Sufficient conditions for starlike functions associated with the lemniscate of Bernoulli

Abstract

Let -1 ≤ B < A ≤ 1. The condition on β is determined so thatA few more problems of the similar flavor are also considered. MSC: 30C80; 30C45

Starlikeness associated with lemniscate of Bernoulli

“…Kumar et al [5] proved that for β > 0 if p(z) + βzp ′ (z)/p n (z) ≺ √ 1 + z (n = 0, 1, 2), then p(z) ≺ √ 1 + z. Extending this, we obtain lower bound for β so that Observe that g(θ) = g(−θ) for all θ ∈ (−π/4, π/4) and the second derivative test shows that the minimum of g occurs at θ = 0 for βm > 1.1874.…”

“…Ali et al [1] studied the class SL with the help of differential subordination and obtained some lower bound on β such that p(z) ≺ √ 1 + z whenever 1 + βzp ′ (z)/p n (z) ≺ √ 1 + z (n = 0, 1, 2), where p is analytic on D with p(0) = 1. Kumar et al [5] proved that whenever β > 0, p(z)+βzp ′ (z)/p n (z) ≺ √ 1 + z (n = 0, 1, 2) implies p(z) ≺ √ 1 + z for p as mentioned above. Motivated by work in [1,3,4,5,7,11,12,13], the method of differential subordination of first and second order has been used to obtain sufficient conditions for the function f ∈ A to belong to class SL.…”

“…Section 4 deals with obtaining sufficient conditions on β and γ, using the method of differential subordination which implies p(z) ≺ √ 1 + z if γzp ′ (z) + βz 2 p ′′ (z) ≺ z/(8 √ 2) and others. Section 5 admits alternate proofs for the results proved in [1] and [5]. The proofs are based on properties of admissible functions formulated by Miller and Mocano [8].…”

q-Analogue of Differential Subordinations

“…Similar kind of differential subordinations are also discussed by various authors. They used these results to find sufficient conditions for starlike functions, see [31][32][33][34][35][36]. Motivated by the above work, we introduce and investigate some q-differential subordinations.…”

Sufficient Conditions for Starlikeness

“…Recently Ali et al [2] determined the condition on β for p(z) ≺ √ 1 + z when 1 + βzp ′ (z)/p n (z) with n = 0, 1, 2 or (1 − β)p(z) + βp 2 (z) + βzp ′ (z) is subordinated to √ 1 + z. Motivated by the works in [1,2,3,9,15,17], in Section 2, we determine the sharp conditions on β so that p(z)…”