Question: This question concerns a seemingly folk lore result, which states that automorphism groups of generic complete intersections are trivial, under certain assumptions.

To state the question, let $r \geq 1$ and let $d\_1,\ldots,d\_r, n \in \mathbb{N}$ be such that

1. $n \geq 3$.
2. $d\_i \geq 2$ for each $i$.
3. $(d\_1,\ldots,d\_r;n) \neq (2;n)$ nor $(2,2;n)$.

> 
> Let $X$ be a generic smooth complete intersection of dimension $n$ in projective space $\mathbb{P}^{n+r}$ over $\mathbb{C}$ with equations of degrees $d\_1,\ldots,d\_r$. Is $\mathrm{Aut}(X)$ trivial?
> 
> 
> 

Remarks:

* The conditions (1), (2), (3) imply that $\mathrm{Aut}(X)$ is finite and preserves the hyperplane class, for each such smooth complete intersection $X$ (not necessarily generic).
* The result is not true without condition (3), as here the generic such complete intersection has a non-trivial automorphism group (hopefully I did not miss any other ''bad'' cases).
* The answer is well-known to be yes when $r=1$, i.e. for generic hypersurfaces.

I need a version of this result in a paper I am currently writing. We are able to use this to show that the action of $\mathrm{Aut}(X)$ on $H^n(X,\mathbb{C})$ is faithful for any such smooth complete intersection (not necessarily generic). I would be most interested if anyone has any ideas on alternative approaches to this problem as well.

We are able to handle many special cases, such as when the degrees $d\_i$ are all different or $X$ is of general type. A critical case is for example when all the degrees are the same and $X$ is Fano of high codimension.

Answer: 
a simple upper bound is $(\sqrt{a\_{\rm max}}-\sqrt{a\_{\rm min}})^2$, with $a\_{\rm max}$ and $a\_{\rm min}$ the largest and smallest of the $a\_i$'s. So for $a\_i=a\_1+(i-1)\varepsilon$ this would give as upper bound $(\sqrt{a\_1+(n-1)\varepsilon}-\sqrt{a\_1})^2$.

see theorem 1 of [Some Inequalities for Elementary Mean Values](http://www.ams.org/journals/mcom/1984-42-165/S0025-5718-1984-0725994-5/S0025-5718-1984-0725994-5.pdf), B. Meyker (1984).
Some results for differences $A\_n-G\_n, G\_n-H\_n$ and so after summing up for $A\_n-H\_n$ you may find in the book: Classical and new inequalities in analysis by D. S. Mitrinovic; J. E. Pecaric; A. M. Fink on pages 25,39.