A Class of Close-to-Convex Functions Satisfying a Differential Inequality: Pardeep Kaur, Sukhwinder Singh Billing - Team, Support (Thai Journal of Mathematics, 2022)
Let H^φ_α(β) denote the class of functions f, analytic in the open unit disk E, which satisfy the condition \[[(1-α)zf'(z)φ(z)+α(2+zf''(z)f'(z)-zφ'(z)φ(z))]>β, zE, \] where α, β are pre-assigned real numbers and φ is a starlike function in E. In the present paper, we prove that members of the class H^φ_α(β) are close-to-convex and hence univalent for real numbers α, β, α≤β<1 and for a starlike function φ.