Spherical derivative of meromorphic function with image of finite spherical area
<p/> <p>Let <inline-formula><graphic file="1029-242X-2000-984181-i1.gif"/></inline-formula> be a domain in the complex plane <inline-formula><graphic file="1029-242X-2000-984181-i2.gif"/></inline-formula> with the Poincare metric &l...
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| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2000-01-01
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| Series: | Journal of Inequalities and Applications |
| Subjects: | |
| Online Access: | http://www.journalofinequalitiesandapplications.com/content/5/984181 |
| Summary: | <p/> <p>Let <inline-formula><graphic file="1029-242X-2000-984181-i1.gif"/></inline-formula> be a domain in the complex plane <inline-formula><graphic file="1029-242X-2000-984181-i2.gif"/></inline-formula> with the Poincare metric <inline-formula><graphic file="1029-242X-2000-984181-i3.gif"/></inline-formula> which is <inline-formula><graphic file="1029-242X-2000-984181-i4.gif"/></inline-formula> if <inline-formula><graphic file="1029-242X-2000-984181-i5.gif"/></inline-formula> is the open unit disk. Suppose that the Riemann sphere <inline-formula><graphic file="1029-242X-2000-984181-i6.gif"/></inline-formula> of radius 1/2, so that it has the area <inline-formula><graphic file="1029-242X-2000-984181-i7.gif"/></inline-formula> and let <inline-formula><graphic file="1029-242X-2000-984181-i8.gif"/></inline-formula>. Let <inline-formula><graphic file="1029-242X-2000-984181-i9.gif"/></inline-formula>, <inline-formula><graphic file="1029-242X-2000-984181-i10.gif"/></inline-formula>, be the supremum of the spherical derivative <inline-formula><graphic file="1029-242X-2000-984181-i11.gif"/></inline-formula> of <inline-formula><graphic file="1029-242X-2000-984181-i12.gif"/></inline-formula> meromorphic in <inline-formula><graphic file="1029-242X-2000-984181-i13.gif"/></inline-formula> such that the spherical area of the image <inline-formula><graphic file="1029-242X-2000-984181-i14.gif"/></inline-formula> is not greater than <inline-formula><graphic file="1029-242X-2000-984181-i15.gif"/></inline-formula>. Then <inline-formula><graphic file="1029-242X-2000-984181-i16.gif"/></inline-formula> The equality holds if and only if <inline-formula><graphic file="1029-242X-2000-984181-i17.gif"/></inline-formula> is simply connected.</p> |
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| ISSN: | 1025-5834 1029-242X |