Sufficient Conditions for Starlikeness

Abstract

Abstract. We obtain the conditions on β so that 1+βzpThese results are applied to obtain sufficient conditions for normalized analytic function f to belong to various subclasses of starlike functions, or to satisfy the condition | log(zf ′ (z)/f (z))| < 1 or |(zf ′ (z)/f (z)) 2 − 1| < 1 or zf ′ (z)/f (z) lying in the region bounded by the cardioid (9x 2 + 9y 2 − 18x + 5) 2 − 16(9x 2 + 9y 2 − 6x + 1) = 0.

On λ-pseudo bi-starlike functions related to some conic domains

“…then the class S * ( φ) is associated with a cardioid [19]. We may consider also the functions φ k,α related to the conic sections, that were introduced and studied by Kanas et al [8] - [13], where (0 ≤ k < ∞, 0 ≤ α < 1) and where…”

Differential subordination for Janowski functions with positive real part

“…Recently, Sharma and Ravichandran [22] established similar type subordination for analytic functions associated to Cardioid. Motivated by this work, the condition on β is computed so that p(z)…”

Some Special Differential Subordinations

“…Remark 4.6. We would like to point out that authors in [26] have obtained conditions on real parameter β so that the subordination 1 + βzp ′ (z)/p j (z), (j = 0, 1, 2) implies p(z)…”