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Majorization-Subordination Theorems for Locally Univalent Functions, II

  • Douglas Michael Campbell (a1)

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Let denote the set of all normalized analytic univalent functions in the open unit disc D. Let f(z), F(z) and φ(z) be analytic in |z| < r. We say that f(z) is majorized by F(z) in we say that f(z) is subordinate to F(z) in where .

Let be the set of all locally univalent (f’(z) ≠ 0) analytic functions in D with order ≦α which are of the form f(z) = z +… . The family is known as the universal linear invariant family of order α [6]. A concise summary of and introduction to properties of linear invariant families which relate to the following material is contained in [1]. The present paper contains the proofs of some of the results announced in [1]

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References

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1. Campbell, D. M., Majorization-subordination theorems for locally univalent functions, Bull. Amer. Math. Soc. 78 (1972), 535538.
2. Lewandowski, Z., Sur les majorants des fonctions holomorphes dans le cercle \z\ < 1, Ann. Univ. Mariae Curie-Sklodowska Sect A 15 (1961), 511.
3. Littlewood, J. E., Lectures on the theory of functions (Oxford University Press, Oxford, 1944).
4. MacGregor, T. H., Majorization by univalent functions, Duke Math. J. 34 (1967), 95102.
5. Nehari, Z., Conformai mapping (McGraw-Hill, New York, 1952).
6. Pommerenke, Ch., Linear-invariante Familien Analytischer Funktionen. J, Math. Ann. 155 (1964), 108154.
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Majorization-Subordination Theorems for Locally Univalent Functions, II

  • Douglas Michael Campbell (a1)

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