# When is alternating sum $\sum_{i}f(a_i)-\sum_{i<j}f(a_i+a_j)+\ldots+(-1)^{n-1}f(a_1+\ldots+a_n)$ always positive?

Let $a_1,a_2,\ldots,a_n\geq 1$, and let $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$. Consider the sum

$$S(f)=\sum_{i}f(a_i)-\sum_{i<j}f(a_i+a_j)+\sum_{i<j<k}f(a_i+a_j+a_k)-\ldots+(-1)^{n-1}f(a_1+\ldots+a_n).$$

This question shows that if $f(x)=\frac1x$, then $S(f)>0$ for all $a_1,\ldots,a_n$. If we perturb $f$ a tiny bit, say $f(x)=\frac{1}{x}-\frac{1}{100x^{100}}$, I would imagine that $S(f)>0$ still always holds. But the proof method for $f(x)=\frac1x$ is hard to generalize to other functions. Can we prove it in some other way?

More generally, is there a theorem out there stating sufficient conditions under which $S(f)>0$ always holds?

• The proof for $1/x$ may be directly generalized to Laplace transforms of non-negative measures supported on non-negative reals: $f(a)=\int e^{-at} d\mu(t)$ – Fedor Petrov Jun 30 '15 at 12:22
• Is this inspired by the recent Popoviciu question mathoverflow.net/questions/210350 ? I remember seeing some criteria involving $n$-th derivatives, but I'm not sure if I still can find them. – darij grinberg Jun 30 '15 at 15:16
• I think your condition is what is called Condition $\left(C_{n,n,n-1}\right)$ in Corollary 6.12 of Pecaric, Proschan, Tong, Convex Functions, Partial Orderings, and Statistical Applications ( books.google.de/… ). I am not fully sure, though, since I might be misreading the $\cdots$ in the formula. – darij grinberg Jun 30 '15 at 15:21
• I expect as Darij that it could be inspired, and so, thanks a lot nan. I had no idea that this actually holds for $x \mapsto \frac{1}{x}$. Actually the reason why I've asked about Popoviciu generalization, is because I need to prove the following inequality mathoverflow.net/questions/210236/… and its generalization for arbitrary number of elements. And since the function I use is basically a fraction, then the above fact seems to be most likely applicable to my problem. – Marek Adamczyk Jul 3 '15 at 20:55
• This paper: unix.cc.wmich.edu/~ledyaev/Spring2013/sendov.pdf studies such functions (and the CM case as noted by Fedor is discussed, as are additional properties such as convexity, harmonic convexity, etc.) – Suvrit Apr 2 '17 at 3:31