Show that for any positive real numbers $x$, $y$, $z$ such that $xy + yz + zx = 3$, we have that: $$x^2(y+z)+y^2(z+x)+z^2(x+y)+2\sqrt{xyz}\left(\sqrt{x^3+3x}+\sqrt{y^3+3y}+\sqrt{z^3+3z}\right) \ge 2xyz\left(x^2+y^2+z^2+6\right)$$
2026-03-26 09:49:45.1774518585
Inequality very very long
92 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail AtRelated Questions in INEQUALITY
- Confirmation of Proof: $\forall n \in \mathbb{N}, \ \pi (n) \geqslant \frac{\log n}{2\log 2}$
- Prove or disprove the following inequality
- Proving that: $||x|^{s/2}-|y|^{s/2}|\le 2|x-y|^{s/2}$
- Show that $x\longmapsto \int_{\mathbb R^n}\frac{f(y)}{|x-y|^{n-\alpha }}dy$ is integrable.
- Solution to a hard inequality
- Is every finite descending sequence in [0,1] in convex hull of certain points?
- Bound for difference between arithmetic and geometric mean
- multiplying the integrands in an inequality of integrals with same limits
- How to prove that $\pi^{e^{\pi^e}}<e^{\pi^{e^{\pi}}}$
- Proving a small inequality
Related Questions in CAUCHY-SCHWARZ-INEQUALITY
- optimization with strict inequality of variables
- Proving a small inequality
- Two Applications of Schwarz Inequality
- Prove $a^2+b^2+c^2\gt \frac {1}{2018}$ given $\left({3a + 28b + 35c}\right)\left({20a + 23b +33c}\right) = 1$
- Prove that $\frac{1}{\sqrt{ab+a+2}}+ \frac{1}{\sqrt{bc+b+2}}+ \frac{1}{\sqrt{ac+c+2}} \leq \frac{3}{2}$
- Prove that $a+b+c\le \frac {a^3}{bc} + \frac {b^3}{ac} + \frac {c^3}{ab}$
- Find the greatest and least values of $(\sin^{-1}x)^2+(\cos^{-1}x)^2$
- Inequality with $ab+bc+ca=3$
- Prove the next cyclic inequality
- How to prove this interesting inequality: $\frac{5x+3y+z}{5z+3y+x}+\frac{5y+3z+x}{5x+3z+y}+\frac{5z+3x+y}{5y+3x+z}\ge 3$?
Related Questions in SUM-OF-SQUARES-METHOD
- An algebraic inequality involving $\sum_{cyc} \frac1{(a+2b+3c)^2}$
- Prove the next cyclic inequality
- I have a inequality, I don't know where to start
- How can one prove that this polynomial is non-negative?
- How to prove expression greater than 0
- Homogeneous fourth degree inequality : $\sum x_i^2x_j^2 +6x_1x_2x_3x_4 \geq\sum x_ix_jx_k^2$
- Find minimum value of $\sum \frac {\sqrt a}{\sqrt b +\sqrt c-\sqrt a}$
- Triangle inequality $\frac{ab}{a^{2}+ b^{2}}+ \frac{bc}{b^{2}+ c^{2}}+ \frac{ca}{c^{2}+ a^{2}}\geq \frac{1}{2}+ \frac{2r}{R}$
- Prove this stronger inequality
- Express $x^4 + y^4 + x^2 + y^2$ as sum of squares of three polynomials in $x,y$
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?