A sufficient condition for univalency for functions with integral representation (Q2724009)

By using the method of Loewner chains, the author proves the following main result:NEWLINENEWLINENEWLINETheorem. Let \(f,g\) be analytic in the unit disc and have the form \(z + a_2z^2+\dots\;\). Let \(\alpha\) be complex, \(\beta > 0\), such that \(|\alpha - \beta |< \beta\). If NEWLINE\[NEWLINE{1-|z |^{2\beta} \over \beta} \left |(\alpha - 1) {zg'(z) \over g(z)} + {zf''(z) \over f'(z)} + 1 - \beta \right |\leq 1,\quad |z |< 1, NEWLINE\]NEWLINE then the function NEWLINE\[NEWLINEH(z) = \left(a \int_0^z g^{a-1}(u)f'(u) du\right)^{1/2}NEWLINE\]NEWLINE is analytic and univalent in the unit disc, where the principal branch is intended.NEWLINENEWLINENEWLINEMany corollaries were also given.