On some univalent integral operators
A new class \(B_{\alpha}[A,B,C,D]\) for \(\alpha >0\), \(-1\leq B<A\leq 1\), \(- 1\leq D<C\leq 1\) a subclass of the class of Bazilevich functions is defined. Closely following the argument in \textit{R. Parvatham} and \textit{T. N. Shanmugam} [Bull Aust. Math. Soc. 28, 207-215 (1983; Zbl 0515.30008)], few lemmas were established. (In fact they were already proved by T. N. Shanmugam and formed part of his Ph. D. thesis.) Using these lemmas some results were established in this paper, mainly it is proved that this class \(B_{\alpha}[A,B,C,D]\) is closed under the transform by the integral operator \[ \{(m+\alpha)z^{-m}\int^{z}_{0}t^{m- 1}f^{\alpha}(t)dt\}^{1/\alpha}. \] The main result was already established in a much more generalized form for the class \(B(\alpha,h_ 1,h_ 2)\) by \textit{T. N. Shanmugam} in Proc. Indian Acad. Sci., Math. Sci. 95, 121-125 (1986; Zbl 0629.30012).
scientific article; zbMATH DE number 4023638