Hanner inequalities

The Hanner inequalities come from functional analysis and are inequalities for L p norms . They have some important consequences, including that the L p spaces are for uniformly convex spaces . ${\ displaystyle 1

They are named after the Swedish mathematician Olof Hanner .

statement

Be . If so , then ${\ displaystyle f, g \ in L ^ {p} (\ Omega, {\ mathcal {A}}, \ mu)}$${\ displaystyle 1 \ leq p \ leq 2}$

${\ displaystyle \ | f + g \ | _ {p} ^ {p} + \ | fg \ | _ {p} ^ {p} \ geq {\ big (} \ | f \ | _ {p} + \ | g \ | _ {p} {\ big)} ^ {p} + {\ big |} \ | f \ | _ {p} - \ | g \ | _ {p} {\ big |} ^ {p }}$

and

${\ displaystyle 2 ^ {p} {\ big (} \ | f \ | _ {p} ^ {p} + \ | g \ | _ {p} ^ {p} {\ big)} \ geq {\ big (} \ | f + g \ | _ {p} + \ | fg \ | _ {p} {\ big)} ^ {p} + {\ big |} \ | f + g \ | _ {p} - \ | fg \ | _ {p} {\ big |} ^ {p}}$.

If so , then the inequality symbols are reversed, that is, becomes off . ${\ displaystyle 2 \ leq p <\ infty}$${\ displaystyle \ geq}$${\ displaystyle \ leq}$

Explanations of the inequalities

The second inequality is obtained from the first by substituting and . Because then the left side becomes too ${\ displaystyle f = f + g}$${\ displaystyle g = fg}$${\ displaystyle \ | f + g \ | _ {p} ^ {p} + \ | fg \ | _ {p} ^ {p}}$

${\ displaystyle \ | (f + g) + (fg) \ | _ {p} ^ {p} + \ | (f + g) - (fg) \ | _ {p} ^ {p} = \ | 2f \ | _ {p} ^ {p} + \ | 2g \ | _ {p} ^ {p} = 2 ^ {p} (\ | f \ | _ {p} ^ {p} + \ | g \ | _ {p} ^ {p})}$

and the right side is reshaped similarly.

For the norm is induced by a scalar product . In this case, the inequalities become equations and are equivalent to the parallelogram equation . ${\ displaystyle p = 2}$

Individual evidence

1. C. Schütt: functional analysis , page 73. Accessed June 24, 2020.