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Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Convexity properties of the Gamma function #

In this file, we prove that Gamma and log ∘ Gamma are convex functions on the positive real line. We then prove the Bohr-Mollerup theorem, which characterises Gamma as the unique positive-real-valued, log-convex function on the positive reals satisfying f (x + 1) = x f x and f 1 = 1.

The proof of the Bohr-Mollerup theorem is bound up with the proof of (a weak form of) the Euler limit formula, Real.BohrMollerup.tendsto_logGammaSeq, stating that for positive real x the sequence x * log n + log n! - ∑ (m : ℕ) in Finset.range (n + 1), log (x + m) tends to log Γ(x) as n → ∞. We prove that any function satisfying the hypotheses of the Bohr-Mollerup theorem must agree with the limit in the Euler limit formula, so there is at most one such function; then we show that Γ satisfies these conditions.

Since most of the auxiliary lemmas for the Bohr-Mollerup theorem are of no relevance outside the context of this proof, we place them in a separate namespace Real.BohrMollerup to avoid clutter. (This includes the logarithmic form of the Euler limit formula, since later we will prove a more general form of the Euler limit formula valid for any real or complex x; see Real.Gamma_seq_tendsto_Gamma and Complex.Gamma_seq_tendsto_Gamma in the file Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean.)

As an application of the Bohr-Mollerup theorem we prove the Legendre doubling formula for the Gamma function for real positive s (which will be upgraded to a proof for all complex s in a later file).

TODO: This argument can be extended to prove the general k-multiplication formula (at least up to a constant, and it should be possible to deduce the value of this constant using Stirling's formula).

theorem ConvexOn.congr {𝕜 : Type u_1} {E : Type u_2} {β : Type u_3} {s : Set E} {f : Eβ} {g : Eβ} [OrderedSemiring 𝕜] [SMul 𝕜 E] [AddCommMonoid E] [OrderedAddCommMonoid β] [SMul 𝕜 β] (hf : ConvexOn 𝕜 s f) (hfg : Set.EqOn f g s) :
ConvexOn 𝕜 s g
theorem ConcaveOn.congr {𝕜 : Type u_1} {E : Type u_2} {β : Type u_3} {s : Set E} {f : Eβ} {g : Eβ} [OrderedSemiring 𝕜] [SMul 𝕜 E] [AddCommMonoid E] [OrderedAddCommMonoid β] [SMul 𝕜 β] (hf : ConcaveOn 𝕜 s f) (hfg : Set.EqOn f g s) :
ConcaveOn 𝕜 s g
theorem StrictConvexOn.congr {𝕜 : Type u_1} {E : Type u_2} {β : Type u_3} {s : Set E} {f : Eβ} {g : Eβ} [OrderedSemiring 𝕜] [SMul 𝕜 E] [AddCommMonoid E] [OrderedAddCommMonoid β] [SMul 𝕜 β] (hf : StrictConvexOn 𝕜 s f) (hfg : Set.EqOn f g s) :
theorem StrictConcaveOn.congr {𝕜 : Type u_1} {E : Type u_2} {β : Type u_3} {s : Set E} {f : Eβ} {g : Eβ} [OrderedSemiring 𝕜] [SMul 𝕜 E] [AddCommMonoid E] [OrderedAddCommMonoid β] [SMul 𝕜 β] (hf : StrictConcaveOn 𝕜 s f) (hfg : Set.EqOn f g s) :
theorem ConvexOn.add_const {𝕜 : Type u_1} {E : Type u_2} {β : Type u_3} {s : Set E} {f : Eβ} [OrderedSemiring 𝕜] [SMul 𝕜 E] [AddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 β] (hf : ConvexOn 𝕜 s f) (b : β) :
ConvexOn 𝕜 s (f + fun (x : E) => b)
theorem ConcaveOn.add_const {𝕜 : Type u_1} {E : Type u_2} {β : Type u_3} {s : Set E} {f : Eβ} [OrderedSemiring 𝕜] [SMul 𝕜 E] [AddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 β] (hf : ConcaveOn 𝕜 s f) (b : β) :
ConcaveOn 𝕜 s (f + fun (x : E) => b)
theorem StrictConvexOn.add_const {𝕜 : Type u_1} {E : Type u_2} {s : Set E} [OrderedSemiring 𝕜] [SMul 𝕜 E] [AddCommMonoid E] {γ : Type u_4} {f : Eγ} [OrderedCancelAddCommMonoid γ] [Module 𝕜 γ] (hf : StrictConvexOn 𝕜 s f) (b : γ) :
StrictConvexOn 𝕜 s (f + fun (x : E) => b)
theorem StrictConcaveOn.add_const {𝕜 : Type u_1} {E : Type u_2} {s : Set E} [OrderedSemiring 𝕜] [SMul 𝕜 E] [AddCommMonoid E] {γ : Type u_4} {f : Eγ} [OrderedCancelAddCommMonoid γ] [Module 𝕜 γ] (hf : StrictConcaveOn 𝕜 s f) (b : γ) :
StrictConcaveOn 𝕜 s (f + fun (x : E) => b)
theorem Real.Gamma_mul_add_mul_le_rpow_Gamma_mul_rpow_Gamma {s : } {t : } {a : } {b : } (hs : 0 < s) (ht : 0 < t) (ha : 0 < a) (hb : 0 < b) (hab : a + b = 1) :
Real.Gamma (a * s + b * t) Real.Gamma s ^ a * Real.Gamma t ^ b

Log-convexity of the Gamma function on the positive reals (stated in multiplicative form), proved using the Hölder inequality applied to Euler's integral.

The function n ↦ x log n + log n! - (log x + ... + log (x + n)), which we will show tends to log (Gamma x) as n → ∞.

Equations
Instances For
    theorem Real.BohrMollerup.f_nat_eq {f : } {n : } (hf_feq : ∀ {y : }, 0 < yf (y + 1) = f y + Real.log y) (hn : n 0) :
    f n = f 1 + Real.log (Nat.factorial (n - 1))
    theorem Real.BohrMollerup.f_add_nat_eq {f : } {x : } (hf_feq : ∀ {y : }, 0 < yf (y + 1) = f y + Real.log y) (hx : 0 < x) (n : ) :
    f (x + n) = f x + Finset.sum (Finset.range n) fun (m : ) => Real.log (x + m)
    theorem Real.BohrMollerup.f_add_nat_le {f : } {x : } {n : } (hf_conv : ConvexOn (Set.Ioi 0) f) (hf_feq : ∀ {y : }, 0 < yf (y + 1) = f y + Real.log y) (hn : n 0) (hx : 0 < x) (hx' : x 1) :
    f (n + x) f n + x * Real.log n

    Linear upper bound for f (x + n) on unit interval

    theorem Real.BohrMollerup.f_add_nat_ge {f : } {x : } {n : } (hf_conv : ConvexOn (Set.Ioi 0) f) (hf_feq : ∀ {y : }, 0 < yf (y + 1) = f y + Real.log y) (hn : 2 n) (hx : 0 < x) :
    f n + x * Real.log (n - 1) f (n + x)

    Linear lower bound for f (x + n) on unit interval

    theorem Real.BohrMollerup.le_logGammaSeq {f : } {x : } (hf_conv : ConvexOn (Set.Ioi 0) f) (hf_feq : ∀ {y : }, 0 < yf (y + 1) = f y + Real.log y) (hx : 0 < x) (hx' : x 1) (n : ) :
    f x f 1 + x * Real.log (n + 1) - x * Real.log n + Real.BohrMollerup.logGammaSeq x n
    theorem Real.BohrMollerup.ge_logGammaSeq {f : } {x : } {n : } (hf_conv : ConvexOn (Set.Ioi 0) f) (hf_feq : ∀ {y : }, 0 < yf (y + 1) = f y + Real.log y) (hx : 0 < x) (hn : n 0) :
    theorem Real.BohrMollerup.tendsto_logGammaSeq_of_le_one {f : } {x : } (hf_conv : ConvexOn (Set.Ioi 0) f) (hf_feq : ∀ {y : }, 0 < yf (y + 1) = f y + Real.log y) (hx : 0 < x) (hx' : x 1) :
    Filter.Tendsto (Real.BohrMollerup.logGammaSeq x) Filter.atTop (nhds (f x - f 1))
    theorem Real.BohrMollerup.tendsto_logGammaSeq {f : } {x : } (hf_conv : ConvexOn (Set.Ioi 0) f) (hf_feq : ∀ {y : }, 0 < yf (y + 1) = f y + Real.log y) (hx : 0 < x) :
    Filter.Tendsto (Real.BohrMollerup.logGammaSeq x) Filter.atTop (nhds (f x - f 1))
    theorem Real.eq_Gamma_of_log_convex {f : } (hf_conv : ConvexOn (Set.Ioi 0) (Real.log f)) (hf_feq : ∀ {y : }, 0 < yf (y + 1) = y * f y) (hf_pos : ∀ {y : }, 0 < y0 < f y) (hf_one : f 1 = 1) :

    The Bohr-Mollerup theorem: the Gamma function is the unique log-convex, positive-valued function on the positive reals which satisfies f 1 = 1 and f (x + 1) = x * f x for all x.

    The Gamma doubling formula #

    As a fun application of the Bohr-Mollerup theorem, we prove the Gamma-function doubling formula (for positive real s). The idea is that 2 ^ s * Gamma (s / 2) * Gamma (s / 2 + 1 / 2) is log-convex and satisfies the Gamma functional equation, so it must actually be a constant multiple of Gamma, and we can compute the constant by specialising at s = 1.

    Auxiliary definition for the doubling formula (we'll show this is equal to Gamma s)

    Equations
    Instances For
      theorem Real.Gamma_mul_Gamma_add_half_of_pos {s : } (hs : 0 < s) :
      Real.Gamma s * Real.Gamma (s + 1 / 2) = Real.Gamma (2 * s) * 2 ^ (1 - 2 * s) * Real.sqrt Real.pi

      Legendre's doubling formula for the Gamma function, for positive real arguments. Note that we shall later prove this for all s as Real.Gamma_mul_Gamma_add_half (superseding this result) but this result is needed as an intermediate step.