On the multivalence of a class of meromorphic functions

Theorem 44). THEOREM A. Let φ{z) be a regular function defined in a convex domain D in the complex plane and let a be an arbitrary constant. Furthermore suppose that (1) For zeD, φ {p)(z)e A, where A is a convex domain; (2) There exists a polar set E (a set of the form Θι ^ arg z ^ θ21

THEOREM 1. The function g(z) defined by (2) is univalent in the disk This estimate is sharp for a = 1 and for all a the best estimate of the radius of the disk about the origin for which all functions of type g(z) are univalent does not exceed the number

THEOREM 2. Let g{z) be a function as defined by (2). Then g(z) is completely p-valent in the disk \z\ <rp where rp is the unique positive root (<1) of the equation

THEOREM 3. Let f{z) and g(z) be as defined by (3) and (2) respec- tively and let conditions (6) be satisfied. Then (a) The function f(z) is univalent in the region | z | rx(l) where rJJ) is the largest positive root of the equation χU+2 _ r2φχ2l _ 1 = 0 and where

Theorem 3(b) are completely p-valent. It seems, however, that these bounds are not sharp for p > 1. The question of the exact bound remains open even in the apparently simpler case of polynomials which we shall take up in the next section. We conclude this section with one estimate on the number R such that the image of \z\ — R by functions of type (1) is starlike with respect to the origin. For simplicity we shall assume that a = 1.

THEOREM 4. Let f{z) be a function defined by (1). Then if R > (27/ll) 1/2 then the image of \z\ = R is starlike with respect to the origin.

THEOREM 5. Suppose that the polynomial f{z) = z p(l + α / + + α»_pS—*) απ_p 7^0, 1 ^ ^ < ^ , l ^ r ^ w — p, does %o£ vanish in the unit disk except for a p-fold zero at the origin. Then f(z) is necessarily p-valent in the disk Furthermore the polynomials. for

THEOREM 6. Suppose that the polynomial f(z) = aqz q + + anz n , aq Φ 0, 1 ^ q < n, vanishes exactly p times (counting multiplicities) in the disk | z | < 1 and assume, furthermore, that the (n — p) zeros of f(z), which lie outside the unit disk, say zίy "*,zn-p possess the symmetry property (11)

Theorem 12', p. 46.

Theorem 6) A = A' and furthermore starting with the basic inequality R 2: A' + r(s + 1) . Then we can set B[ = A' + 2α0 ^ ZA' R'2 = A' + 3R[ ^ 4R[ ^ SΛA' < 3.4.5A' etc. This leads to the inequality Thus /(») is p-valent in the disk 2 < + 1)! where A' - Max {(n - p) llk,

THEOREM 7. If the polynomial of degree n f(z) = aLz + α2z 2 + + an-qz n~ q + anz n has (n — 1) zeros in the unit disk, then f(z) is at most (n — T)-valent in the disk \z\ < x(q) where x{q) is as defined in Theorem D.