Radius problem associated with certain ratios and linear combinations of analytic functions

Abstract For normalized starlike functions f : 𝔻 → ℂ, we consider the analytic functions g : 𝔻 → ℂ defined by g ( z ) = (1 + z ( f ″( z ))/ f ′( z ))/( zf ′( z )/ f ( z )) and g ( z ) = (1 − α )( zf ′( z ))/ f ( z ) + α (1 + ( zf ″( z ))/ f ′( z )), 0 ≤ α ≤ 1. We determine the largest radius ρ with 0 < ρ ≤ 1 such that g ( ρ z ) is subordinate to various functions with positive real part.