Univalence criteria for two integral operators (Q410212)

Let \(A\) denote the class of functions \(f(z)=z+\sum_{n=2}^{\infty}a_nz^n\), \(|z|<1\). The authors prove two theorems and numerous corollaries giving univalence conditions for functions represented by the integral operators NEWLINE\[NEWLINEI(f_1,\dots,f_n;g_1,\dots,g_n)(z)= \left(\beta\int_0^zt^{\beta-1}\prod_{k=1}^n\left(\frac{f_k(t)}{t}\right)^{(a_k-1)/M_k}(g_k'(t))^{\gamma_k}dt\right)^{1/\beta}NEWLINE\]NEWLINE and NEWLINE\[NEWLINEJ(f_1,\dots,f_n;g_1,\dots,g_n)(z) = \left(\left(1+\sum_{k=1}^na_k\right)\int_0^z\prod_{k=1}^n(f_k(t))^{a_k}(g_k'(t))^{\gamma_k}dt\right)^{1/\left(1+\sum_{k=1}^na_k\right)},NEWLINE\]NEWLINE where, for \(k=1,\dots,n\), \(a_k,\gamma_k\in\mathbb C\), \(\beta\in\mathbb C\setminus\{0\}\), \(f_k,g_k\in A\) and \(M_k\geq1\).