lean_workbook_plus_13019

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Disproved status carries no information about that problem. Do not import this node or use it as a dependency.

Prove that\n\n

$$\sum_\text{cyc} \sqrt[4]{\frac{1}{\frac1a + \frac1b + \frac1c}} \leq \sqrt[16]{\frac{4^{15}}{81}} \sqrt[16]{a^4 + b^4 + c^4 + d^4}$$

\n\nor, by letting $x = \frac1a, y = \frac1b, z = \frac1c, t = \frac1d$ and raising both sides to the $16$ th power\n\n

$$\left( \sum_\text{cyc} \sqrt[4]{\frac{1}{x + y + z}} \right)^{16} \leq \frac{4^{15}}{81} \left( \sum_\text{cyc} \frac{1}{x^4} \right)$$

\n\nBy the Power-Mean inequality\n\n

$$\left( \sum_\text{cyc} \sqrt[4]{\frac{1}{x + y + z}} \right)^{16} \leq 4^{15} \left( \sum_\text{cyc} \frac{1}{(x + y + z)^4} \right)$$

\n\nSo it remains to prove that\n\n

$$81 \sum_\text{cyc} \frac{1}{(x + y + z)^4} \leq \sum_\text{cyc} \frac{1}{x^4}$$

\n\nIf we can prove that\n\n

$$\frac{243}{(x + y + z)^4} \leq \frac{1}{x^4} + \frac{1}{y^4} + \frac{1}{z^4} \tag{1}$$

\n\nthen we can arrive at the desired conclusion by summing similar inequalities. Indeed, $(1)$ is equivalent to\n\n

$$(x^4y^4 + y^4z^4 + z^4x^4)(x + y + z)^4 \geq 243x^4y^4z^4$$

\n\nbut the last one is true by $AM-GM$\n\n

$$(x^4y^4 + y^4z^4 + z^4x^4)(x + y + z)^4 \geq 3x^{\frac83}y^{\frac83}z^{\frac83} \cdot \left(3x^{\frac13}x^{\frac13}x^{\frac13} \right)^4 = 243x^4y^4z^4$$

\n\nso we are done.

Why this node was retired

The posted statement is

theorem lean_workbook_plus_13019 :
  ∀ x y z : ℝ, (x + y + z) ^ 4 ≠ 0 → 1 / (x + y + z) ^ 4 * 81 ≤ 1 / x ^ 4 + 1 / y ^ 4 + 1 / z ^ 4   :=  by sorry

Nonzero total x+y+z does not ensure nonzero positive individual reciprocals; the source AM–GM argument requires them.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

For x,y,z>0 prove the stronger243/(x+y+z)^4≤1/x^4+1/y^4+1/z^4; recover the four-variable cyclic conclusion by summing.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.

import Mathlib.Analysis.Complex.Basic

theorem lean_workbook_plus_13019 :
  ∀ x y z : ℝ, (x + y + z) ^ 4 ≠ 0 → 1 / (x + y + z) ^ 4 * 81 ≤ 1 / x ^ 4 + 1 / y ^ 4 + 1 / z ^ 4   :=  by sorry