GoodmanConjecture.lean`

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7 declarations, 1 contain sorry, 0 declared as axioms

Goodman's conjecture on coefficients of $p$-valent functions

A regular (analytic) function is $p$-valent on the unit disk if it assumes each value at most $p$ times there, and some value exactly $p$ times. For such a function $f(z) = \sum_{n \ge 1} b_n z^n$ (so $f(0) = 0$), Goodman's conjecture (1948) asserts that its coefficients satisfy $$|b_n| \le \sum_{k=1}^{p} \frac{2k (n+p)!}{(p-k)!\,(p+k)!\,(n-p-1)!\,(n^2-k^2)} |b_k| \qquad (n > p).$$ This generalises the classical coefficient bounds for univalent functions (the case $p = 1$, where the bound reads $|b_n| \le n\,|b_1|$, i.e. $|b_n| \le n$ under the classical normalisation $b_1 = 1$) to $p$-valent functions. It has been verified in several special cases, including $p$-valent typically-real functions.

References:

Wikipedia
Goodman, A. W., On some determinants related to $p$-valent functions, Trans. Amer. Math. Soc. 63 (1948), 175–192.

Imports (1)

FormalConjecturesUtil

structure GoodmanConjecture.IsPValent Documented declaration: IsPValent line 52 source
Source documentation
A function f : ℂ → ℂ is $p$-valent on the open unit disk $\mathbb{D} = \{z : |z| < 1\}$ if it is analytic there, attains every value at most $p$ times, and attains some value exactly $p$ times. Here $p \ge 1$. Values are counted at distinct points. For a non-constant analytic function this agrees with counting with multiplicity: near a point where $f - w$ has a zero of order $m$, every value close to $w$ is attained at $m$ distinct points, so the maximal number of distinct preimages equals the maximal number of preimages counted with multiplicity.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Complex, Nat
```
structure IsPValent (f : ℂ → ℂ) (p : ℕ) : Prop where
…
```
theorem GoodmanConjecture.isPValent_id_one The identity is $1$-valent (univalent) on the unit disk line 62 source
Source documentation
The identity is $1$-valent (univalent) on the unit disk.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: GoodmanConjecture.IsPValent, id, Complex, OfNat.ofNat, Nat
Proof uses: GoodmanConjecture.IsPValent.mk, le_rfl, AnalyticOnNhd.analyticOn, Metric.ball, Real, analyticOnNhd_id, Trans.trans, ENat, LE.le, Eq, Set.encard, Set.ofPred, And, Membership.mem, Set, Singleton.singleton, Set.encard_le_encard, Eq.mpr, And.right, of_eq_true and 19 more
50 instances, coercions and auxiliary constants left out.

Tactic trace: 14 steps
Open to load the checked before-and-after goal states.
```
theorem isPValent_id_one : IsPValent id 1 where
…
```
Mentions: GoodmanConjecture.IsPValent
def GoodmanConjecture.coeff The $n$-th Taylor coefficient of $f$ at $0$, i.e. $b_n = f^{(n)}(0) / n!$ line 78 source
Source documentation
The $n$-th Taylor coefficient of $f$ at $0$, i.e. $b_n = f^{(n)}(0) / n!$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Complex, Nat
Proof uses: HDiv.hDiv, iteratedDeriv, OfNat.ofNat, Nat.factorial
14 instances, coercions and auxiliary constants left out.
```
noncomputable def coeff (f : ℂ → ℂ) (n : ℕ) : ℂ := iteratedDeriv n f 0 / n.factorial
```
structure GoodmanConjecture.IsNormalized Goodman's normalisation $f(z) = \sum_{n \ge 1} b_n z^n$, i.e. $f(0) = 0$ line 81 source
Source documentation
Goodman's normalisation $f(z) = \sum_{n \ge 1} b_n z^n$, i.e. $f(0) = 0$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Complex
```
structure IsNormalized (f : ℂ → ℂ) : Prop where
…
```
def GoodmanConjecture.goodmanBound The Goodman bound $$\sum_{k=1}^{p} \frac{2k (n+p)!}{(p-k)!\,(p+k)!\,(n-p-1)!\,(n^2-k^2)} |b_k|,$$ the conjectured upper bound for $|b_n|$ line 87 source
Source documentation
The Goodman bound $$\sum_{k=1}^{p} \frac{2k (n+p)!}{(p-k)!\,(p+k)!\,(n-p-1)!\,(n^2-k^2)} |b_k|,$$ the conjectured upper bound for $|b_n|$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Complex, Nat, Real
Proof uses: Finset.sum, Finset.Icc, OfNat.ofNat, HMul.hMul, HDiv.hDiv, Nat.factorial, HAdd.hAdd, HSub.hSub, HPow.hPow, Norm.norm, GoodmanConjecture.coeff
23 instances, coercions and auxiliary constants left out.
```
noncomputable def goodmanBound (f : ℂ → ℂ) (p n : ℕ) : ℝ :=
…
```
theorem GoodmanConjecture.goodman_conjecture Goodman's conjecture sorry line 99 source
Source documentation
Goodman's conjecture. For every $p$-valent normalised function $f$ on the unit disk and every $n > p$, the $n$-th coefficient is bounded by the Goodman bound: $$|b_n| \le \sum_{k=1}^{p} \frac{2k (n+p)!}{(p-k)!\,(p+k)!\,(n-p-1)!\,(n^2-k^2)} |b_k|.$$

depends on sorry what this means
Statement uses: Complex, Nat, GoodmanConjecture.IsPValent, GoodmanConjecture.IsNormalized, LT.lt, LE.le, Real, Norm.norm, GoodmanConjecture.coeff, GoodmanConjecture.goodmanBound
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
7 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem goodman_conjecture (f : ℂ → ℂ) (p n : ℕ) (hf : IsPValent f p)
    (hf' : IsNormalized f) (hn : p < n) :
    ‖coeff f n‖ ≤ goodmanBound f p n := by
…
```
Mentions: GoodmanConjecture.IsPValent, GoodmanConjecture.IsNormalized, GoodmanConjecture.coeff, GoodmanConjecture.goodmanBound
theorem GoodmanConjecture.goodmanBound_one Sanity check: in the univalent case $p = 1$, the Goodman bound reduces to the classical Bieberbach-type bound $|b_n| \le n\,|b_1|$ line 111 source
Source documentation
Sanity check: in the univalent case $p = 1$, the Goodman bound reduces to the classical Bieberbach-type bound $|b_n| \le n\,|b_1|$. Concretely the single $k = 1$ summand is $$\frac{2 (n+1)!}{0!\,2!\,(n-2)!\,(n^2-1)} = \frac{(n+1)!}{(n-2)!\,(n^2-1)} = n,$$ so the bound equals $n\,|b_1|$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Complex, Nat, LT.lt, OfNat.ofNat, Eq, Real, GoodmanConjecture.goodmanBound, HMul.hMul, Norm.norm, GoodmanConjecture.coeff
Proof uses: HAdd.hAdd, Exists, Exists.intro, HSub.hSub, Decidable.byContradiction, Not, Eq.mpr, Finset.sum, Finset.Icc, HDiv.hDiv, Nat.factorial, HPow.hPow, id, congrArg, GoodmanConjecture.goodmanBound.eq_1, Singleton.singleton, Finset, Finset.Icc_self, Finset.sum_singleton, HMul and 120 more
124 instances, coercions and auxiliary constants left out.

Tactic trace: 29 steps
Open to load the checked before-and-after goal states.
```
theorem goodmanBound_one (f : ℂ → ℂ) (n : ℕ) (hn : 1 < n) :
    goodmanBound f 1 n = n * ‖coeff f 1‖ := by
…
```
Mentions: GoodmanConjecture.goodmanBound, GoodmanConjecture.coeff