Neighbourhoods and partial sums of starlike functions based on Ruscheweyh derivatives. (English) Zbl 1054.30014

Rosy, Thomas; Subramanian, K. G.; Murugusundaramoorthy, G.

JIPAM, J. Inequal. Pure Appl. Math. 4, No. 4, Paper No. 64, 8 p. (2003).

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

Reviewer: V. Karunakaran (Madurai)

MSC:

30C45

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