Iterated integral transforms of Caratheodory functions and their applications to analytic and univalent functions

Abstract

In this paper we develop and study some integral transforms of Caratheodory functions. We apply the transforms to study certain other classes of analytic and univalent functions both to obtain new results and provide new proofs of some known ones.

Some remarks on certain Bazilevic functions

“…In fact for 0 ≤ β < 1, ψ 1 (u, v) = v/ξu, with ξ > 0 real and Ω = [C − 0] × C is found to be contained in the set Ψ β , 0 ≤ β < 1, defined in [2]. Thus the above theorem and corollaries can be extended easily to the class T α n (β).…”

Quasi-convolution of analytic functions with applications

“…. Several examples of members of the set Ψ have been mentioned in [4,11] and [12, p.27]. We shall need the following member:…”

“…is convex null. In fact it has been mentioned in [4] that the sequence preserves many geometric structures of analytic functions, particularly starlikeness, convexity and subordination. In this article we would make use of the convex null sequence {d k } ∞ k=0 in the investigation of convolution properties of functions of the class T α n (β).…”

“…where α > 0 is real, 0 ≤ β < 1, D n (n ∈ N) is the Salagean derivative operator defined as: D n f (z) = D(D n−1 f (z)) = z[D n−1 f (z)] ′ with D 0 f (z) = f (z) and powers in (2) meaning principal determinations only. The geometric condition (2) slightly modifies the one given originally in [15] (see [4]). If β = 0, we have the class B n (α) studied by Abdulhalim in [1].…”

“…It is natural, therefore, to desire to investigate the convolution properties of classes of functions. However, readers familiar with studies in Bazilevic functions of type α (see [1,3,4,5,10,15,18,19]) would appreciate the challenges posed by the index α in some characterizations of those functions, particularly when α = 1 or is generally non-integers. Of note, in particular, is in their convolution.…”