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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 𝒦𝒮ℒ of functions f analytic in the open unit disc Δ={z:|z|<1} (which is related to a shell-like curve and associated with Fibonacci numbers) satisfying the condition that

f(0)=0,f ' (0)=1andzf '' (z) f ' (z)τz+2τ 2 z 2 1-τz-τ 2 z 2 (zΔ),

where the number τ=(1-5)/2 is such that |τ| fulfils the golden section of the segment [0,1]. Some relevant remarks and useful connections of the main results are also pointed out.

MSC:
30C45Special classes of univalent and multivalent functions
30C80Maximum principle; Schwarz’s lemma, Lindelöf principle, etc. (one complex variable)
References:
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