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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.

30C45Special classes of univalent and multivalent functions
30C80Maximum principle; Schwarz’s lemma, Lindelöf principle, etc. (one complex variable)
Full Text: DOI
[1] Sokół, J.: On starlike functions connected with Fibonacci numbers, Folia sci. Univ. tech. Resoviensis 175, 111-116 (1999) · Zbl 0995.30012
[2] J. Dziok, R.K. Raina, J. Sokół, On a class of starlike functions related to shell-like curve connected with Fibonacci numbers (submitted for publication).
[3] Ma, W.; Minda, D.: A unified treatment of some special classes of univalent functions, Conf. proc. Lecture notes anal., 157-169 (1994) · Zbl 0823.30007
[4] J. Dziok, R.K. Raina, J. Sokół, On {$\alpha$}-convex functions related to shell-like functions connected with Fibonacci numbers, Appl. Math. Comput., 2011, in press (doi:10.1016/j.amc.2011.01.059).
[5] Robertson, M. S.: Certain classes of starlike functions, Michigan math. J. 76, No. 1, 755-758 (1954)
[6] Janowski, W.: Extremal problems for a family of functions with positive real part and some related families, Ann. polon. Math. 23, 159-177 (1970) · Zbl 0199.39901
[7] Janowski, W.: Some extremal problems for certain families of analytic functions, Ann. polon. Math. 28, 297-326 (1973) · Zbl 0275.30009
[8] Goodman, A. W.: Univalent functions, Univalent functions (1983)
[9] Stankiewicz, J.: Quelques problèmes extrémaux dans LES classes des fonctions ${\alpha}$-angulairement étoilées, Ann. univ. Mariae Curie-skłodowska sect. A 20, 59-75 (1966) · Zbl 0247.30007