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Integral transforms of functions with the derivative in a halfplane

Authors:

Institution:

1. Department of Mathematics, Indian Institute of Technology IIT-Madras, 600 36, Chennai, India
2. School of Teacher Education, S?r-Tr?ndelag College, N-7004, Trondheim, Norway

Abstract:

LetA be the class of normalized analytic functions in the unit disk Δ and define the class

$\mathcal{P}_\beta = \left\{ {f \in \mathcal{A}|\exists \alpha \in \mathbb{R}|\operatorname{Re} \{ e^{ia} (f'(z) - \beta )\} > 0,z \in \Delta } \right\}$

For a functionf εA the Alexander transformF ₀ is given by

$F_0 (z) = \int {_0^1 \frac{{f(tz)}}{t}dt.}$

Our main object is to establish a sharp relation betweenβ andγ such thatf εP _β implies thatF ₀ is starlike of orderγ, 0 ≤γ ≤ 1/2. A corresponding result for the Libera transformF ₁(z) = 2∫ ₀ ¹ f(tz)dt is also given.

Keywords:

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