Starlike Functions in the Hornich Space

Abstract

We will show that the set of starlike univalent functions in D is starlike in the Hornich space, i.e. for starlike functions f and 0 ≤ α ≤ 1 the function \(\int_0^z(f'(\zeta))^\alpha d\zeta\) is also starlike. This solves a problem given by Kim, Ponnusamy and Sugawa in [6]. An important step in proving this result will be to show that for starlike functions f and z ∈ D we have \(\bigg|\int_0^1{\rm arg}(z/\gamma'(t))dt\bigg| < \pi /2/,\) where γ(t):= f−1(t f (z)), 0 ≤ t ≤ 1.