The Schwarzian derivative and univalent functions

Abstract

In this paper we prove under certain conditions the function w = f ( z ) w = f(z) is univalent in | z | > 1 |z| > 1 .

Some characteristic properties of analytic functions

“…It may be noted that for λ = 0, α = 0, µ = 3, the class B (0, 3; 0) = U was earlier studied by Ozaki and Nunukawa in [11] (see also Obradovic et al [10] and Singh [19]), where it is proved that the functions f ∈ U are univalent. For two analytic functions p, q such that p(0) = 1 = q(0), we say that p is subordinate to q in U and write p(z) ≺ q(z), z ∈ U, if there exists a Schwarz function w, analytic in U with w(0) = 0, and |w(z)| < 1, z ∈ U such that p(z) = q(w(z)), z ∈ U.…”

Some Starlikeness Conditions for the Analytic Functions and Integral Transforms

“…, n = 1, µ = 1 and λ = 1, this class becomes U, which was introduced by Ozaki and Nunokawa [8]. They shown that U(1, 1) = U ⊂ S. For α = 1, n = 1, 0 < λ ≤ 1 and µ < 0 this class becomes U(λ, µ) which is a subclass of the class of Bazilevič functions.…”

Properties of a new integral operator

“…Recently, Frasin and Jahangiri [4] define the family B(µ, λ), µ ≥ 0, 0 ≤ λ < 1 consisting of functions f ∈ A satisfying the condition Frasin and Darus [5]); (iv) B(2, 0) = S (see Ozaki and Nunokawa [3]). Let N (ρ) be the subclass of A that contains all the functions f which satisfy the inequality…”