All through ,i have oberved that (in mathlinks) the new inequalities that come up are alluring to think easy and with sufficient thought they exactly reverse our impressions. I observed that ,they mostly involved some well known expressions and had been devised to particularly make them look tough. Take for example the following inequality –
or this –
The first i did get the solution using schur’ and it turned out to be not as tough as it looks. The second was easy too..but i didnt get a solution when i tried it some months back. The point is, i wanted to make such type of inequalities too and find some elegant solutions for them (to make up for my nature to get faulty with huge expressions :P). With such notions i thought , why not go from the basics.
What is the most basic inequality ? Ofcourse the idea of SOS ought to be the most basic but it certainly isnt what a beginner would learn. So, AM-GM has to be the top of this list.
I considered the inequality –
Hmm…. are you thinking ,what i am thinking?
Lets assume the truth of the following inequality –
It should be noted that the first term is greater than two and the second term less than 2 which made the inequality *not obvious*.
I checked some values of $a,b$ and saw it to be true and it is easily noted that the equality occurs for $a=b$. In no time, i did come up with a proof for the inequality –
By AM-GM for four positive reals,
Hurray! the first such inequality of mine! 😀
I hope i get better and tougher results then such namesake easy things.