and the Bieberbach conjecture

Theorem 2.1. f ∈S if and only if the corresponding Grunsky coefficients satisfy the inequalities

Theorem 2.2 (Sufficient condition). A function g(z) holomorphic on the open disc D will be univalent if its Schwarzian derivative satisfies the inequality |{g(z); z}| ≤ 2 (1 −|z|2)2 . (2.16) 7

Theorem 2.3 (Necessary condition). If a holomorphic function g(z) is univalent in the open disc D then |{g(z); z}| ≤ 6 (1 −|z|2)2 . (2.17) 2.3 Koebe growth theorem A very important theorem that shines in our analysis of scattering amplitude is the Koebe Growth Theorem. This theorem, essentially, puts upper and lower bound on the magnitude of a schlicht function f ∈S.

Theorem 2.4. If f ∈S and |z| < 1, then |z| (1 + |z|)2 ≤|f(z)| ≤ |z| (1 −|z|)2 . (2.18) One of the equalities holds at some point z ̸= 0 if and only if f is a rotation of the Koebe function. We want to emphasize that the bounds are the consequence of f being univalent. Thus, the con- verse of the theorem need not be true, i.e. a function defined on the unit disc D with the normalization same as that of a schlicht function satisfying any one of the four bounding relations above need not be univalen

Theorem 2.5. Let f be an arbitrary schlicht function, f ∈S, with the power series representation defined by (2.1). Then the Bieberbach conjecture holds true, i.e. |bn| ≤n, ∀n ≥2; (2.21) with the equality holding if and only if f is a rotation of the Koebe function k(z) defined in (2.5), i.e. if and only if f(z) = e−iθk(eiθz), ∀θ ∈R. (2.22) While de Branges proved the Bieberbach conjecture in its full generality only in 1985, various special cases have been proved earlier. One particular case relev

Theorem 2.6 (Szeg¨o theorem). Define the numbers {rn ∈R+} such that the mth section of a schlicht function f ∈S, fm, is univalent in the disc Drn for all m ≥n. Then, r1 = 1 4, (2.24) i.e., each section remains univalent in the disc D1/4, and the number 1/4 can’t be replaced by a higher one. The statement of the number 1/4 not being replaceable by a higher number needs some explana- tion. Consider an arbitrary f ∈S, and let the domain over which the nth section fn is univalent be Dfn. Then the abo

Lemma 3.1 (Positivity lemma). For a unitary theory, if a ∈  −2µ 9 , 2µ 3  then the absorptive part of the amplitude, A  s1; s(+) 2 (s1, a)  , is non-negative for s1 ∈ h

Lemma 4.1. Consider the kernel H(˜z; s1, a) of the dispersion relation given by (3.4), H(˜z; s1, a) = 27a2˜z (2s1 −3a) 27a3˜z −27a2˜zs1 −(˜z −1)2 (s1)3 . (4.1) Define the function F(˜z; s1, a) := H(˜z; s1, a) β1(a, s1) . (4.2) 11For the string case, however, we will find that for a < 0 the bounds we will consider will still hold. We do not have a general explanation for this apart from observing that α1 < 0 for certain −1/3 < a < 2/3, which is the range of a we will be interested in. 13

Corollary 4.1.1. For a ∈  −2µ 9 , 0  ∪  0, 4µ 9  and s1 ∈ h 2µ 3 , ∞ 

Corollary 4.1.2. For a ∈  −2µ 9 , 0  ∪  0, 4µ 9  and s1 ∈ h 2µ 3 , ∞ 

Theorem 4.2. For non-zero M(˜z, a) and a ∈  −2µ 9 , 0  ∪  0, 4µ 9  , with µ > 0,

Theorem 6.1. Let M(˜z, a) be a unitary and crossing-symmetric scattering amplitude admitting the dispersive representation (3.3) and admits the power series expansion (3.5) about ˜z = 0 which con- verges in the open disc |˜z| < 1. Define the function f(˜z, a) := M(˜z, a) −α0 α1(a)a2 , α0 = M(˜z = 0, a). (6.1) Then for a ∈  −2µ 9 , 0  ∪