Abstract
For some classes of analytic functions f, g, h and k in the open unit disk U, we consider the general integral operator Tn, that was introduced in a recent work [1] and we obtain new conditions of univalence for this integral operator. The key tools in the proofs of our results are the Pascu’s and the Pescar’s univalence criteria, as well as the Mocanu’s and erb’s Lemma. Some corollaries of the main results are also considered. Relevant connections of the results presented here with various other known results are briefly indicated.
Highlights
Introduction and preliminariesLet A denote the class of the functions of the form: ∞f (z) = z + anzn, (1)n=2 which are analytic in the open unit diskU = {z ∈ C :| z |< 1}C
N=2 which are analytic in the open unit disk
Zfi (z) − 1 fi(z) zgi (z) gi (z) i=1 zhi(z) − 1 + zki(z) − 1 hi(z) zhi (z) + zki (z). We find from this last inequality (6) that 1 − |z|2c c zTn (z) Tn(z)
Summary
Let A denote the class of the functions of the form:. C. Vi) For αi − 1 = βi = 0, ki(z) = z and ki(z) = 1 we obtain the integral operator which was defined and studied by Pescar [25]. Z2 2 we obtain the integral operator which was defined and studied by Bucur and Breaz in [6]. Xi) For δ = 1, αi − 1 = δi = 0, βi = γi and hi(z) = fi(z) we obtain the integral operator which was defined and studied by Nguyen, Oprea and Breaz in [18]. Eiθ zm, The problem of univalence for some generalized integral operators using functions from the class B (μ, λ) were recently obtained in papers[5], [7],[8], [11], [19]
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