Iterating the Minimum Modulus: Functions of Order Half, Minimal Type
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AbstractFor a transcendental entire function f, the property that there exists $$r>0$$ r > 0 such that $$m^n(r)\rightarrow \infty $$ m n ( r ) → ∞ as $$n\rightarrow \infty $$ n → ∞ , where $$m(r)=\min \{|f(z)|:|z|=r\}$$ m ( r ) = min { | f ( z ) | : | z | = r } , is related to conjectures of Eremenko and of Baker, for both of which order 1/2 minimal type is a significant rate of growth. We show that this property holds for functions of order 1/2 minimal type if the maximum modulus of f has sufficiently regular growth and we give examples to show the sharpness of our results by using a recent generalisation of Kjellberg’s method of constructing entire functions of small growth, which allows rather precise control of m(r).
1993 ◽
Vol 114
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pp. 43-55
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2001 ◽
Vol 63
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pp. 367-377
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1999 ◽
Vol 19
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pp. 1281-1293
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2000 ◽
Vol 20
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pp. 1577-1582
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2018 ◽
Vol 40
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pp. 89-116
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2016 ◽
Vol 37
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pp. 1291-1307
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1974 ◽
Vol 10
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pp. 67-70
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