Intereting Posts

A vector space V is an irreducible End(V)-module
Show that holomorphic $f_1, . . . , f_n $ are constant if $\sum_{k=1}^n \left| f_k(z) \right|$ is constant.
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Find the sum of all the multiples of 3 or 5 below 1000
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Number of distinct numbers picked after $k$ rounds of picking numbers with repetition from $$
Infinite Prime Numbers: With Fermat Numbers
Why isn't there interest in nontrivial, nondiscrete topologies on finite groups?
Combinatorial interpretation of sum of squares, cubes
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Books for starting with analysis
What does Universal mapping property for a free monoid mean?

I’m interested why this is true:

$$ \text{Considering }\forall (x,y,z) \in (1,\infty) $$

The following holds:

- Method to eliminate $x$ between the equation $x^2 + ax + b = 0$ and $xy+ l(x + y) + m = 0$
- Equivalence of geometric and algebraic definitions of conic sections
- How to prove $\cos 36^{\circ} = (1+ \sqrt 5)/4$?
- Why does cancelling change this equation?
- How do you factor a quadratic expression, without using the formula?
- How to solve $x + \sqrt{x + \sqrt{11+x}}=11$ algebraically?

$$\log_xy^z+\log_x{z^y}+log_y{z^x} \geq \frac{3}{2}$$

This is taken from a high school textbook of mine. I tried finding a meaningful manipulation by using AM-GM, but that got pretty messy. I’d like to avoid Lagrange multipliers since this is meant to be a pretty basic problem.

Any progress would be appreciated.

- Strategies to denest nested radicals.
- Are these solutions of $2 = x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}}$ correct?
- Polynomial uniqueness proof
- An oddity in some linear equations
- Ramsey Number Inequality: $R(\underbrace{3,3,…,3,3}_{k+1}) \le (k+1)(R(\underbrace{3,3,…3}_k)-1)+2$
- Non-probabilistic proofs of a binomial coefficient identity from a probability question
- How to compute $\prod\limits^{\infty}_{n=2} \frac{n^3-1}{n^3+1}$
- What are functions used for?
- $a,b,c$ are positive reals and distinct with $a^2+b^2 -ab=c^2$. Prove $(a-c)(b-c)<0$
- Compound interest formula with regular deposits, solve for time

CW answer to remove it from unanswered queue:

For $(x,y,z)=(1.1, 1.01, 1.001)$, the expression is clearly $\simeq 0.22<\frac{3}{2}$, contradicting the inequality at hand.

**Hint**

Set

$$f(x,y,z)=z\frac{\ln y}{\ln x}+y\frac{\ln z}{\ln x}+x\frac{\ln z}{\ln y} $$

look at the stationary point and conclude.

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- Proof of Stokes' Theorem in $\mathbb{R}^n$
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- Proof that classifying spaces for discrete groups are the Eilenberg-MacLane spaces