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

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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>f</m:mi> <m:mo>(</m:mo> <m:mn>0</m:mn> <m:mo>)</m:mo> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mtext> </m:mtext> <m:msup> <m:mi>f</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo>(</m:mo> <m:mn>0</m:mn> <m:mo>)</m:mo> <m:mo>=</m:mo> <m:mn>1</m:mn> <m:mtext> </m:mtext> <m:mtext>and</m:mtext> <m:mtext> </m:mtext> <m:mfrac> <m:mrow> <m:mi>z</m:mi> <m:msup> <m:mi>f</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo>(</m:mo> <m:mi>z</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mrow> <m:mi>f</m:mi> <m:mo>(</m:mo> <m:mi>z</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mfrac> <m:mo>≺</m:mo> <m:mfrac> <m:mrow> <m:mn>1</m:mn> <m:mo>+</m:mo> <m:msup> <m:mi>τ</m:mi> <m:mn>2</m:mn> </m:msup> <m:msup> <m:mi>z</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> <m:mrow> <m:mn>1</m:mn> <m:mo>−</m:mo> <m:mi>τ</m:mi> <m:mi>z</m:mi> <m:mo>−</m:mo> <m:msup> <m:mi>τ</m:mi> <m:mn>2</m:mn> </m:msup> <m:msup> <m:mi>z</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:mfrac> <m:mtext> </m:mtext> <m:mo>(</m:mo> <m:mi>z</m:mi> <m:mo>∈</m:mo> <m:mtext>Δ</m:mtext> <m:mo>)</m:mo> <m:mo>,</m:mo> </m:mrow> </m:math> f(0) = 0, f'(0) = 1 and zf'(z)f(z) 1 + τ ²z²1 - τ z - τ ²z² (z ∈ Δ ), where ≺ denotes the subordination and the number <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>τ</m:mi> <m:mo>=</m:mo> <m:mo>(</m:mo> <m:mn>1</m:mn> <m:mo>−</m:mo> <m:msqrt> <m:mn>5</m:mn> </m:msqrt> <m:mo>)</m:mo> <m:mo>/</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> τ = (1 - √ 5 )/2 is such that |τ| fulfils the golden section of the segment [0,1].