Inequalities
7.0 Tchebychef's Inequality
7.0 Tchebychef's Inequality
If ${a_1},{a_2},...,{a_n}$ and ${b_1},{b_2},...,{b_n}$ are any real numbers such that
(i) If ${a_1} \leqslant {a_2} \leqslant ... \leqslant {a_n}{\text{ and }}{b_1} \leqslant {b_2} \leqslant ... \leqslant {b_n}$, then $$({a_1} + {a_2} + ... + {a_n}{\text{)(}}{b_1} + {b_2} + ... + {b_n}) \leqslant n({a_1}{b_1} + {a_2}{b_2} + ... + {a_n}{b_n})$$
(ii) If ${a_1}{a_2}...{a_n}{\text{ and }}{b_1} \geqslant {b_2} \geqslant ... \geqslant {b_n}$, then $$\begin{equation} \begin{aligned} ({a_1} + {a_2} + ... + {a_n}{\text{)(}}{b_1} + {b_2} + ... + {b_n}) \geqslant n({a_1}{b_1} + {a_2}{b_2} + ... + {a_n}{b_n}) \\ \\\end{aligned} \end{equation} $$
Question 8. If $a$, $b$, $c$ and $d$ are positive real numbers, prove that $$({a^5} + {b^5} + {c^5} + {d^5}) \geqslant abcd(a + b + c + d)$$
Solution: Assume $a<b<c<d$, then $${a^4} < {b^4} < {c^4} < {d^4}$$
Now, apply Tchebychef's Inequality for two sets $a,b,c,d{\text{ and }}{a^4},{b^4},{c^4},{d^4}$, we get $$\begin{equation} \begin{aligned} (a + b + c + d)({a^4} + {b^4} + {c^4} + {d^4})4(a.{a^4} + b.{b^4} + c.{c^4} + d.{d^4}) \\ ({a^5} + {b^5} + {c^5} + {d^5})\frac{{(a + b + c + d)({a^4} + {b^4} + {c^4} + {d^4})}}{4}\quad ...(1) \\\end{aligned} \end{equation} $$
Since, A.M.$ \geqslant $G.M. $$\begin{equation} \begin{aligned} \frac{{{a^4} + {b^4} + {c^4} + {d^4}}}{4} \geqslant {({a^4}{b^4}{c^4}{d^4})^{\frac{1}{4}}} \\ \frac{{{a^4} + {b^4} + {c^4} + {d^4}}}{4} \geqslant abcd \\ (a + b + c + d)\left( {\frac{{{a^4} + {b^4} + {c^4} + {d^4}}}{4}} \right) \geqslant abcd(a + b + c + d)\quad ...(2) \\ \\\end{aligned} \end{equation} $$
From $(1)$ and $(2)$, $$({a^5} + {b^5} + {c^5} + {d^5}) \geqslant abcd(a + b + c + d)$$
