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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 $$f(z)\in S^\lambda_p(\alpha, \beta)$$.

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