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analysis.special_functions.gamma.bohr_mollerup

Convexity properties of the Gamma function #

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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.bohr_mollerup.tendsto_log_gamma_seq, 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.bohr_mollerup 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 analysis.special_functions.gamma.beta.)

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 convex_on.congr {𝕜 : Type u_1} {E : Type u_2} {β : Type u_3} {s : set E} {f g : E β} [ E] [ β] (hf : s f) (hfg : g s) :
s g
theorem concave_on.congr {𝕜 : Type u_1} {E : Type u_2} {β : Type u_3} {s : set E} {f g : E β} [ E] [ β] (hf : s f) (hfg : g s) :
s g
theorem strict_convex_on.congr {𝕜 : Type u_1} {E : Type u_2} {β : Type u_3} {s : set E} {f g : E β} [ E] [ β] (hf : f) (hfg : g s) :
g
theorem strict_concave_on.congr {𝕜 : Type u_1} {E : Type u_2} {β : Type u_3} {s : set E} {f g : E β} [ E] [ β] (hf : f) (hfg : g s) :
g
theorem convex_on.add_const {𝕜 : Type u_1} {E : Type u_2} {β : Type u_3} {s : set E} {f : E β} [ E] [ β] (hf : s f) (b : β) :
s (f + λ (_x : E), b)
theorem concave_on.add_const {𝕜 : Type u_1} {E : Type u_2} {β : Type u_3} {s : set E} {f : E β} [ E] [ β] (hf : s f) (b : β) :
s (f + λ (_x : E), b)
theorem strict_convex_on.add_const {𝕜 : Type u_1} {E : Type u_2} {s : set E} [ E] {γ : Type u_3} {f : E γ} [ γ] (hf : f) (b : γ) :
(f + λ (_x : E), b)
theorem strict_concave_on.add_const {𝕜 : Type u_1} {E : Type u_2} {s : set E} [ E] {γ : Type u_3} {f : E γ} [ γ] (hf : f) (b : γ) :
(f + λ (_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) ^ a * ^ 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.

noncomputable def real.bohr_mollerup.log_gamma_seq (x : ) (n : ) :

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
theorem real.bohr_mollerup.f_nat_eq {f : } {n : } (hf_feq : {y : }, 0 < y f (y + 1) = f y + ) (hn : n 0) :
f n = f 1 + real.log ((n - 1).factorial)
theorem real.bohr_mollerup.f_add_nat_eq {f : } {x : } (hf_feq : {y : }, 0 < y f (y + 1) = f y + ) (hx : 0 < x) (n : ) :
f (x + n) = f x + (finset.range n).sum (λ (m : ), real.log (x + m))
theorem real.bohr_mollerup.f_add_nat_le {f : } {x : } {n : } (hf_conv : (set.Ioi 0) f) (hf_feq : {y : }, 0 < y f (y + 1) = f y + ) (hn : n 0) (hx : 0 < x) (hx' : x 1) :
f (n + x) f n + x *

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

theorem real.bohr_mollerup.f_add_nat_ge {f : } {x : } {n : } (hf_conv : (set.Ioi 0) f) (hf_feq : {y : }, 0 < y f (y + 1) = f 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.bohr_mollerup.log_gamma_seq_add_one (x : ) (n : ) :
= + - (x + 1) * (real.log (n + 1) -
theorem real.bohr_mollerup.le_log_gamma_seq {f : } {x : } (hf_conv : (set.Ioi 0) f) (hf_feq : {y : }, 0 < y f (y + 1) = f y + ) (hx : 0 < x) (hx' : x 1) (n : ) :
f x f 1 + x * real.log (n + 1) - x * +
theorem real.bohr_mollerup.ge_log_gamma_seq {f : } {x : } {n : } (hf_conv : (set.Ioi 0) f) (hf_feq : {y : }, 0 < y f (y + 1) = f y + ) (hx : 0 < x) (hn : n 0) :
f x
theorem real.bohr_mollerup.tendsto_log_gamma_seq_of_le_one {f : } {x : } (hf_conv : (set.Ioi 0) f) (hf_feq : {y : }, 0 < y f (y + 1) = f y + ) (hx : 0 < x) (hx' : x 1) :
(nhds (f x - f 1))
theorem real.bohr_mollerup.tendsto_log_gamma_seq {f : } {x : } (hf_conv : (set.Ioi 0) f) (hf_feq : {y : }, 0 < y f (y + 1) = f y + ) (hx : 0 < x) :
(nhds (f x - f 1))
theorem real.bohr_mollerup.tendsto_log_Gamma {x : } (hx : 0 < x) :
theorem real.eq_Gamma_of_log_convex {f : } (hf_conv : (set.Ioi 0) (real.log f)) (hf_feq : {y : }, 0 < y f (y + 1) = y * f y) (hf_pos : {y : }, 0 < y 0 < f y) (hf_one : f 1 = 1) :
(set.Ioi 0)

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.

theorem real.Gamma_two  :
= 1

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.

noncomputable def real.doubling_Gamma (s : ) :

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

Equations
theorem real.doubling_Gamma_add_one (s : ) (hs : s 0) :
theorem real.log_doubling_Gamma_eq  :
(λ (s : ), real.log (real.Gamma (s / 2)) + real.log (real.Gamma (s / 2 + 1 / 2)) + s * - real.log (2 * real.pi)) (set.Ioi 0)
theorem real.doubling_Gamma_eq_Gamma {s : } (hs : 0 < s) :
theorem real.Gamma_mul_Gamma_add_half_of_pos {s : } (hs : 0 < s) :
* real.Gamma (s + 1 / 2) = real.Gamma (2 * s) * 2 ^ (1 - 2 * s) *

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.