Abstract
is univalent in E, then |ft„| ^«|fti|. This conjecture has been proved in many special cases and has a long history (3). To the best of our knowledge it has not been generalized to the class of ?-valent functions. This is done in §3. In §4 it is shown that the truth of this conjecture would imply a set of trigonometric inequalities, Theorem 3, which are generalizations of the elementary | sin »0/sin d ^n. A proof of these inequalities is given in §5. Conversely it is shown, Theorem 6, that these inequalities have an implication which tends to strengthen the conjecture slightly. Theorem 5 gives a second set of trigonometric inequalities which are generalizations of the trivial |cos nd iSI. Finally in §6 we note that the same methods may be used to obtain bounds for analogous algebraic expressions. This last result, Theorem 7, is not new(4) but the method of proof is different. 2. Precise statements of theorems. A recent result(6) is:
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