Some results on convex meromorphic functions

In this paper, we define a function F : D×D×D → C in terms of f and show that ReF > 0 for all ζ, z, w ∈ D if and only if f belongs to the class of convex meromorphic functions.


Introduction and preliminaries.
Let us denote by S(p) with 0 < p < 1 the set of univalent functions in the unit disk D = {z ∈ C : |z| < 1} such that f (0) = 0, f (0) = 1 and f (p) = ∞. We denote by K the subset of functions in S(p) which omits a convex set in the extended plane C = C ∪ {∞}, that is, f ∈ K if and only if the set C \ f (D) = {w ∈ C : f (z) = w} is convex. Functions in K are called convex meromorphic functions. Many people have worked on convex holomorphic functions and the results obtained have already found their place in many books; see, for examle Ruscheweyh and Sheil-Small [6], Sheil-Small [8], Schober [7] and Duren [1]. So far, several works on convex meromorphic functions have appeared in the literature; for further reading see Ohno [5] and the references therein including Yulin and Owa [9]. It should be remarked that the functions that omit a convex set are called concave functions, nowadays. The set K we described above is a subset of concave functions that belong to the set S(p). Therefore, we prefer to call the functions in K convex meromorphic functions. In this paper, we also consider these functions and we think our results in a way unify the earlier results. That is, as Duren [1, p. 250] comments "We shall digress briefly to establish some global properties of convex functions. Everything is a consequence of the following proposition ..."; many earlier results will follow from ours. However, a slight modification in the statement and the proof of our Theorem 1 below yields that it is true for all concave functions. To see this, it is enough to assume p ∈ D and replace the factor 1−pz 1−pζ by Then, F has a positive real part with F (0, z, w) = 1.
Proof. Without loss of generality, we can assume that the function f can be extended to the boundary as a continuous function. The function F (ζ, z, w) has a holomorphic extension to D 3 , that is, all the singularities of F are removable. To see this, it is enough to observe the following three evaluations: Now, we consider the function for |z| = |ζ| = 1. We set z = αw and ζ = βw in (2.2), where α = e ia and β = e ib , 0 < a, b < 2π are distinct constants. Thus we have is purely imaginary and and thus where f (w)−f (e ia w) f (w)−f (e ib w) = Re iϕ for |w| = 1. Note that for a < b, the images of w, e ia w and e ib w under f have the same order on ∂F (D) in positive direction. Therefore in this case ϕ ∈ (0, π). Similarly, ϕ ∈ (π, 2π) for b < a. It follows that Re {F (βw, αw, w)} > 0 on (∂D) 3 . Furthermore, it is a consequence of the Cauchy integral formula in a polydisk that the absolute value of the holomorphic function e −F (ζ,z,w) attains its maximum on the distinguished boundary (∂D) 3 of the polydisk D 3 (see, for example, Gunning [3] Theorem 4, p. 6 or Hörmander [4] Section 2.2, p. 25). It follows that Re {F (ζ, z, w)} > 0 throughout the polydisk D × D × D. Thus we get the desired result. Obviously F (0, z, w) = 1.
Now we can prove that the converse of our theorem is also true: has a positive real part, then f ∈ K.
Proof. Observe that so for z → ζ, we have 0 0 and applying L'Hospital's rule, we find .

A set of useful corollaries.
As we pointed out above, the function Dividing both sides by z and then integrating both sides with respect to z, we find h(z, w) = z/H 2 (z, w). Proof. It follows from Corollary 5 that there exists a probability measure µ on ∂D such that Subtracting 1 from both sides and dividing by −2z, we obtain Integrating both sides with respect to z and noting that the integral constant is zero, we arrive at log H(z, w) = |η|=1 log(1 − ηz)dµ.

Conclusions.
In this paper we focused on the main results and their implications that are listed in a series of corollaries. Of course, each of these results has implications about the coefficients of the convex meromorphic functions. For example, the absolute values of the coefficients of the function H(z, w) = 1 + c 1 (w)z + c 2 (w)z 2 + . . . are bounded by 1, i.e., Here we note that many coefficient inequalities for concave functions can be found in the references. We think that applications of our results to coefficient inequalities will be the subject of another paper.