Find $f$ on $[1,a]$ such that $f \ge x\log x$, $f$ is strictly convex, analytic, and touches $x \log x$ only $n$ times.

Question

I am looking for a function $f$ defined on $[1,a]$ that satisfies the following:

1. $f(x) \ge x \log x$ on $[1,a]$
2. $f(x)$ is strictly convex
3. $f(x)$ is analytic on $[1,a]$.
4. $f(x)-x\log(x)=0$ only $n$ times on $[1,a]$.

My attempt was to construct $f$ as follows:

$f(x)=x\log(x)+g(x)$

where $g(x)$ would have oscillatory behavior (e.g., $\cos(x)$). However, every time I try to add oscillations I violate strict convexity condition.

Answer 1

Example, $n=3$ on the interval $[1,3]$,

$$ f(x) = x\log x+\frac{(x-1)^2(x-2)^2(x-3)^2}{10} $$

It is analytic on $(0,+\infty)$.

Here is $f(x)$ on $[1,3]$

Here is $f(x) - x\log x$. It is of course nonnegative and vanishes only at $1,2,3$.

Here is $f''(x)$. It is positive, so $f$ is strictly convex.

The coefficient $1/10$ was chosen so that $f''(x)$ is strictly positive.

If we consider $f(x)= x\log x+k(x-1)^2(x-2)^2(x-3)^2$ for $k>0$, then $f''(x)$ converges uniformly to $1/x$ as $k \to 0$, so of course we can choose $k$ so close to zero that $f''(x)$ is positive on $[1,3]$.