Fekete–Szegö problem for some starlike functions related to shell-like curves

Abstract

Abstract In this note, we investigate the Fekete-Szegö problem for a class 𝓢𝒽 of functions f analytic in the open unit disc Δ = {z: |z| < 1} (and which is related to a shell-like curve associated with Fibonacci numbers) satisfying the conditions that f ( 0 ) = 0 , f ′ ( 0 ) = 1 and z f ′ ( z ) f ( z ) ≺ 1 + τ 2 z 2 1 − τ z − τ 2 z 2 ( z ∈ Δ ) , $$f(0) = 0,\qquad f'(0) = 1\qquad {\text{and}}\qquad \frac{{zf'(z)}}{{f(z)}} \prec \frac{{1 + {\tau ^2}{z^2}}}{{1 - \tau z - {\tau ^2}{z^2}}}\quad (z \in \Delta ),$$ where ≺ denotes the subordination and the number τ = ( 1 − 5 ) / 2 $\tau = (1 - \sqrt 5 )/2$ is such that |τ| fulfils the golden section of the segment [0,1].