Univalence criterion and convexity for an integral operator (Q427700)

Let \(f\) and \(g\) be normalized analytic functions defined on the open unit disk of the complex plane satisfying the conditions NEWLINE\[NEWLINE|z^2f'(z)/f^2(z) -1|<1,\quad |f''(z)/f'(z)|<1\quad\text{and}\quad |g(z)|<MNEWLINE\]NEWLINE for some \(M>0\). For a complex number \(\alpha\) with non-negative real part satisfying \(2\sqrt{3}(1+2M^2)|\alpha|\leq 9\), the authors show that the integral NEWLINE\[NEWLINEF(z):=\int_0^z[f'(t)\exp(g(t))]^\alpha dtNEWLINE\]NEWLINE is univalent in the unit disk. They also determine the order of convexity of the function \(F\) when \(f\) and \(g\) satisfy the conditions NEWLINE\[NEWLINE|f'(z) (z/f(z))^\mu -1|<1-\beta,\quad |f''(z)/f'(z)|<1\quad\text{and}\quad |g(z)|<M.NEWLINE\]