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Majorizations and quasi-subordinations for certain analytic functions. (English) Zbl 0767.30014

This paper contains two theorems about functions analytic in the open unit disk Δ. The first theorem yields a value of r depending on α and β such that the condition |f(z)||g(z)| for |z|<1 implies |f ' (z)||g ' (z)| for |z|r. Here it is assumed that f(z)=a 1 z- n=2 a n z n where a 1 0 and a n 0, h(z)=zg ' (z) g(z)=1- n=1 c n z n where c n 0 and Re{h(z)+αzh ' (z)}>β for |z|<1 (Reα0 and 0β1).

The second theorem asserts that if f(z)=z+ n=2 a n z n is quasi-subordinate to g and Re p g(z) s(z) > 1 2 for |z|<1 where s is a normalized univalent function, then |a n |(p+n)! (p+1)!(n-1)! for n2. Here p1, and the function f(z)=z (1-z) p+2 exhibits sharpness for the coefficient estimate.

30C45Special classes of univalent and multivalent functions
30C80Maximum principle; Schwarz’s lemma, Lindelöf principle, etc. (one complex variable)
30C50Coefficient problems for univalent and multivalent functions