File: Mathlib/Analysis/MeanInequalitiesPow.lean

full_name: Real.pow_arith_mean_le_arith_mean_pow
code: theorem pow_arith_mean_le_arith_mean_pow (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
    (hw' : ∑ i ∈ s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (n : ℕ) :
    (∑ i ∈ s, w i * z i) ^ n ≤ ∑ i ∈ s, w i * z i ^ n 
goal: (∑ i ∈ s, w i * z i) ^ n ≤ ∑ i ∈ s, w i * z i ^ n

full_name: Real.pow_arith_mean_le_arith_mean_pow_of_even
code: theorem pow_arith_mean_le_arith_mean_pow_of_even (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
    (hw' : ∑ i ∈ s, w i = 1) {n : ℕ} (hn : Even n) :
    (∑ i ∈ s, w i * z i) ^ n ≤ ∑ i ∈ s, w i * z i ^ n 
goal: (∑ i ∈ s, w i * z i) ^ n ≤ ∑ i ∈ s, w i * z i ^ n

full_name: Real.pow_sum_div_card_le_sum_pow
code: /-- Specific case of Jensen's inequality for sums of powers -/
theorem pow_sum_div_card_le_sum_pow {f : ι → ℝ} (n : ℕ) (hf : ∀ a ∈ s, 0 ≤ f a) :
    (∑ x ∈ s, f x) ^ (n + 1) / (s.card : ℝ) ^ n ≤ ∑ x ∈ s, f x ^ (n + 1) 
goal: (∑ x ∈ s, f x) ^ (n + 1) / ↑s.card ^ n ≤ ∑ x ∈ s, f x ^ (n + 1)

full_name: Real.zpow_arith_mean_le_arith_mean_zpow
code: theorem zpow_arith_mean_le_arith_mean_zpow (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
    (hw' : ∑ i ∈ s, w i = 1) (hz : ∀ i ∈ s, 0 < z i) (m : ℤ) :
    (∑ i ∈ s, w i * z i) ^ m ≤ ∑ i ∈ s, w i * z i ^ m 
goal: (∑ i ∈ s, w i * z i) ^ m ≤ ∑ i ∈ s, w i * z i ^ m

full_name: Real.rpow_arith_mean_le_arith_mean_rpow
code: theorem rpow_arith_mean_le_arith_mean_rpow (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
    (hw' : ∑ i ∈ s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) {p : ℝ} (hp : 1 ≤ p) :
    (∑ i ∈ s, w i * z i) ^ p ≤ ∑ i ∈ s, w i * z i ^ p 
goal: (∑ i ∈ s, w i * z i) ^ p ≤ ∑ i ∈ s, w i * z i ^ p

full_name: Real.arith_mean_le_rpow_mean
code: theorem arith_mean_le_rpow_mean (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i) (hw' : ∑ i ∈ s, w i = 1)
    (hz : ∀ i ∈ s, 0 ≤ z i) {p : ℝ} (hp : 1 ≤ p) :
    ∑ i ∈ s, w i * z i ≤ (∑ i ∈ s, w i * z i ^ p) ^ (1 / p) 
goal: ∑ i ∈ s, w i * z i ≤ (∑ i ∈ s, w i * z i ^ p) ^ (1 / p)

full_name: NNReal.pow_arith_mean_le_arith_mean_pow
code: /-- Weighted generalized mean inequality, version sums over finite sets, with `ℝ≥0`-valued
functions and natural exponent. -/
theorem pow_arith_mean_le_arith_mean_pow (w z : ι → ℝ≥0) (hw' : ∑ i ∈ s, w i = 1) (n : ℕ) :
    (∑ i ∈ s, w i * z i) ^ n ≤ ∑ i ∈ s, w i * z i ^ n 
goal: (∑ i ∈ s, w i * z i) ^ n ≤ ∑ i ∈ s, w i * z i ^ n

full_name: NNReal.pow_sum_div_card_le_sum_pow
code: theorem pow_sum_div_card_le_sum_pow (f : ι → ℝ≥0) (n : ℕ) :
    (∑ x ∈ s, f x) ^ (n + 1) / (s.card : ℝ) ^ n ≤ ∑ x ∈ s, f x ^ (n + 1) 
goal: ↑(∑ x ∈ s, f x) ^ (n + 1) / ↑s.card ^ n ≤ ↑(∑ x ∈ s, f x ^ (n + 1))

full_name: NNReal.rpow_arith_mean_le_arith_mean_rpow
code: /-- Weighted generalized mean inequality, version for sums over finite sets, with `ℝ≥0`-valued
functions and real exponents. -/
theorem rpow_arith_mean_le_arith_mean_rpow (w z : ι → ℝ≥0) (hw' : ∑ i ∈ s, w i = 1) {p : ℝ}
    (hp : 1 ≤ p) : (∑ i ∈ s, w i * z i) ^ p ≤ ∑ i ∈ s, w i * z i ^ p 
goal: (∑ i ∈ s, w i * z i) ^ p ≤ ∑ i ∈ s, w i * z i ^ p

full_name: NNReal.rpow_arith_mean_le_arith_mean2_rpow
code: /-- Weighted generalized mean inequality, version for two elements of `ℝ≥0` and real exponents. -/
theorem rpow_arith_mean_le_arith_mean2_rpow (w₁ w₂ z₁ z₂ : ℝ≥0) (hw' : w₁ + w₂ = 1) {p : ℝ}
    (hp : 1 ≤ p) : (w₁ * z₁ + w₂ * z₂) ^ p ≤ w₁ * z₁ ^ p + w₂ * z₂ ^ p 
goal: (w₁ * z₁ + w₂ * z₂) ^ p ≤ w₁ * z₁ ^ p + w₂ * z₂ ^ p

full_name: NNReal.rpow_add_le_mul_rpow_add_rpow
code: /-- Unweighted mean inequality, version for two elements of `ℝ≥0` and real exponents. -/
theorem rpow_add_le_mul_rpow_add_rpow (z₁ z₂ : ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
    (z₁ + z₂) ^ p ≤ (2 : ℝ≥0) ^ (p - 1) * (z₁ ^ p + z₂ ^ p) 
goal: (z₁ + z₂) ^ p ≤ 2 ^ (p - 1) * (z₁ ^ p + z₂ ^ p)

full_name: NNReal.arith_mean_le_rpow_mean
code: /-- Weighted generalized mean inequality, version for sums over finite sets, with `ℝ≥0`-valued
functions and real exponents. -/
theorem arith_mean_le_rpow_mean (w z : ι → ℝ≥0) (hw' : ∑ i ∈ s, w i = 1) {p : ℝ} (hp : 1 ≤ p) :
    ∑ i ∈ s, w i * z i ≤ (∑ i ∈ s, w i * z i ^ p) ^ (1 / p) 
goal: ∑ i ∈ s, w i * z i ≤ (∑ i ∈ s, w i * z i ^ p) ^ (1 / p)

full_name: _private.Mathlib.Analysis.MeanInequalitiesPow.0.NNReal.add_rpow_le_one_of_add_le_one
code: private theorem add_rpow_le_one_of_add_le_one {p : ℝ} (a b : ℝ≥0) (hab : a + b ≤ 1) (hp1 : 1 ≤ p) :
    a ^ p + b ^ p ≤ 1 
goal: a ^ p + b ^ p ≤ 1

full_name: NNReal.add_rpow_le_rpow_add
code: theorem add_rpow_le_rpow_add {p : ℝ} (a b : ℝ≥0) (hp1 : 1 ≤ p) : a ^ p + b ^ p ≤ (a + b) ^ p 
goal: a ^ p + b ^ p ≤ (a + b) ^ p

full_name: NNReal.rpow_add_rpow_le_add
code: theorem rpow_add_rpow_le_add {p : ℝ} (a b : ℝ≥0) (hp1 : 1 ≤ p) :
    (a ^ p + b ^ p) ^ (1 / p) ≤ a + b 
goal: (a ^ p + b ^ p) ^ (1 / p) ≤ a + b

full_name: NNReal.rpow_add_rpow_le
code: theorem rpow_add_rpow_le {p q : ℝ} (a b : ℝ≥0) (hp_pos : 0 < p) (hpq : p ≤ q) :
    (a ^ q + b ^ q) ^ (1 / q) ≤ (a ^ p + b ^ p) ^ (1 / p) 
goal: (a ^ q + b ^ q) ^ (1 / q) ≤ (a ^ p + b ^ p) ^ (1 / p)

full_name: NNReal.rpow_add_le_add_rpow
code: theorem rpow_add_le_add_rpow {p : ℝ} (a b : ℝ≥0) (hp : 0 ≤ p) (hp1 : p ≤ 1) :
    (a + b) ^ p ≤ a ^ p + b ^ p 
goal: (a + b) ^ p ≤ a ^ p + b ^ p

full_name: ENNReal.rpow_arith_mean_le_arith_mean_rpow
code: /-- Weighted generalized mean inequality, version for sums over finite sets, with `ℝ≥0∞`-valued
functions and real exponents. -/
theorem rpow_arith_mean_le_arith_mean_rpow (w z : ι → ℝ≥0∞) (hw' : ∑ i ∈ s, w i = 1) {p : ℝ}
    (hp : 1 ≤ p) : (∑ i ∈ s, w i * z i) ^ p ≤ ∑ i ∈ s, w i * z i ^ p 
goal: (∑ i ∈ s, w i * z i) ^ p ≤ ∑ i ∈ s, w i * z i ^ p

full_name: ENNReal.rpow_arith_mean_le_arith_mean2_rpow
code: /-- Weighted generalized mean inequality, version for two elements of `ℝ≥0∞` and real
exponents. -/
theorem rpow_arith_mean_le_arith_mean2_rpow (w₁ w₂ z₁ z₂ : ℝ≥0∞) (hw' : w₁ + w₂ = 1) {p : ℝ}
    (hp : 1 ≤ p) : (w₁ * z₁ + w₂ * z₂) ^ p ≤ w₁ * z₁ ^ p + w₂ * z₂ ^ p 
goal: (w₁ * z₁ + w₂ * z₂) ^ p ≤ w₁ * z₁ ^ p + w₂ * z₂ ^ p

full_name: ENNReal.rpow_add_le_mul_rpow_add_rpow
code: /-- Unweighted mean inequality, version for two elements of `ℝ≥0∞` and real exponents. -/
theorem rpow_add_le_mul_rpow_add_rpow (z₁ z₂ : ℝ≥0∞) {p : ℝ} (hp : 1 ≤ p) :
    (z₁ + z₂) ^ p ≤ (2 : ℝ≥0∞) ^ (p - 1) * (z₁ ^ p + z₂ ^ p) 
goal: (z₁ + z₂) ^ p ≤ 2 ^ (p - 1) * (z₁ ^ p + z₂ ^ p)

full_name: ENNReal.add_rpow_le_rpow_add
code: theorem add_rpow_le_rpow_add {p : ℝ} (a b : ℝ≥0∞) (hp1 : 1 ≤ p) : a ^ p + b ^ p ≤ (a + b) ^ p 
goal: a ^ p + b ^ p ≤ (a + b) ^ p

full_name: ENNReal.rpow_add_rpow_le_add
code: theorem rpow_add_rpow_le_add {p : ℝ} (a b : ℝ≥0∞) (hp1 : 1 ≤ p) :
    (a ^ p + b ^ p) ^ (1 / p) ≤ a + b 
goal: (a ^ p + b ^ p) ^ (1 / p) ≤ a + b

full_name: ENNReal.rpow_add_rpow_le
code: theorem rpow_add_rpow_le {p q : ℝ} (a b : ℝ≥0∞) (hp_pos : 0 < p) (hpq : p ≤ q) :
    (a ^ q + b ^ q) ^ (1 / q) ≤ (a ^ p + b ^ p) ^ (1 / p) 
goal: (a ^ q + b ^ q) ^ (1 / q) ≤ (a ^ p + b ^ p) ^ (1 / p)

full_name: ENNReal.rpow_add_le_add_rpow
code: theorem rpow_add_le_add_rpow {p : ℝ} (a b : ℝ≥0∞) (hp : 0 ≤ p) (hp1 : p ≤ 1) :
    (a + b) ^ p ≤ a ^ p + b ^ p 
goal: (a + b) ^ p ≤ a ^ p + b ^ p