CERTAIN DIFFERENTIAL-OPERATORS FOR MEROMORPHIC FUNCTIONS

Let J(n)-(alpha) be the class of functions of the form f(z) = a-1/z + SIGMA-k = 0 infinity a(k)z(k) regular in the punctured disk E = {z : 0 < \z\ < 1} with a simple pole at z = 0 and satisfying Re {(D(n+1)f(z))'/(D(n)f(z))' - 2} < - n+alpha/n+1, for n-epsilon-N0 = {0, 1, 2, ...}, \z\ < 1, 0 less-than-or-equal-to alpha < 1 where D(n)f(z) = a-1/z + SIGMA-m = 2 infinity m(n)am-2z(m-2). It is proved that J(n+1)(alpha) subset-of J(n)(alpha). Since J0(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.