Analytic functions with polar and logarithmic singularities and locally convex boundary values (Q1933373)

Denote by \(\Sigma^0_m(\log)\) a family of functions of the form \[ F(z)= z^m+ a_{m-1} z^{m-1}+\cdots+ a_1 z+\lambda Lnz+ \sum^\infty_{n=0} \alpha_n z^{-n},\;|z|> 1, \] such that \(F(z)- (z^m+ a_{m-1} z^{m-1}+\cdots+ a_1 z+\lambda Lnz)\) is holomorphic in \(\mathbb{D}^*= \{z\in\overline{\mathbb{C}}:|z|> 1\}\) and for all \(z\in \mathbb{D}^*\) the inequality \[ \text{Re}\Biggl(1+ {zF''(z)\over F'(z)}\Biggr)> 0 \] is fulfilled. In this paper some sharp estimates for the derivative and the Schwarzian are obtained. Some geometric properties of such convex functions with some logarithmic normalization are described.