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On an integral operator for convex univalent functions

Published online by Cambridge University Press:  17 April 2009

Vinod Kumar
Affiliation:
Department of Mathematics, Christ Church College, Kanpur-208001, India.
S. L. Shukla
Affiliation:
Department of Mathematics, Janta College, Bakewar-206124, Etawah, India.
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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 |m1| < Mm. 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| ≤ (1b)/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−|m1|)/(M + |m1|), or a complex number satisfying |α| ≤ φ. In both cases the bounds for α are sharp.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1986

References

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