Subordination by convex functions
Let K(α), 0≤α<1, denote the class of functions g(z)=z+Σn=2∞anzn 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)+zf′(z)<g(z)+zg′(z) in...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2000-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.1155/S016117120000140X |
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| Summary: | Let K(α), 0≤α<1, denote the class of functions g(z)=z+Σn=2∞anzn 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)+zf′(z)<g(z)+zg′(z) in U, then (i) f(z)<g(z) at least in |z|<r0,r0=5/3=0.745… if f∈K; and (ii) f(z)<g(z) at least in |z|<r1,r1((51−242)/23)1/2=0.8612… if
g∈K(1/2). |
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| ISSN: | 0161-1712 1687-0425 |