Inequality 57 (George Basdekis)

Problem:

If x_1,x_2,\ldots, x_n are non-negative real numbers with sum equal to 1 then find the maximum value of the expression

\displaystyle P(x_1,x_2,\ldots,x_n)=x_1^4x_2^2+x_2^4x_3^2+\cdots+x_n^4x_1^2+n^{3(n-1)}x_1^4x_2^4\cdots x_n^4.

Solution:

First we may, without loss of generality, assume that x_1=\max\{x_1,x_2,\ldots,x_n\}. Now, we shall prove that

\displaystyle P(x_1,x_2,\ldots,x_n)\leq P(x_1,x_2+x_3+\cdots+x_n,0,0,\ldots,0).

It holds that

\displaystyle \begin{aligned} P(x_1&,x_2+x_3+\cdots+x_n,0,0,\ldots,0)=x_1^4(x_2+x_3+\cdots+x_n)^2\geq\\&\geq 2x_1^4(x_2x_3+x_3x_4+\cdots+x_nx_1)+x_1^4(x_2^2+x_n^2)\\&\geq (x_1^4x_2x_3+x_1^4x_3x_4+\cdots+x_1^4x_nx_1)+(x_2^4x_3^2+\cdots+x_{n-1}^4x_n^2)+x_1^4x_2^2+x_n^4x_1^2\\&\geq x_1^4x_2x_3+(x_1^4x_2^2+x_1^4x_3^2+x_2^4x_3^2+\cdots+x_n^4x_1^2).\end{aligned}

Therefore, we only need to prove that

\displaystyle x_1^4x_2x_3\geq n^{3(n-1)}x_1^4x_2^4\cdots x_n^4.

For n=3 we get

\displaystyle x_1^4x_2x_3\geq 2^6x_1^4x_2^4x_3^4

or equivalently the above Inequality reduces to the \displaystyle x_2^3x_3^3\leq\frac{1}{2^6}. But this one holds due to the AM-GM Inequality since we have

\displaystyle x_2^3x_3^3\leq\left[\frac{x_2+x_3}{2}\right]^6\leq\left[\frac{x_1+x_2+x_3}{2}\right]^6=\frac{1}{2^6}.

Now, for n>3 on the other side, we only need to show that

\displaystyle n^{3(n-1)}x_1^4x_2^4\cdots x_n^4\leq x_1^4x_2x_3

or

\displaystyle x_2^3x_3^3x_4^4\cdots x_n^4\leq\frac{1}{n^{3(n-1)}}.

But, again, from the AM-GM Inequality we acquire

\displaystyle \begin{aligned}x_2^3x_3^3x_4^4\cdots x_n^4\leq (x_2x_3\cdots x_n)^3&\leq\left[\frac{x_2+x_3+\cdots+x_n}{n-1}\right]^{3(n-1)}\\&\leq\left[\frac{x_1+x_2+\cdots+x_n}{n-1}\right]^{3(n-1)}\\&=\frac{1}{n^{3(n-1)}}.\end{aligned}

So, finally, we are left to find the maximum value of the expression

\displaystyle P(x_1,x_2+x_3+\cdots+x_n,0,0,\ldots,0)=x_1^4(1-x_1)^2.

We have according to the AM-GM Inequality

\displaystyle \begin{aligned} x_1^4(1-x_1)^2&=4^4\cdot 2^2\cdot\frac{x_1}{4}\cdot\frac{x_1}{4}\cdot\frac{x_1}{4}\cdot\frac{x_1}{4}\cdot\frac{1-x_1}{2}\cdot\frac{1-x_1}{2}\\&\leq 4^4\cdot 2^2\cdot\left[\frac{4\cdot\frac{x_1}{4}+2\cdot\frac{1-x_1}{2}}{4+2}\right]^6\\&=\frac{16}{729}.\end{aligned}

Thus, the maximum value of the expression P is equal to 16/729 and it is obtained if and only if x_1=2/3,\, x_2=1/3,\, x_3=x_4=\cdots=x_n=0 Q.E.D.

Inequality 56 (George Basdekis)

Problem:

Let a,b,c be positive real numbers. Prove the Inequality

\displaystyle \frac{a\sqrt{a}}{2a+b}+\frac{b\sqrt{b}}{2b+c}+\frac{c\sqrt{c}}{2c+a}\leq\frac{(a+b+c)\sqrt{a+b+c}}{3\sqrt{3}\sqrt[3]{abc}}.

Solution:

First, we observe that from the AM-GM Inequality we have that

\displaystyle 3a(a+2b)\leq\left[\frac{4a+2b}{2}\right]^2=(2a+b)^2

or equivalently

\displaystyle \frac{3a}{(2a+b)^2}\leq\frac{1}{a+2b}\Leftrightarrow \frac{3a^3}{(2a+b)^2}\leq\frac{a^2}{a+2b}

and now taking square roots for both sides, we see that

\displaystyle \sqrt{3}\cdot\frac{a\sqrt{a}}{2a+b}\leq\frac{a}{\sqrt{a+2b}}.

Therefore, we acquire

\displaystyle \sqrt{3}\sum_{cyc}\frac{a\sqrt{a}}{2a+b}\leq\sum_{cyc}\frac{a}{\sqrt{a+2b}}.

Now, from the Cauchy-Schwarz Inequality we have for the right hand side

\displaystyle \begin{aligned}\left(\sum_{cyc}\frac{a}{\sqrt{a+2b}}\right)^2&\leq\sum_{cyc}[a(a+2c)]\sum_{cyc}\frac{a}{(a+2b)(a+2c)}\\&=(a+b+c)^2\sum_{cyc}\frac{a}{(a+2b)(a+2c)}.\end{aligned}

Moreover, once again, from the AM-GM Inequality we have for the denominators of the above sum that

\displaystyle (a+2b)(a+2c)\geq 9\sqrt[3]{a^2b^2c^2}

and thus, it will hold that

\displaystyle \begin{aligned}(a+b+c)^2\sum_{cyc}\frac{a}{(a+2b)(a+2c)}&\leq (a+b+c)^2\sum_{cyc}\frac{a}{9\sqrt[3]{a^2b^2c^2}}\\&=\frac{(a+b+c)^3}{9\sqrt[3]{a^2b^2c^2}}.\end{aligned}

Finally, we have proved that

\displaystyle \left(\sum_{cyc}\frac{a}{\sqrt{a+2b}}\right)^2\leq\frac{(a+b+c)^3}{9\sqrt[3]{a^2b^2c^2}}\Leftrightarrow \sum_{cyc}\frac{a}{\sqrt{a+2b}}\leq\frac{(a+b+c)\sqrt{a+b+c}}{3\sqrt[3]{abc}}.

The Inequality we have proved is

\displaystyle \sqrt{3}\cdot\sum_{cyc}\frac{a\sqrt{a}}{2a+b}\leq\frac{(a+b+c)\sqrt{a+b+c}}{3\sqrt[3]{abc}}

which also is the Inequality we are given to prove. The proof is completed Q.E.D.

Inequality 55 (Unknown Author)

Problem:

If x,y are positive real numbers then prove that

\displaystyle x^y+y^x>1.

Solution:

The Inequality is obvious for x,y\geq 1. So, let’s consider the case where x,y\in (0,1). We will now divide the Inequality into two cases.

  • Case 1st: x^{y-1}>y^{x-1}. Then it holds that

\displaystyle (x^y+y^x)^{1/x}=\left[y^x\left(\frac{x^y}{y^x}+1\right)\right]^{1/x}=y\cdot\left[\frac{x^y}{y^x}+1\right]^{1/x}

therefore, according to Bernulli’s Inequality it will hold that

\displaystyle \begin{aligned}y\cdot\left[\frac{x^y}{y^x}+1\right]^{1/x}&\geq y\left(1+\frac{x^{y-1}}{y^x}\right)\\&\geq y\left(1+\frac{y^{x-1}}{y^x}\right)\\&=y+1>1.\end{aligned}

  • Case 2nd: y^{x-1}>x^{y-1}. Then it is

\displaystyle (x^y+y^x)^{1/y}=\left[x^y\left(\frac{y^x}{x^y}+1\right)\right]^{1/y}=x\cdot\left[\frac{y^x}{x^y}+1\right]^{1/y}

therefore, once again from Bernulli’s Inequality we get

\displaystyle \begin{aligned}x\cdot\left[\frac{y^x}{x^y}+1\right]^{1/y}&\geq x\left(1+\frac{y^{x-1}}{x^y}\right)\\&\geq x\left(1+\frac{x^{y-1}}{x^y}\right)\\&=x+1>1\end{aligned}

and the proof is completed Q.E.D.

Inequality 54 (George Basdekis)

Problem:

Let a,b and c be real numbers. Prove the Inequality

\displaystyle\frac{1}{(a-b)^2}+\frac{1}{(b-c)^2}+\frac{1}{(c-a)^2}\geq\frac{9}{2(a^2+b^2+c^2)}.

Solution:

Without loss of generality, let us assume that a>b>c. Then according to the AM-GM Inequality we have that

\displaystyle\begin{aligned}\frac{1}{(a-b)^2}+\frac{1}{(b-c)^2}\geq\frac{2}{(a-b)(b-c)}&\geq\frac{2\cdot 4}{[(a-b)+(b-c)]^2}\\&=\frac{8}{(a-c)^2}.\end{aligned}

Therefore, it holds that

\displaystyle\frac{1}{(a-b)^2}+\frac{1}{(b-c)^2}+\frac{1}{(c-a)^2}\geq\frac{9}{(c-a)^2}.

It remains now to prove that

\displaystyle \frac{9}{(c-a)^2}\geq\frac{9}{2(a^2+b^2+c^2)},

which reduces to the obvious, after expansions, Inequality (a+c)^2+2b^2\geq 0. Equality holds if and only if (a,b,c)\sim (-b,b,0) or any cyclic permutations of the previous equality Q.E.D.

Inequality 53 (Vo Quoc Ba Can)

Problem:

Let \displaystyle a,b,c>0 such that \displaystyle a+b+c=3 and \displaystyle a^2+b^2+c^2=4. Find the minimum and the maximum value of the expression

\displaystyle Q=\frac{a}{b}.

Solution:

From the Cauchy – Schwarz Inequality we have that

\displaystyle (a^2+b^2+c^2)\cdot\left[\frac{(a+b)^2}{a^2+b^2}+1\right]\geq (a+b+c)^2,

or due to the hypothesis

\displaystyle \frac{(a+b)^2}{a^2+b^2}\geq \frac{5}{4}.

From this last Inequality, we acquire

\displaystyle a^2+b^2\leq 8ab,

Let us now divide with \displaystyle ab. Then we get that

\displaystyle \frac{a}{b}+\frac{b}{a}\leq 8.

Solving this Inequality, gives us

\displaystyle 4-\sqrt{15}\leq \frac{a}{b}\leq 4+\sqrt{15}.

Thus, \displaystyle Q_{\min}=4-\sqrt{15} and \displaystyle Q_{\max}=4+\sqrt{15}, Q.E.D.

Inequality 52 (Unknown Author)

Problem:

Let \displaystyle x_1,x_2,...,x_n be positive real numbers with sum equal to \displaystyle 1. Find the minimum value of the expression

\displaystyle A=\max\left\{\frac{x_1}{1+x_1},\frac{x_2}{1+x_1+x_2},...,\frac{x_n}{1+x_1+x_2+...+x_n}\right\}.

Solution:

From the definition of \displaystyle A, we have that

\displaystyle A\geq \frac{x_1}{1+x_1}, A\geq \frac{x_2}{1+x_1+x_2},...,A\geq \frac{x_n}{1+x_1+x_2+...+x_n}.

Solving the 1st Inequality we get that

\displaystyle x_1\leq \frac{A}{1-A}.

Solving now the 2nd Inequality we have that

\displaystyle A\geq \frac{x_2}{1+x_1+x_2}\geq \frac{x_2}{1+\frac{A}{1-A}+x_2}=\frac{x_2}{\frac{1}{1-A}+x_2}.

The last relation reduces to the

\displaystyle x_2\leq \frac{A}{(1-A)^2}.

So, we see now that \displaystyle x_k\leq \frac{A}{(1-A)^k} for \displaystyle k=1,2,...,n.

Summing up these \displaystyle n Inequalities we acquire

\displaystyle x_1+x_2+...+x_n\leq \frac{A}{1-A}+\frac{A}{(1-A)^2}+...+\frac{A}{(1-A)^n}

or, due to the hypothesis we now have that

\displaystyle \frac{A}{1-A}+\frac{A}{(1-A)^2}+...+\frac{A}{(1-A)^n}\geq 1.

If we solve this last Inequality it gives us the result \displaystyle A\geq 1-\frac{1}{\sqrt[n]{2}}, which is also and the value we are searching, Q.E.D.