Glasnik Matematicki, Vol. 39, No.2 (2004), 265-272.

ON THE LINEAR COMBINATION OF THE REPRESENTATIONS OF STARLIKENESS AND CONVEXITY

Nikola Tuneski and Roza Aceska

Faculty of Mechanical Engineering, Karpos II b.b., 1000 Skopje, Republic of Macedonia
e-mail: nikolat@mf.ukim.edu.mk
e-mail: aroza@mf.ukim.edu.mk

Abstract.   Let A be the class of analytic functions in the unit disk U = {z : |z| < 1} that are normalized with f(0) = f'(0) - 1 = 0. Also, let S*[A,B], -1 ≤ B < A ≤ 1, be the class of functions f ∈ A, such that zf'(z) / f(z) < (1 + Az) / (1 + Bz), where "<" denotes the usual subordination. In this paper we investigate the linear combination of the analytic representations of starlikeness and convexity and give sharp sufficient conditions over the differential operator

a zf'(z) / f(z) + b (1 + zf''(z) / f'(z))

that imply f ∈ S*[A,B]. In that purpose we use the method of differential subordinations. Several corollaries and examples for different choices of A, B, a and b are given and comparison with previous known results is done.

2000 Mathematics Subject Classification.   30C45.

Key words and phrases.   Starlike function, starlike function of order α, criteria, differential subordination, Jack lemma.

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