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Typically-Real Functions

Published online by Cambridge University Press:  20 November 2018

Richard K. Brown*
Affiliation:
Rutgers University
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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
Copyright
Copyright © Canadian Mathematical Society 1959

References

1. 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
2. Goodman, A. W., Typically real functions with assigned zeros, Proc. Amer. Math. Soc, 2 (1951), 349–57.Google Scholar
3. Goodman, A. W. and Robertson, M. S., A Class of Multivalent Functions, Trans. Amer. Math. Soc, 70 (1951), 127-36.Google Scholar
4. Robertson, M. S., A representation of all analytic functions in terms of functions with positive real parts, Ann. Math., 88 (1937), 770-83.Google Scholar
5. 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