# An Inequality with a Generic Proof

### Problem

Dorin Marghidanu has kindly posted a problem with solution at the CutTheKnotMath facebook page. The method of solution points naturally to a generalization.

$x,y,z,a,b\ge 0,\;$ with $x+y+z=S,\;$ prove the inequality

$2\sqrt{b}+\sqrt{aS+b}\le \sqrt{ax+b}+ \sqrt{ay+b}+ \sqrt{az+b}\le \sqrt{3aS+9b}.$

### Lemma

$x,y,p\ge 0,\;$ with $x+y+z=S,\;$ the following inequality holds

$\sqrt{p}+\sqrt{p+x+y}\le \sqrt{p+x}+ \sqrt{p+y}.$

Indeed, by squaring, one obtains

$p+(p+x+y)+2\sqrt{p(p+x+y)}\le (p+x)+(p+y)+ 2\sqrt{(p+x)(p+y)}$

which is equivalent to $xy\ge 0.$ The equality only occurs when either $x=0\;$ or $y=0,\;$ i.e., for the pairs $(0,k)\;$ and $(k,0),\;$ with $k\ge 0.$

### Solution

We start with the left inequality:

$\begin{align} (\sqrt{ax+b}+ \sqrt{ay+b})+ \sqrt{az+b} &\ge (\sqrt{b}+ \sqrt{ax+ay+b})+ \sqrt{az+b}\\ &=\sqrt{b} +(\sqrt{ax+ay+b})+ \sqrt{az+b})\\ &\ge \sqrt{b}+ (\sqrt{b}+\sqrt{ax+ay+az+b})\\ &=2\sqrt{b}+\sqrt{aS+b}. \end{align}$

The right inequality follows in one step from either QM-AM inequality $\displaystyle\frac{u+v+w}{3}\le\sqrt{\frac{u^2+v^2+w^2}{3}},$ or, given that $t=\sqrt{s}\;$ is a concave function, from Jensen's inequality $\displaystyle\frac{f(u)+f(v)+f(w)}{3}\le f\left(\frac{u+v+w}{3}\right).$ Both ways lead to

$\displaystyle \sqrt{ax+b}+ \sqrt{ay+b}+ \sqrt{az+b}\le 3\sqrt{\frac{(ax+b)+(ay+b)+(az+b)}{3}}=\sqrt{3aS+9b}.$

The equality only holds when $\sqrt{ax+b}+ \sqrt{ay+b}+ \sqrt{az+b},\;$ i.e., when $x=y=z.$

### Generalization

The problem and the solution by Professor Marghidanu are of the remarkable kind where a formulation and a proof of a particular case reveal all the trappings of a general formulation and its proof. Problems like that help acclimate the student to working with general case by taking a few steps from one particular case to the next. Examples like that are known as

The general formulation is suggestive:

**References**

- U. Leron and O. Zaslavsky,
__Generic Proofs: Reflections on Scope and Method__, in The Best Writing on Mathematics 2014 (Mircea Pitici, ed.), Princeton University Press, 2014

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