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Certain results for a class of convex functions related to a shell-like curve connected with Fibonacci numbers. (English) Zbl 1222.30009
Summary: This paper investigates some basic geometric properties for the class $\cal{KSL}$ of functions $f$ analytic in the open unit disc $\Delta =\{z:|z|<1\}$ (which is related to a shell-like curve and associated with Fibonacci numbers) satisfying the condition that $$f(0)=0, \quad f'(0)=1\quad \text{and}\quad \frac{zf''(z)}{f'(z)}\prec\frac{\tau z + 2\tau^2 z^2}{1 - \tau z - \tau^2 z^2} \quad (z\in\Delta),$$ where the number $\tau=(1-\sqrt{5})/2$ is such that $|\tau |$ fulfils the golden section of the segment $[0,1]$. Some relevant remarks and useful connections of the main results are also pointed out.

##### MSC:
 30C45 Special classes of univalent and multivalent functions 30C80 Maximum principle; Schwarz’s lemma, Lindelöf principle, etc. (one complex variable)
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##### References:
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