Tight estimate for a log-linear inequality Given q>0 and p, how do we get a tight estimate for the smallest x such that xlog(x)+px>=q? (such an x always exists).
Tight estimate for a log-linear inequality
Given and , how do we get a tight estimate for the smallest such that ? (such an always exists).
Answer & Explanation
Consider the function
The first derivative
cancels at a single place and at this point the value of the function is . The second derivative being always positive, then this point corresponds to a minimum.
So, the function (which is defined for ) starts from , goes to a minimum and increases again. This means that the inequality
will hold for any larger than the root of
The solution of this equation is given by
where appears Lambert function.
Since , the solution is larger than and the first iterate of Newton method would be . But makes an overestimate of the solution (from Darboux theorem).
Since, in a comment, you tell that your problem is to find n such as
If ound in old notes of mine a similar problem that I adapted to your. Considering , I was able to find this simple correlation
which I established for
So, if you want to refine more, use this as an estimate from which you start Newton method using Stirling approximation for