[Retracted] Partial sums of functions of bounded turning

Abstract

We determine conditions under which the partial sums of the Libera integral operator of functions of bounded turning are also of bounded turning

Partial sums of analytic functions of bounded turning with applications

“…The number 3 8 is sharp ( [13]). In [14], it was also shown that the partial sums of the Libera integral operator of functions of bounded turning are also of bounded turning. We determine conditions under which the partial sums (5-7) of the multiplier integral operators (2-4) of analytic univalent functions of bounded turning are also of bounded turning.…”

Quasi Partial Sums of Harmonic Univalent Functions

“…In [1], Babalola defined a new concept of quasi-partial sums of the generalized Bernardi integral operator for analytic univalent functions and he extended an earlier result of Jahangiri and Farahmand [5]. Yet analogous results on harmonic univalent functions have not been so far explored.…”

Quasi-Partial sums of the generalized Bernard integral of certain analytic functions

“…Denote by A the class of functions: f (z) = z + a 2 z 2 + · · · (1.1) which are analytic in the unit disk E = {z : |z| < 1}. In [5] Jahangiri and Farahmand studied the partial sums of the Liberal integral of the class B(β), which consists of functions in A satisfying Re f ′ (z) > β, 0 ≤ β < 1. Functions in B(β) are called functions of bounded turning.…”

“…The result of Jahangiri and Farahmand [5] naturally leads to inquistion about a more general class of functions (including B(β) as a special case), which was introduced in [7] by Opoola, and has been studied extensively in [2]. This is the class T α n (β) consisting of functions f ∈ A which satisfy the inequality:…”