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Neighbourhoods and partial sums of starlike functions based on Ruscheweyh derivatives. (English) Zbl 1054.30014
Let \(S\) denote the class of all analytic functions \(f(z)\) in the open unit disc \(U\) of the complex plane with \(f(0)= 0= 1- f'(0)\). Let \(T\subset S\) be the class of functions of the form \(f(z)= z- \sum^\infty_{k=2}| a_k| z^k\). Let \(*\) denote the Hadamard convolution product. For \(\lambda\geq 0\), \(-1\leq\alpha< 1\), \(\beta\geq 0\), let \(S^\lambda_p(\alpha, \beta)\) denote the class of functions \(f(z)\in S\) satisfying \[ \text{Re}\Biggl\{{z(D^\lambda f)'(z)\over (D^\lambda f)(z)}- \alpha\Biggr\}> \beta\Biggl|{z(D^\lambda f)'(z)\over (D^\lambda f)(z)}- 1\Biggr| \] for all \(z\in U\) where \[ (D^\lambda f)(z)= f(z)* {z\over (1- z)^{\lambda+ 1}}. \] Let \(TS^\lambda_p(\alpha, \beta)= S^\lambda_p(\alpha, \beta)\cap T\).
In this paper the authors obtain coefficient bounds and extreme points of the families \(S^\lambda_p(\alpha, \beta)\) and \(TS^\lambda_p(\alpha, \beta)\). They also obtain lower bounds for \(\text{Re}({f(z)\over f_n(z)})\) and \(\text{Re}({f'(z)\over f_n'(z)})\) where \(f_n(z)\) is the \(n\)th partial sum of \(f(z)\in S^\lambda_p(\alpha, \beta)\).

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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