Applications of Jack’s lemma

In the Jack’s lemma it is considered q(z), an analytic function in \(|z|<1\) $|z|<1$ with \(q(0)=0\) $q(0)=0$ for which |q(z)| attains its maximum value on the disc \(|z|\le r<1\) $|z|\le r<1$ at the point \(z_0\) $z_0$ , \(|z_0|=r\) $|z_0|=r$ . Then \(z_0q'(z_0)=kq(z_0)\) $z_0{q}^{\prime }(z_0)=kq(z_0)$ and \(k\ge 1\) $k\ge 1$ . In this paper we try to say more about the number k in a generalizations of this lemma, where we consider \(\max |\arg \{q(z)\}|\) $max|arg\{q(z)\}|$ or \(\min |\mathfrak {Re} \{q(z)\}|\) $min|\mathfrak{Re}\{q(z)\}|$ instead of |q(z)|. In these cases q(z) has another normalization at the origin.