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Typically-Real Functions
Published online by Cambridge University Press: 20 November 2018
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A function
an real, is called typically-real of order one in the closed region ∣z∣ ≦ R if it satisfies the following conditions (6).
(1) f(z) is regular in ∣z∣ ≦ R
(2) ∮{f(z)} > 0 if and only if ∮{z} >0.
The same function is called typically-real of order p, p a positive integer greater than one, if it satisfies condition (1) above and in addition the followingcondition (4; 5):
(2’) there exists a constant ρ, 0 < p < R, such that on every circle ∣z∣ = r, ρ, 0 < p < R, ∮{f(z)} changes sign exactly 2p times.
- Type
- Research Article
- Information
- Copyright
- Copyright © Canadian Mathematical Society 1959
References
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Brown, R. K., Some mapping properties of the mean sums of power series and a problem in typically-real functions of order p, Ph.D. thesis (Rutgers University, 1952).Google Scholar
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Goodman, A. W. and Robertson, M. S., A Class of Multivalent Functions, Trans. Amer. Math. Soc, 70 (1951), 127-36.Google Scholar
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Robertson, M. S.
The variation of the sign of V for an analytic function U + iV, Duke Math. ]., 5 (1939), 512-19.Google Scholar
6.
Rogosinski, W., Ueber positive harmonische Entwicklungen und typisch-reelle Potenzreihen
Math. Zeit., 35 (1932), 73-5.Google Scholar
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