An application of Miller and Mocanu's result (Q921161)

For a fixed c and for an analytic f(z), \(f(0)=f'(0)-1=0\), \(| z| <1\), the operator \(I_ c(f)\) is defined by the formula \[ I_ c(f)=(1+c)z^{-c}\int^{z}_{0}t^{-c}f(t)dt. \] It is known that if Re\(\{\) \(c\}\geq 0\) and f(z) is starlike then \(I_ c(f)\) is also starlike [\textit{Z. Lewandowski}, \textit{S. Miller} and the reviewer, Proc. Am. Math. Soc. 56, 111-117 (1976; Zbl 0298.30008)]. In this note it is shown that if f(z) is analytic, \(c>-1\) and some complicated conditions are met then \(| I_ c(f)| <1\).