On an integral operator for convex univalent functions
1986 ◽
Vol 34
(2)
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pp. 211-218
Keyword(s):
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.
2015 ◽
Vol 1
(2)
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pp. 35-37
1992 ◽
Vol 15
(4)
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pp. 719-726
Keyword(s):
2017 ◽
Vol 148
(2)
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pp. 281-291
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Keyword(s):
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2019 ◽
Vol 113
(3)
◽
pp. 2407-2420
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Keyword(s):
2010 ◽
Vol 5
(3)
◽
pp. 955-966
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