On the Arithmetic-Geometric Mean Inequality

Concerning the arithmetic-geometric mean inequality, H. Z. Chuan ([C]) inserted a continuum of additional terms between the two sides of the inequality as follows. Theorem A([C]).

If n is a natural number, s > 0, a_j > 0, q_j ≥ 0 (j = 1,...,n), and q ₁ + ... + q_n = 1 then Π j = 1 n a j q j ≤ ( s ∫ 0 ∞ [ Π j = 1 n ( x + a j ) q j ] − s − 1 d x ) − 1 s ≤ Σ j = 1 n q j a j . https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429187681/dd6f40d9-b60a-425b-8c69-a9c7f3359253/content/unequ26_1.tif"/>

Theorem A can be generalized as

Theorem B([K, <xref ref-type="boxed-text" rid="the26_2">Theorem 2</xref>]).

If s > 0, μ(X) = 1, and f is a positive function of L ¹(μ), then G X f ≤ { s ∫ 0 ∞ exp ⁡ ( ∫ X log ( y + f ( x ) ) − s − 1 d μ ( x ) ) d y } − 1 / s ≤ A X f . ⁡ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429187681/dd6f40d9-b60a-425b-8c69-a9c7f3359253/content/unequ26_2.tif"/>

234Either of the equalities holds if and only if f(x) = constant a. e. [μ]. Here G X f = exp ⁡ ( ∫ X log ⁡ f ( x ) d μ ( x ) ) https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429187681/dd6f40d9-b60a-425b-8c69-a9c7f3359253/content/unequ26_3.tif"/>

and A X f = ∫ X f ( x ) d μ ( x ) https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429187681/dd6f40d9-b60a-425b-8c69-a9c7f3359253/content/unequ26_4.tif"/>

are respectively the geometric mean and the arithmetic mean of f over X.

If 0 ≤ x ≤ 1 then Holder’s inequality says that ∫ Y f 1 ( y ) x f 2 ( y ) 1 − x d υ ( y ) ≤ ( ∫ Y f 1 ( y ) d υ ( y ) ) x ( ∫ Y f 2 ( y ) d υ ( y ) ) 1 − x https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429187681/dd6f40d9-b60a-425b-8c69-a9c7f3359253/content/unequ26_5.tif"/>

for all positive functions f ₁ and f ₂ of L ¹(v). It is known that the inequality can be extended to the case of a multiple product of functions (See, for example, [BB], [HLP], and [RV]), and even to a countable product of functions (provided the product converges). The following is a continuous form of Holder’s inequality.

Theorem C([K, <xref ref-type="boxed-text" rid="the26_1">Theorem 1</xref>]).

Let μ(X) = 1. Let f(x,y) be a positive measurable function defined X × X. Then ∫ Y exp ( ∫ X log f d μ ) d v ≤ exp { ∫ X log ( ∫ Y f d v ) ) d μ } . https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429187681/dd6f40d9-b60a-425b-8c69-a9c7f3359253/content/unequ26_6.tif"/>

Equality holds as a nonzero finite value if and only if f ( x , y ) = g ( x ) h ( x ) a . e . μ × v https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429187681/dd6f40d9-b60a-425b-8c69-a9c7f3359253/content/unequ26_7.tif"/>

for a positive μ-measurable function g with -∞ < ∫ _X log g dμ < ∞ and a positive v-measurable h with ∫ _Y h dv = 1.

If we take X = { 1 , 2 } , f ( x , y ) = f x ( y ) , x ∈ X , https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429187681/dd6f40d9-b60a-425b-8c69-a9c7f3359253/content/unequ26_8.tif"/>

and d μ = ( t χ { 1 } + ( 1 − t ) χ { 2 } ) d m , https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429187681/dd6f40d9-b60a-425b-8c69-a9c7f3359253/content/unequ26_9.tif"/>

where dm is the counting measure and χ _{·} is the corresponding characteristic function, then the continuous form reduces to the discrete form.

Theorem 1.

If n is a natural number, s > 0, 0 < p ≤ 1, q_j > 0, q_j > 0 (j = 1,...,n), and q ₁ + ... + q_n = 1 then Π j = 1 n a j q j ≤ ( s ∫ 0 ∞ [ Σ j = 1 n ( x + a j ) q j p ] − ( s + 1 ) / p d x ) − 1 s ≤ Σ j = 1 n q j a j . https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429187681/dd6f40d9-b60a-425b-8c69-a9c7f3359253/content/unequ26_10.tif"/>

235 Theorem 1 reduces to Theorem A by letting p → 0. A continuous form of Theorem 1 is as follows.

Theorem 2.

If s > 0, 0 < p ≤ 1, μ(X) = 1, and f is a positive function of L ¹(μ), then G X f ≤ { s ∫ 0 ∞ ( ∫ X ( y + f ( x ) ) p d μ ( x ) ) − ( s + 1 ) / p d y } − 1 / s ≤ A X f . https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429187681/dd6f40d9-b60a-425b-8c69-a9c7f3359253/content/unequ26_11.tif"/>

Either of the equalities holds if and only if f(x) = constant a. e. [μ].