Prove or disprove inequality $\frac{a^2}{b+c}+\frac{b^2}{a+c}+\frac{c^2}{a+b}\le\frac{a^4+b^4+c^4}{2abc}$.

$2abc\cdot \dfrac{a^2}{b+c} = 2a^3\cdot \dfrac{bc}{b+c} \leq 2a^3\cdot \dfrac{b+c}{4} = \dfrac{a^3(b+c)}{2}$. Thus:

$2abc\cdot \dfrac{a^2}{b+c} + 2abc\cdot \dfrac{b^2}{c+a} + 2abc\cdot \dfrac{c^2}{a+b} \leq \dfrac{(a^3b + ab^3) + (a^3c + ac^3) + (b^3c + bc^3)}{2} \leq \dfrac{(a^4 + b^4) + (b^4 + c^4) + (c^4 + a^4)}{2} = a^4 + b^4 + c^4$, and this is true because:

$x^3y + xy^3 \leq x^4 + y^4 \iff (x^3 - y^3)(x - y) \geq 0$ is true $\forall x,y \geq 0$

Tags:

Inequality

Real Analysis

Cauchy Schwarz Inequality

Algebra Precalculus

A.M. G.M. Inequality

Related

Let $|G|=p^nm$ where $p$ is a prime and $\gcd(p,m)=1$.Suppose that $H$ is a normal subgroup of $G$ of order $p^n$. For $x_{n+1}=x_n^2-2$, show $\lim_{n\to\infty}\frac{x_n}{x_0x_1\cdots x_{n-1}}=2$ If $h : Y \to X$ is a covering map and $Y$ is connected, then the cardinality of the fiber $h^{-1}(x)$ is independent of $x \in X$. Equivalent ideas of absolute continuity of measures A non-abelian group such that $G/Z(G)$ is abelian. Remarkable integral: $\int_0^{\infty} x \left(1 - \frac{\sinh x}{\cosh x-\sqrt 3/2} \right) = -\frac{13 \pi ^2}{72}$? Applications of Stein spaces in Algebraic Geometry Do line integrals along non-piecewise-smooth curves exist? The meaning of $\lambda$ in Lagrange Multipliers Show that an integer of the form $8k + 7$ cannot be written as the sum of three squares. Prove that if $A$ is normal, then eigenvectors corresponding to distinct eigenvalues are necessarily orthogonal (alternative proof) In what order does mathematics build upon itself?

Recent Posts