <TEX>${\mid}b{\mid}=1$</TEX>.