Abstract

Different versions of the boundary Schwarz lemma for the &#x1D4A9; (&#x1D70C;) class are discussed in this study. Also, for the function g(z) = z+b₂z²+b₃z³+... defined in the unit disc D such that g &#x2208; &#x1D4A9; (&#x1D70C;), we estimate a modulus of the angular derivative of g(z) function at the boundary point 1 &#x2208; &#x1D715;D with g'(1) = 1 + &#x1D70E; (1 - &#x1D70C;), where <TEX>${\rho}={\frac{1}{n}}{\sum\limits_{i=1}^{n}}g(c_i)={\frac{g^{\prime}(c_1)+g^{\prime}(c_2)+{\ldots}+g^{\prime}(c_n)}{n}}{\in}g^{\prime}(D)$</TEX> and &#x1D70C;&#x2260;1, &#x1D70E; > 1 and c₁, c₂, ..., c_n &#x2208; &#x1D715;D. That is, we shall give an estimate below |g"(1)| according to the first nonzero Taylor coefficient of about two zeros, namely z = 0 and z &#x2260; 0. Estimating is made by using the arithmetic average of n different derivatives g'(c₁), g'(c₂), ..., g'(c_n).