Applications of k-Fibonacci numbers for the starlike analytic functions

Abstract The k-Fibonacci numbers F k;n (k>0), defined recursively by F k;0 =0;F k;1 = 1 and F k;n = kF k;n + F k;n 1 for n1 are used to definea new class SL k . The purpose of this paper is to apply properties ofk-FibonaccinumberstoconsidertheclassicalproblemofestimationoftheFekete–SzegoproblemfortheclassSL k . Anapplicationforinversefunctionsisalsogiven. 2000AMSClassification: Primary 30C45, secondary 30C80Keywords: univalent functions; convex functions; starlike functions; subordina-tion; k-Fibonacci numbers. Received27/12/2012 : Accepted04/12/2013Doi: 10.15672/HJMS.2015449091 1. Introduction Let D = fz: jzj<1gdenote the unit disc on the complex plane. The class of allholomorphicfunctionsfintheopenunitdiscD withnormalizationf(0) = 0,f 0 (0) = 1isdenotedbyA andtheclassS ˆA istheclasswhichconsistsofunivalentfunctionsinD.WesaythatfissubordinatetoFinD,writtenasf˚F,ifandonlyiff(z) = F(!(z))forsome!2A,j!(z)j<1,z2D.Recently, N. Yilmaz Ozgur and J. Sokol [5] defined and introduced the class SL