Subordination by convex functions

Ram Singh () and Sukhjit Singh

International Journal of Mathematics and Mathematical Sciences, 2000, vol. 24, 1-6

Abstract:

Let K ( α ) , 0 ≤ α < 1 , denote the class of functions g ( z ) = z + Σ n = 2 ∞ a n z n which are regular and univalently convex of order α in the unit disc U . Pursuing the problem initiated by Robinson in the present paper, among other things, we prove that if f is regular in U , f ( 0 ) = 0 , and f ( z ) + z f ′ ( z ) < g ( z ) + z g ′ ( z ) in U , then (i) f ( z ) < g ( z ) at least in | z | < r 0 , r 0 = 5 / 3 = 0.745 … if f ∈ K ; and (ii) f ( z ) < g ( z ) at least in | z | < r 1 , r 1 ( ( 51 − 24 2 ) / 23 ) 1 / 2 = 0.8612 … if g ∈ K ( 1 / 2 ) .

Date: 2000
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:370941

DOI: 10.1155/S016117120000140X

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