Which Orlicz functions $f$ make the function $f^{-1}\left(\frac{\sum_{j=1}^s f(x_j)}{s}\right)$ convex?

Question

Let $f:\mathbb{R}_+\to\mathbb{R}_+$ be an Orlicz function, or sometimes referred to as an Young function, i.e. it is a convex, non-decreasing function such that $f(0)=0$. I am trying to study the convexity of the function $\phi:\mathbb{R}^n\to\mathbb{R}$, such that $\phi(\mathbf{x})=f^{-1}\left(\frac{1}{s}\sum_{j=1}^s f(|{x}_j|)\right)$. Note that when $f(x)=|x|^p,\ p\ge 1$, then $\phi(\mathbf{x})=\frac{\|\mathbf{x}\|_p}{s^{1/p}}$, which is a convex function. Similarly, a bit of calculation shows that the function $\phi$ is also convex when $f(x)=e^{ax}-1$ for any $a>0$. So intuitively I thought that this result might hold for all Orlicz functions $f$. However, I have not been able to prove this result, and in fact, I think this does not hold for many functions, for example, $f(x)=e^{x^2}-1$. Then, I am left with the investigation of such Orlicz functions $f$, that make the corresponding $\phi$ convex. However, assuming that $f$ is double differentiable, the Hessian of $\phi$ is turning out to be too difficult to analyze. At this point, I am not sure how to proceed to find the properties of $f$ that make $\phi$ convex. Can anyone kindly suggest some ideas, or point me to some relevant references that can help me proceed in this investigation? Thanks in advance.

Answer 1

It follows from Hardy--Littlewood--Polya (see section 3.16, page 86), that if for $x>0$ we have $f, f', f''>0$, and $f'/f''$ is concave, and $d\mu$ is a probability measure on the probability space $\Omega$, then the functional

$$ h \mapsto f^{-1}\left(\int_{\Omega} f(h(\omega)) d\mu(\omega) \right) \qquad (*) $$ is convex on the set of nonnegative functions $h :\Omega \to \mathbb{R}_{+}$. In fact these is "if and only if" characterization (see the reference for the details).

Now to obtain the convexity that you are looking for, consider $\Omega = [0,1]$, and let $d\mu = dx$ be a standard Lebesgue measure on $[0,1]$. Next, partition $[0,1]$ into $s$ subintervals $I_{1}, \ldots, I_{s}$ of equal length $\frac{1}{s}$. Given $p=(p_{1}, \ldots, p_{s}) \in \mathbb{R}^{s}$, define $h_{p}(x) =|p_{j}|$ if $s \in I_{j}$. Next, let $q = (q_{1}, \ldots, q_{s})\in \mathbb{R}^{s}$. Then convexity of the functional (*) means $$ f^{-1}\left(\int_{0}^{1} f\left(\frac{h_{p}(x)+h_{q}(x)}{2}\right)dx \right) \leq \frac{1}{2}\left(f^{-1}\left(\int_{0}^{1} f\left(h_{p}(x)\right)dx \right)+f^{-1}\left(\int_{0}^{1} f\left(h_{q}(x)\right)dx \right) \right). $$ On the other hand we have $$ f^{-1}\left(\int_{0}^{1} f\left(h_{p}(x)\right)dx \right) = f^{-1}\left(\frac{1}{s}\sum_{1\leq j \leq s} f(|p_{j}|) \right)\\ f^{-1}\left(\int_{0}^{1} f\left(h_{q}(x)\right)dx \right) = f^{-1}\left(\frac{1}{s}\sum_{1\leq j \leq s} f(|q_{j}|) \right)\\ $$ and $$ f^{-1}\left(\int_{0}^{1} f\left(\frac{h_{p}(x)+h_{q}(x)}{2}\right)dx \right) = f^{-1}\left(\frac{1}{s}\sum_{1\leq j \leq s} f\left(\frac{|p_{j}|+|q_{j}|}{2}\right) \right) \geq f^{-1}\left(\frac{1}{s}\sum_{1\leq j \leq s} f\left(\frac{|p_{j}+q_{j}|}{2}\right) \right) $$ where the last inequality follows because $|a+b|\leq |a|+|b|$, and $f, f^{-1}$ are strictly increasing.

Hardy--Littlewood--Polya PDF