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On an integral operator for convex univalent functions
Published online by Cambridge University Press: 17 April 2009
Abstract
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Let K (m,M) denote the class of functions 
regular and satisfying |1 + zf″(z)/f′(z)− m| < M in |z| < 1, where |m−1| < M ≤ m. Recently, R.K. Pandey and G. P. Bhargava have shown that if f ε K (m,M), then the function 
du also belongs to K (m,M) provided α is a complex number satisfying the inequality |α| ≤ (1−b)/2, where b = (m-1)/M. In this paper we show by a counterexample that their inequality is in general wrong, and prove a corrected version of their result. We show that F ε K (m,M) provided that α is a real number satisfying −φ ≤ α ≤1, φ = (M−|m−1|)/(M + |m−1|), or a complex number satisfying |α| ≤ φ. In both cases the bounds for α are sharp.
- Type
- Research Article
- Information
- Bulletin of the Australian Mathematical Society , Volume 34 , Issue 2 , October 1986 , pp. 211 - 218
- Copyright
- Copyright © Australian Mathematical Society 1986
References
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