Equality holds in Minkowski's Inequality when

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In summary, the conversation discusses the necessary and sufficient conditions for equality to hold in Minkowski's Inequality for complex functions in L^{p}(X,\mu). The inequality is proved using Holder's Inequality, and the conditions for equality to hold in Holder's Inequality are also discussed. It is stated that for equality to hold in Minkowski's Inequality, one function must be a constant multiple of the other.
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benorin
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I would like to determine necessary and sufficient conditions for equality to hold in Minkowski's Inequality in [itex]L^{p}(X,\mu)[/itex].

For [itex]1\leq p\leq \infty,[/itex] we have [itex]\forall f,g\in L^{p}(X,\mu)[/itex]

[tex]\left\{\int_X \left| f+g\right| ^{p} d\mu\right\} ^{\frac{1}{p}} \leq \left\{\int_X \left| f\right| ^{p} d\mu\right\} ^{\frac{1}{p}} + \left\{\int_X \left| g\right| ^{p} d\mu\right\} ^{\frac{1}{p}}[/tex]

here I wish to allow f and g to be complex. Any help would be nice.
 
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  • #2
So I think I have all the pieces, but I lack your brain power.

Minkowski's Inequality is proved (in Rudin's text) via Holder's Inequality, namely

For [itex]\frac{1}{p}+\frac{1}{q}=1[/itex], [itex]1< p< \infty,1<q< \infty,[/itex], and let f and g be measurable functions on X with range in [itex]\left[ 0, \infty\right] [/itex]. Then

[tex]\left\{\int_X fg d\mu\right\}\leq \left\{\int_X f^{p} d\mu\right\} ^{\frac{1}{p}}\left\{\int_X g^{q} d\mu\right\} ^{\frac{1}{q}}[/tex]

To prove Minkowski's put [itex](f+g)^{p}=f(f+g)^{p-1}+g(f+g)^{p-1}[/itex] and apply Holder's inequality to each of the terms on the right, that is

[tex]\left\{\int_X f(f+g)^{p-1} d\mu\right\}\leq \left\{\int_X f^{p} d\mu\right\} ^{\frac{1}{p}}\left\{\int_X (f+g)^{(p-1)q} d\mu\right\} ^{\frac{1}{q}}[/tex]

and

[tex]\left\{\int_X g(f+g)^{p-1} d\mu\right\}\leq \left\{\int_X g^{p} d\mu\right\} ^{\frac{1}{p}}\left\{\int_X (f+g)^{(p-1)q} d\mu\right\} ^{\frac{1}{q}}[/tex]

note that (p-1)q=p and that 1/q=1-1/p then add the above inequalities to get Minkowski's inequality in the following form:

[tex]\left\{\int_X \left( f+g\right) ^{p} d\mu\right\} ^{\frac{1}{p}} \leq \left\{\int_X f^{p} d\mu\right\} ^{\frac{1}{p}} + \left\{\int_X g^{p} d\mu\right\} ^{\frac{1}{p}}[/tex]

for non-negative f and g. Equality holds in Holder's inequality if, and only if, there are constants a and b such that [itex]af^p=bg^q[/itex].

To arrive at the form of Minkowski's inequality I have posted above, note that

[tex]\left| f + g \right| \leq \left| f \right| + \left| g \right| \Rightarrow \left| f + g \right| ^{p} \leq \left( \left| f \right| + \left| g \right| \right) ^{p}[/tex]

by the triangle inequality. For complex f and g, their magnitudes |f| and |g| are non-negative and measurable if f and g are (so it is ok to generalize as such.) Based on the conditions for equality to hold in Holder's Inequality, what are the conditions for equality to hold in Minkowski's Ineqaulity for complex f and g?
 
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  • #3
when one function is a constant multiple of the other, ie f = ag or somethng like that
 

What is Minkowski's Inequality?

Minkowski's Inequality is a mathematical concept that states the sum of the pth powers of two real numbers is greater than or equal to the pth power of the sum of the numbers, for any value of p greater than or equal to 1. This inequality is commonly used in functional analysis and measure theory.

What is the significance of "Equality holds" in Minkowski's Inequality?

When we say "equality holds" in Minkowski's Inequality, it means that both sides of the inequality are equal. This occurs when the two numbers being compared have the same value, or when one of the numbers is zero.

When does equality hold in Minkowski's Inequality?

Equality holds in Minkowski's Inequality when the two numbers being compared have the same value, or when one of the numbers is zero. This can be represented mathematically as |a + b| = |a| + |b|, where a and b are real numbers.

What is the relationship between Minkowski's Inequality and the triangle inequality?

Minkowski's Inequality is closely related to the triangle inequality, which states that the sum of the lengths of any two sides of a triangle must be greater than or equal to the length of the third side. In fact, Minkowski's Inequality is sometimes referred to as the "generalized triangle inequality" as it extends this concept to higher dimensions.

How is Minkowski's Inequality used in real-world applications?

Minkowski's Inequality has many practical applications in fields such as physics, engineering, and economics. It is commonly used to prove the existence and uniqueness of solutions in differential equations, to analyze the convergence of numerical methods, and to measure the accuracy of statistical models. It is also used in optimization problems to find the shortest paths between points and in image processing to measure the similarity between images.

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