Certain differential operators for meromorphic functions. (English) Zbl 0749.30016

Summary: Let \(J_ n(\alpha)\) be the class of functions of the form \(f(z)={a_{- 1}\over z}+\sum^ \infty_{k=0}a_ kz^ k\) regular in the punctured disk \(E=\{z:0<| z|<1\}\) with a simple pole at \(z=0\) and satisfying \[ \text{Re}\{{(D^{n+1}f(z))'\over (D^ nf(z))'}-2\}<- {n+\alpha\over n+1},\text{ for } n\in\mathbb{N}_ 0=\{0,1,2,\ldots\},\;| z|<1,\;0\leq\alpha<1, \] where \(D^ n f(z)={a_{-1}\over z}+\sum^ \infty_{m=2}m^ na_{m-2}z^{m-2}\).
It is proved that \(J_{n+1}(\alpha)\subset J_ n(\alpha)\). Since \(J_ 0(\alpha)\) is the class of meromorphic convex functions of order \(\alpha\), all functions in \(J_ n(\alpha)\) are convex. Further property preserving integrals are considered.


30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)