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Young's inequality at $p = q = 2$ and at $p = 3$ , $q = 3 / 2$

Example

(i) At the conjugate pair $p = q = 2$ , Young's inequality for products (rational conjugate exponents) reads $a b \leq \frac{a^{2}}{2} + \frac{b^{2}}{2}$ for all $a, b \geq 0$ , with equality exactly when $a = b$ ; the instance $a : = 4$ , $b : = 9$ gives $36 \leq 97 / 2$ .

(ii) At $p : = 3$ and $q : = 3 / 2$ , conjugate since $\frac{1}{3} + \frac{2}{3} = 1$ , it reads $a b \leq \frac{a^{3}}{3} + \frac{2}{3} b^{3 / 2}$ ; the instance $a : = 2$ , $b : = 4$ gives $8$ on both sides.

Facts & Assumptions

Given: Reals $a, b \geq 0$ for the general statements, the instance $a = 4$ , $b = 9$ for (i), and the instance $a = 2$ , $b = 4$ for (ii).

[L1]

Young's inequality (Young's inequality for products (rational conjugate exponents)): for conjugate rationals $p, q > 1$ , that is $\frac{1}{p} + \frac{1}{q} = 1$ , and all reals $a, b \geq 0$ , $a b \leq \frac{a^{p}}{p} + \frac{b^{q}}{q}$ , with the rational powers of Rational powers $a^{r}$ of a positive base and its supplementary clause $0^{p} = 0$ for $p > 0$ .

[L2]

AM-GM (The arithmetic mean, geometric mean inequality): for $n = 2$ and nonnegative entries $u, v$ , $\sqrt{u v} \leq (u + v) / 2$ , equivalently $u v \leq ((u + v) / 2)^{2}$ , with equality exactly when $u = v$ . On positive entries, Weighted AM-GM inequality with rational weights with weights $1 / 2, 1 / 2$ gives the same inequality; its statement does not include an equality clause or zero bases.

[L3]

Roots and powers (Existence and uniqueness of $n$ -th roots: a unique $a^{1 / n} \geq 0$ with $(a^{1 / n})^{n} = a$ , Rational powers $a^{r}$ of a positive base, Laws of rational exponents): for $c \geq 0$ , $c^{1 / 2}$ is the unique nonnegative $s$ with $s^{2} = c$ ; $u^{2} v^{2} = (u v)^{2}$ by the product law; and $(4^{1 / 2})^{3} = 4^{3 / 2} = 8$ since $4^{1 / 2} = 2$ and $2^{3} = 8$ .

[L4]

Monotonicity of squaring (Squaring is monotone on the nonnegatives): for $u, v \geq 0$ , $u \leq v$ holds exactly when $u^{2} \leq v^{2}$ ; in particular $a^{2} = b^{2}$ with $a, b \geq 0$ gives $a = b$ .

[L5]

Integer powers (Laws of integer exponents): $a^{3} = a^{2} a$ and $2^{3} = 8$ , $9^{2} = 81$ , $4^{2} = 16$ by evaluation.

Verification

technique · direct

1.1L1algebra

Part (i), the inequality: $p = q = 2$ are conjugate since $\frac{1}{2} + \frac{1}{2} = 1$ , so [L1] gives $a b \leq \frac{a^{2}}{2} + \frac{b^{2}}{2}$ for all $a, b \geq 0$ .

1.2L2L3L4

Part (i), the equality clause, by AM-GM: for $a, b \geq 0$ the entries $a^{2}, b^{2}$ are nonnegative, their geometric mean is $(a^{2} b^{2})^{1 / 2} = a b$ , because $a b \geq 0$ and $(a b)^{2} = a^{2} b^{2}$ by [L3] make $a b$ the unique nonnegative square root of $a^{2} b^{2}$ , and their arithmetic mean is $(a^{2} + b^{2}) / 2$ ; [L2] gives $(a^{2} + b^{2}) / 2 \geq a b$ with equality exactly when $a^{2} = b^{2}$ , which for $a, b \geq 0$ holds exactly when $a = b$ by [L4]; so equality in (i) is exactly $a = b$ .

1.3L1L3L5algebra

Part (ii), the inequality and its instance: $\frac{1}{3} + \frac{2}{3} = 1$ by rational arithmetic, so [L1] applies at $p = 3$ , $q = 3 / 2$ ; at $a = 2$ , $b = 4$ the first term is $a^{3} / 3 = 8 / 3$ and the second is $(2 / 3) b^{3 / 2} = (2 / 3) (4^{1 / 2})^{3} = (2 / 3) \cdot 8 = 16 / 3$ by [L3] and [L5], and the right-hand sum is $8 / 3 + 16 / 3 = 8 = a b$ , so both sides are equal at this instance.

2.1step 1.2L5algebra

Part (i), the instance: $a = 4$ , $b = 9$ give left-hand side $4 \cdot 9 = 36$ and right-hand side $(16 + 81) / 2 = 97 / 2$ , with $36 = 72 / 2 \leq 97 / 2$ , and $4 \neq 9$ so the inequality is strict, consistent with the equality clause of step 1.2.

3.1step 1.1step 1.2step 2.1step 1.3L1L2∎

Summary: (i) holds with equality exactly at $a = b$ (steps 1.1, 1.2) and the instance $4, 9$ is strict (step 2.1); (ii) holds at the conjugate pair $3, 3 / 2$ and its instance $2, 4$ has equality on both sides (step 1.3); the equality clause proved in (i) is a property of that instance of [L1] via AM-GM, not a claim about the general Young inequality.

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Sources

John K. Hunter, An Introduction to Real Analysis (author-hosted complete lecture notes, University of California at Davis) (standard reference, not scraped)

Young's inequality at p=q=2p=q=2 and at p=3p=3, q=3/2q=3/2