Starlikeness of nonlinear integral transforms
For \(n \geq 1\), \(\lambda>0\), \(\mu>0\), let \(\mathcal{U}_n(\lambda,\mu)\) be the class of functions \(f\) of the form \(f(z)=z+a_nz^{n+1}+\cdots\) satisfying the condition \(f(z)/z\neq 0\), \( | f'(z)(z/f(z))^{\mu+1}-1 | < \lambda \). For a real-valued, nonnegative, integrable function \(\varphi(t)\) normalized by the condition \(\int_0 ^1 \varphi(t) dt =1\), and for \(\mu > 0\), let \([V_\varphi^\mu(f)](z)= z [\int_0 ^1 \varphi(t) (tz/f(tz))^\mu dt]^{1/\mu}\), where \(f\in \mathcal{U}_n(\lambda, \mu)\). The authors obtain conditions on the parameters \(\lambda\) and \(\mu\) and on the function \(\varphi(t)\) such that the transform \(V_\varphi^\mu(f)\) is univalent or starlike. Also they investigate some special integral transforms related to hypergeometric functions, polylogarithms, and Hurwitz functions.
scientific article; zbMATH DE number 5635044