Bohnenblust-Hille inequalities: analytical and computational aspects

The Bohnenblust-Hille polynomial and multilinear inequalities were proved in 1931 and the determination of exact values of their constants is still an open and challenging problem, pursued by various authors. The present paper briefly surveys recent attempts to attack/solve this problem; it also presents new results, like connections with classical results of the linear theory of absolutely summing operators, and new perspectives.

Keywords
Bohnenblust-Hille inequality; Hardy-Littlewood inequality; extreme points; Banach spaces

Authorship

WASTHENNY V. CAVALCANTE

Universidade Federal de Pernambuco/UFPE, Av. Jornalista Aníbal Fernandes, s/n, Cidade Universitária, Departamento de Matemática, 50740-560 Recife, PE, BrazilUniversidade Federal de PernambucoBrazilRecife, PE, BrazilUniversidade Federal de Pernambuco/UFPE, Av. Jornalista Aníbal Fernandes, s/n, Cidade Universitária, Departamento de Matemática, 50740-560 Recife, PE, Brazil

DANIEL M. PELLEGRINO

Universidade Federal da Paraíba/UFPB, CCEN, Departamento de Matemática, Cidade Universitária, 58051-900 João Pessoa, PB, BrazilUniversidade Federal da ParaíbaBrazilJoão Pessoa, PB, BrazilUniversidade Federal da Paraíba/UFPB, CCEN, Departamento de Matemática, Cidade Universitária, 58051-900 João Pessoa, PB, Brazil

Correspondence to: Daniel Marinho Pellegrino. E-mail: pellegrino@pq.cnpq.br / dmpellegrino@gmail.com

SCIMAGO INSTITUTIONS RANKINGS

Formulas

Formulas (142)

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{(\sum_{i, j = 1}^{n} {| T (e_{i}, e_{j}) |}^{\frac{4}{3}})}^{\frac{3}{4}} \leq \sqrt{2} sup_{∥ x ∥, ∥ y ∥ \leq 1} | T (x, y) |

{(\sum_{j_{1}, \dots, j_{m} = 1}^{n} {| T (e_{j_{1}}, \dots, e_{j_{m}}) |}^{\frac{2 m}{m + 1}})}^{\frac{m + 1}{2 m}} \leq B_{ℂ, m} (n) ∥ T ∥,

B_{ℂ, m} := sup_{n} B_{ℂ, m} (n) < \infty .

∥ T ∥ := sup_{∥ x^{(1)} ∥, \dots, ∥ x^{(m)} ∥ \leq 1} | T (x^{(1)}, \dots, x^{(m)}) | .

{(\sum_{| α | = m} {| c_{α} (P) |}^{\frac{2 m}{m + 1}})}^{\frac{m + 1}{2 m}} \leq C_{ℂ, m} sup_{∥ x ∥ \leq 1} | P (x) |

1

\leq B_{ℂ, m} \leq m^{\frac{m + 1}{2 m}} {(\sqrt{2})}^{m - 1},

1

\leq C_{ℂ, m} \leq B_{ℂ, m} \frac{m^{\frac{m}{2}} {(m + 1)}^{\frac{m + 1}{2}}}{2^{m} {(m!)}^{\frac{m + 1}{2 m}}}

{\begin{matrix} 1 \leq B_{ℂ, m} \leq m^{\frac{1 - γ}{2}} < m^{0.212} \\ {(1 + 2^{1 - m})}^{\frac{m - 1}{4 m}} \leq C_{ℂ, m} \leq κ {(1 + ε)}^{m}, \end{matrix}

{\begin{matrix} 2^{1 - \frac{1}{m}} \leq B_{ℝ, m} \leq 2^{\frac{446381}{55440} - \frac{m}{2}} \prod_{j = 14}^{m} {(\frac{Γ (\frac{3}{2} - \frac{1}{j})}{\sqrt{π}})}^{\frac{j}{2 - 2 j}} < 1.3 m^{0.365} (for m \geq 14) \\ {(1.17)}^{m} < C_{ℝ, m} \leq {(2 + ε)}^{m}, \end{matrix}

{(\sum_{j_{1}, \dots, j_{m} = 1}^{n} {| T (e_{j_{1}}, \dots, e_{j_{m}}) |}^{\frac{2 m p}{m p + p - 2 m}})}^{\frac{m p + p - 2 m}{2 m p}} \leq C_{𝕂, m, p} ∥ T ∥,

∥ T ∥ = \max {\begin{matrix} | a_{11} + a_{21} | + | a_{12} + a_{22} |, | a_{11} - a_{21} | + | a_{12} - a_{22} |, \\ \sqrt{a_{11}^{2} + a_{21}^{2} + 2 a_{11} a_{21} \frac{\frac{a_{11}^{2} a_{21}^{2}}{a_{12}^{2} a_{22}^{2}} (a_{12}^{2} + a_{22}^{2}) - (a_{11}^{2} + a_{21}^{2})}{2 a_{11} a_{21} (1 - \frac{a_{11} a_{21}}{a_{12} a_{22}})}} \\ + \sqrt{a_{12}^{2} + a_{22}^{2} + 2 a_{12} a_{22} \frac{\frac{a_{11}^{2} a_{21}^{2}}{a_{12}^{2} a_{22}^{2}} (a_{12}^{2} + a_{22}^{2}) - (a_{11}^{2} + a_{21}^{2})}{2 a_{11} a_{21} (1 - \frac{a_{11} a_{21}}{a_{12} a_{22}})}} \end{matrix}}

| \frac{a_{11}^{2} a_{21}^{2}}{a_{12}^{2} a_{22}^{2}} (a_{12}^{2} + a_{22}^{2}) - (a_{11}^{2} + a_{21}^{2}) | \leq | 2 a_{11} a_{21} (1 - \frac{a_{11} a_{21}}{a_{12} a_{22}}) |;

∥ T ∥ = \max {| a_{11} + a_{21} | + | a_{12} + a_{22} |, | a_{11} - a_{21} | + | a_{12} - a_{22} |}

∥ T ∥ = sup {∥ T_{z} ∥ : {∥ z ∥}_{\infty} \leq 1},

T_{z} (w) = (a_{11} z_{1} + a_{21} z_{2}) w_{1} + (a_{12} z_{1} + a_{22} z_{2}) w_{2} .

∥ T ∥ = sup {∥ T_{z} ∥ : {∥ z ∥}_{\infty} \leq 1} = sup {| a_{11} z_{1} + a_{21} z_{2} | + | a_{12} z_{1} + a_{22} z_{2} | : {∥ z ∥}_{\infty} \leq 1} .

f (z_{1}, z_{2}) = | a_{11} z_{1} + a_{21} z_{2} | + | a_{12} z_{1} + a_{22} z_{2} |

f (z_{1}, z_{2})

= \sqrt{{(a_{11} x_{1} + a_{21} x_{2})}^{2} + {(a_{11} y_{1} + a_{21} y_{2})}^{2}}

+ \sqrt{{(a_{12} x_{1} + a_{22} x_{2})}^{2} + {(a_{12} y_{1} + a_{22} y_{2})}^{2}} .

f (θ_{1}, θ_{2}) = \sqrt{a_{11}^{2} + a_{21}^{2} + 2 a_{11} a_{21} \cos (θ_{1} - θ_{2})} + \sqrt{a_{12}^{2} + a_{22}^{2} + 2 a_{12} a_{22} \cos (θ_{1} - θ_{2})} .

f (t) = \sqrt{a_{11}^{2} + a_{21}^{2} + 2 a_{11} a_{21} \cos t} + \sqrt{a_{12}^{2} + a_{22}^{2} + 2 a_{12} a_{22} \cos t} .

f^{'} (t) = \frac{- a_{11} a_{21} \sin t}{\sqrt{a_{11}^{2} + a_{21}^{2} + 2 a_{11} a_{21} \cos t}} + \frac{- a_{12} a_{22} \sin t}{\sqrt{a_{12}^{2} + a_{22}^{2} + 2 a_{12} a_{22} \cos t}} = 0

\frac{- a_{11} a_{21}}{a_{12} a_{22}} = \frac{\sqrt{a_{11}^{2} + a_{21}^{2} + 2 a_{11} a_{21} \cos t}}{\sqrt{a_{12}^{2} + a_{22}^{2} + 2 a_{12} a_{22} \cos t}} .

2 a_{11} a_{21} (1 - \frac{a_{11} a_{21}}{a_{12} a_{22}}) \cos t = \frac{a_{11}^{2} a_{21}^{2}}{a_{12}^{2} a_{22}^{2}} (a_{12}^{2} + a_{22}^{2}) - (a_{11}^{2} + a_{21}^{2})

| \frac{a_{11}^{2} a_{21}^{2}}{a_{12}^{2} a_{22}^{2}} (a_{12}^{2} + a_{22}^{2}) - (a_{11}^{2} + a_{21}^{2}) | \leq | 2 a_{11} a_{21} (1 - \frac{a_{11} a_{21}}{a_{12} a_{22}}) |,

\cos t_{0} = \frac{\frac{a_{11}^{2} a_{21}^{2}}{a_{12}^{2} a_{22}^{2}} (a_{12}^{2} + a_{22}^{2}) - (a_{11}^{2} + a_{21}^{2})}{2 a_{11} a_{21} (1 - \frac{a_{11} a_{21}}{a_{12} a_{22}})} .

∥ T ∥ = \max f = \max {\begin{matrix} | a_{11} + a_{21} | + | a_{12} + a_{22} |, | a_{11} - a_{21} | + | a_{12} - a_{22} |, \\ \sqrt{a_{11}^{2} + a_{21}^{2} + 2 a_{11} a_{21} \frac{\frac{a_{11}^{2} a_{21}^{2}}{a_{12}^{2} a_{22}^{2}} (a_{12}^{2} + a_{22}^{2}) - (a_{11}^{2} + a_{21}^{2})}{2 a_{11} a_{21} (1 - \frac{a_{11} a_{21}}{a_{12} a_{22}})}} \\ + \sqrt{a_{12}^{2} + a_{22}^{2} + 2 a_{12} a_{22} \frac{\frac{a_{11}^{2} a_{21}^{2}}{a_{12}^{2} a_{22}^{2}} (a_{12}^{2} + a_{22}^{2}) - (a_{11}^{2} + a_{21}^{2})}{2 a_{11} a_{21} (1 - \frac{a_{11} a_{21}}{a_{12} a_{22}})}} \end{matrix}} .

f (t_{0}) = | a_{11} + a_{21} | + | a_{12} + a_{22} |

f (t_{0}) = | a_{11} - a_{21} | + | a_{12} - a_{22} | .

s g n (\frac{a_{11} a_{21}}{a_{12} a_{22}}) = 1 .

∥ T ∥ = \max f = \max {| a_{11} + a_{21} | + | a_{12} + a_{22} |, | a_{11} - a_{21} | + | a_{12} - a_{22} |} .

s g n (\frac{a_{11} a_{21}}{a_{12} a_{22}}) = - 1

| \frac{a_{11}^{2} a_{21}^{2}}{a_{12}^{2} a_{22}^{2}} (a_{12}^{2} + a_{22}^{2}) - (a_{11}^{2} + a_{21}^{2}) | > | 2 a_{11} a_{21} (1 - \frac{a_{11} a_{21}}{a_{12} a_{22}}) | .

∥ T ∥ = \max f = \max {| a_{11} + a_{21} | + | a_{12} + a_{22} |, | a_{11} - a_{21} | + | a_{12} - a_{22} |} .

f (t) = | a_{11} + a_{21} | + | a_{12} + a_{22} | = | a_{11} - a_{21} | + | a_{12} - a_{22} |

f (t) = | a_{21} | + \sqrt{a_{12}^{2} + a_{22}^{2} + 2 a_{12} a_{22} \cos t}

f (t_{0}) = | a_{21} | \leq | a_{11} + a_{21} | + | a_{12} + a_{22} | .

f^{'} (t) = \frac{- a_{12} a_{22} \sin t}{\sqrt{a_{12}^{2} + a_{22}^{2} + 2 a_{12} a_{22} \cos t}}

f (t_{1}) = | a_{21} | + | a_{12} + a_{22} | = | a_{11} + a_{21} | + | a_{12} + a_{22} |

f (t_{1}) = | a_{21} | + | a_{12} - a_{22} | = | a_{11} - a_{21} | + | a_{12} - a_{22} | .

∥ T ∥ = \max {| a_{11} + a_{21} | + | a_{12} + a_{22} |, | a_{11} - a_{21} | + | a_{12} - a_{22} |} .

f (λ x + (1 - λ) y) \leq λ f (x) + (1 - λ) f (y),

k_{0} = \sum_{j = 1}^{n} λ_{j} k_{j},

f (k_{0}) \leq \sum_{j = 1}^{n} λ_{j} f (k_{j}) < \sum_{j = 1}^{n} λ_{j} f (k_{0}) = f (k_{0}),

∥ T ∥ = \max {| T (x_{1}, \dots, x_{m}) | : x_{1}, \dots, x_{m} \in e x t B_{E}} .

∥ T ∥ = | T (y_{1}, \dots, y_{m}) | .

B_{E} = c o n v (e x t (B_{E})),

x_{i} = \sum_{j_{i} = 1}^{k_{i}} λ_{j_{i}}^{i} y_{j_{i}}^{i},

| T (x_{1}, \dots, x_{m}) |

= | T (\sum_{j_{1} = 1}^{k_{1}} λ_{j_{1}}^{1} y_{j_{1}}^{1}, \dots, \sum_{j_{m} = 1}^{k_{m}} λ_{j_{n}}^{m} y_{j_{m}}^{m}) |

\leq \sum_{j_{1} = 1}^{k_{1}} \dots \sum_{j_{m} = 1}^{k_{m}} λ_{j_{1}}^{1} \dots λ_{j_{m}}^{m} | T (y_{j_{1}}^{1}, \dots, y_{j_{m}}^{m}) |

< ∥ T ∥,

sup {(\sum_{j_{1}, \dots, j_{m} = 1}^{n} {| T (e_{j_{1}}, \dots, e_{j_{m}}) |}^{\frac{2 m}{m + 1}})}^{\frac{m + 1}{2 m}}

f : B_{ℒ (^{m} ℓ_{\infty}^{n} (ℝ))} \to ℝ

f (T) = {(\sum_{j_{1}, \dots, j_{m} = 1}^{n} {| T (e_{j_{1}}, \dots, e_{j_{m}}) |}^{\frac{2 m}{m + 1}})}^{\frac{m + 1}{2 m}}

| a + b | + | c + d | = | a - b | + | c - d | .

a + b + | c + d | \geq | a - b | + c - d

a \geq - d, b \geq - d, c \geq - d or a \geq c, b \geq c, - d \geq c .

| a - b | + c - d \geq a + b + | c + d |

a \geq b, c \geq b, - d \geq b or b \geq a, c \geq a, - d \geq a .

\pm x_{1} y_{1}, \pm x_{2} y_{1}, \pm x_{1} y_{2}, \pm x_{2} y_{2},

\frac{1}{2} (\pm x_{1} y_{1} \pm x_{2} y_{1} \pm x_{1} y_{2} \mp x_{2} y_{2}),

\frac{1}{2} (\mp x_{1} y_{1} \pm x_{2} y_{1} \pm x_{1} y_{2} \pm x_{2} y_{2}),

\frac{1}{2} (\pm x_{1} y_{1} \mp x_{2} y_{1} \pm x_{1} y_{2} \pm x_{2} y_{2}),

\frac{1}{2} (\pm x_{1} y_{1} \pm x_{2} y_{1} \mp x_{1} y_{2} \pm x_{2} y_{2}) .

A (x, y) = (a + ε) x_{1} y_{1}

B (x, y) = (a - ε) x_{1} y_{1},

A (x, y) = α x_{1} y_{1} + β x_{2} y_{1} + γ x_{1} y_{2} + δ x_{2} y_{2}

B (x, y) = α^{'} x_{1} y_{1} + β^{'} x_{2} y_{1} + γ^{'} x_{1} y_{2} + δ^{'} x_{2} y_{2},

∥ T ∥ = | a | + | b | \leq 1 .

A (x, y) = (a + s g n (a) ε) x_{1} y_{1} + (b - s g n (b) ε) x_{2} y_{1}

B (x, y) = (a - s g n (a) ε) x_{1} y_{1} + (b + s g n (b) ε) x_{2} y_{1},

∥ T ∥ = \max {| a + b | + | c |, | a - b | + | c |} .

| a + b | + | c | = | a - b | + | c | \Leftrightarrow a = 0 or b = 0 .

A (x, y) = (a + ε) x_{1} y_{1} + (b - ε) x_{2} y_{1} + c x_{1} y_{2}

B (x, y) = (a - ε) x_{1} y_{1} + (b + ε) x_{2} y_{1} + c x_{1} y_{2},

A (x, y) = (a + ε) x_{1} y_{1} + (b + ε) x_{2} y_{1} + c x_{1} y_{2}

B (x, y) = (a - ε) x_{1} y_{1} + (b - ε) x_{2} y_{1} + c x_{1} y_{2},

A (x, y) = (a + ε) x_{1} y_{1} + (b - ε) x_{2} y_{1} + c x_{1} y_{2} + d x_{2} y_{2}

B (x, y) = (a - ε) x_{1} y_{1} + (b + ε) x_{2} y_{1} + c x_{1} y_{2} + d x_{2} y_{2},

A (x, y) = (a + ε) x_{1} y_{1} + (b + ε) x_{2} y_{1} + c x_{1} y_{2} + d x_{2} y_{2}

B (x, y) = (a - ε) x_{1} y_{1} + (b - ε) x_{2} y_{1} + c x_{1} y_{2} + d x_{2} y_{2},

∥ T ∥ = | a + b | + | c + d |

∥ T ∥ = | a - b | + | c - d | .

c a r d {a, b, c, - d} \neq 1,

a + b + c + d > a - b + c - d,

| a + b | + | c + d | > | a - b | + | c - d |

A (x, y) = (a + ε) x_{1} y_{1} + (b - ε) x_{2} y_{1} + c x_{1} y_{2} + d x_{2} y_{2}

B (x, y) = (a - ε) x_{1} y_{1} + (b + ε) x_{2} y_{1} + c x_{1} y_{2} + d x_{2} y_{2},

| a + b | + | c + d | = | a - b | + | c - d | = a + c

∥ T ∥ = a + c \leq 1 .

A (x, y) = (a + ε) x_{1} y_{1} + b x_{2} y_{1} + c x_{1} y_{2} - b x_{2} y_{2}

B (x, y) = (a - ε) x_{1} y_{1} + b x_{2} y_{1} + c x_{1} y_{2} - b x_{2} y_{2},

T (x, y) = a x_{1} y_{1} + b x_{2} y_{1} + (1 - a) x_{1} y_{2} - b x_{2} y_{2} .

1 - a \geq b .

T (x, y) = (1 - b) x_{1} y_{1} + b x_{2} y_{1} + b x_{1} y_{2} - b x_{2} y_{2} .

| (1 - b) - b | + | b - (- b) | = | 1 - 2 b | + 2 b .

A (x, y) = (1 - b + ε) x_{1} y_{1} + (b - ε) x_{2} y_{1} + (b - ε) x_{1} y_{2} + (- b + ε) x_{2} y_{2}

B (x, y) = (1 - b - ε) x_{1} y_{1} + (b + ε) x_{2} y_{1} + (b + ε) x_{1} y_{2} + (- b - ε) x_{2} y_{2},

A (x, y) = (a + ε) x_{1} y_{1} + b x_{2} y_{1} + (1 - a - ε) x_{1} y_{2} - b x_{2} y_{2}

B (x, y) = (a - ε) x_{1} y_{1} + b x_{2} y_{1} + (1 - a + ε) x_{1} y_{2} - b x_{2} y_{2},

a > - d \Rightarrow a + d > - a - d \Rightarrow a + b + c + d > b - a + c - d .

| a + b | + | c + d | > | a - b | + | c - d | .

A (x, y) = (a + ε) x_{1} y_{1} + (b - ε) x_{2} y_{1} + c x_{1} y_{2} + d x_{2} y_{2}

B (x, y) = (a - ε) x_{1} y_{1} + (b + ε) x_{2} y_{1} + c x_{1} y_{2} + d x_{2} y_{2},

T (x, y) = a x_{1} y_{1} + b x_{2} y_{1} + c x_{1} y_{2} - a x_{2} y_{2}

c a r d {a, b, c, - d} = 1 .

T (x, y) = a x_{1} y_{1} + a x_{2} y_{1} + a x_{1} y_{2} - a x_{2} y_{2} .

A (x, y) = α x_{1} y_{1} + β x_{2} y_{1} + γ x_{1} y_{2} + δ x_{2} y_{2},

B (x, y) = α^{'} x_{1} y_{1} + β^{'} x_{2} y_{1} + γ^{'} x_{1} y_{2} + δ^{'} x_{2} y_{2},

(α + α^{'}, β + β^{'}, γ + γ^{'}, δ + δ^{'}) = (1, 1, 1, - 1) .

0 \leq α < \frac{1}{2}

\Rightarrow - \frac{1}{2} < - α \leq 0

\Rightarrow \frac{1}{2} < 1 - α \leq 1

\Rightarrow \frac{1}{2} < α^{'} \leq 1

\frac{1}{2} < α \leq 1 \Rightarrow 0 \leq α^{'} < \frac{1}{2} .

0 \leq β < \frac{1}{2} \Rightarrow \frac{1}{2} < β^{'} \leq 1

B ((1, 1), (1, 0)) = α^{'} + β^{'} > 1

∥ B ∥ = α^{'} + β^{'} + γ^{'} + δ^{'} > 1 .

∥ B ∥ = α^{'} - β^{'} + γ^{'} - δ^{'} > 1 .

∥ B ∥ = α^{'} - β^{'} + γ^{'} - δ^{'} > 1 .

A ((1, - 1), (0, 1)) = - δ + γ > 1

∥ A ∥ = α + β + γ + δ > 1 .

∥ A ∥ = β - α + γ - δ > 1 .

A ((1, 1), (1, 0)) = α + β > 1

∥ A ∥ = α + β - δ - γ > 1 .

∥ A ∥ = α - β + γ - δ > 1 .

B ((1, - 1), (0, 1)) = - δ^{'} + γ^{'} > 1

A ((1, - 1), (0, 1)) = - δ + γ > 1,

∥ A ∥ = α + β + γ + δ > 1 .

∥ A ∥ = α - β + γ - δ > 1 .

\frac{1}{p} + \frac{1}{p^{*}} = 1 .

ℓ_{s}^{w} (E) := {{(x_{j})}_{j = 1}^{\infty} \subset E : \sum_{j = 1}^{\infty} {| φ (x_{j}) |}^{s} < \infty for all φ \in E^{*}},

{∥ {(x_{j})}_{j = 1}^{\infty} ∥}_{w, s} = sup_{φ \in B_{E^{*}}} {(\sum_{j = 1}^{\infty} {| φ (x_{j}) |}^{s})}^{1 / s} .

{(\sum_{i = 1}^{\infty} {(\sum_{j = 1}^{\infty} {| A (e_{i}, e_{j}) |}^{q})}^{\frac{1}{q} r})}^{\frac{1}{r}} \leq C ∥ A ∥

{(\sum_{j = 1}^{\infty} {∥ x_{j} ∥}_{ℓ_{q}}^{r})}^{\frac{1}{r}} = {(\sum_{j = 1}^{\infty} {(\sum_{i = 1}^{\infty} {| v (e_{j}) (e_{i}) |}^{q})}^{\frac{1}{q} r})}^{\frac{1}{r}} .

{(\sum_{j = 1}^{\infty} {∥ x_{j} ∥}_{ℓ_{q}}^{r})}^{\frac{1}{r}} = {(\sum_{j = 1}^{\infty} {(\sum_{i = 1}^{\infty} {| A (e_{j}, e_{i}) |}^{q})}^{\frac{1}{q} r})}^{\frac{1}{r}} \leq C ∥ A ∥ = C ∥ v ∥ = {∥ {(x_{j})}_{j = 1}^{\infty} ∥}_{w, s} .

\frac{1}{u} + \frac{1}{v} < 1,

{(\sum_{i = 1}^{\infty} {(\sum_{j = 1}^{\infty} {| A (e_{i}, e_{j}) |}^{q})}^{\frac{1}{q} r})}^{\frac{1}{r}} \leq C ∥ A ∥

{\begin{matrix} q \geq \frac{v}{v - 1} \\ r \geq \frac{u v}{u v - u - v} \\ \frac{1}{q} + \frac{1}{r} \leq \frac{3}{2} - (\frac{1}{u} + \frac{1}{v}) . \end{matrix}

\frac{1}{u} + \frac{1}{v} \geq 1

{\begin{matrix} q \geq \frac{p^{*}}{p^{*} - 1} = p \\ r \geq \frac{p^{*} s^{*}}{p^{*} s^{*} - p^{*} - s^{*}} = \frac{p s}{p + s - p s} \\ \frac{1}{q} + \frac{1}{r} \leq \frac{3}{2} - (\frac{1}{u} + \frac{1}{v}) = \frac{3}{2} - (\frac{1}{p^{*}} + \frac{1}{s^{*}}), \\ \frac{1}{s^{*}} + \frac{1}{p^{*}} < 1, i.e., s < p^{*}, \end{matrix}

{\begin{matrix} r \geq \max {\frac{2 p q s}{2 q s - 2 p s + 2 p q - p q s}, \frac{p s}{p + s - p s}}, \\ s < p^{*} . \end{matrix}

r

\geq \max {\frac{2 p q s}{2 q s - 2 p s + 2 p q - p q s}, \frac{p s}{p + s - p s}},

s

< p^{*} .

r

\geq p when q \leq 2

r

\geq \frac{2 p q}{2 q - 2 p + p q} when q \geq 2 .

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Bohnenblust-Hille inequalities: analytical and computational aspects

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[1] Correspondence to: Daniel Marinho Pellegrino. E-mail: pellegrino@pq.cnpq.br / dmpellegrino@gmail.com