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Hi. I have a question related to Turan`s problem, that is

Find a sequence of polynomial $P_n(x)$ satisfying $P_{n+1}(x)P_{n-1}(x) < P_{n}^2(x)$.

I am considering the generalized question for positive multivarible, i.e.

Let $k$ be a positive integer and let $x_1, ... x_k$ be k indeterminates. For $x_1, ... x_n > 0$, find a sequence of polynomial $P_n(x_1, ... , x_k)$ satisfying

$P_{n+1}(x_1, ... , x_k) P_{n-1}(x_1, ... , x_k) < A(n)P_n(x_1, ... , x_k)^2$ where

$A(n)$ is some fixed function for $n$.

Is there any result or some reference related to this problem?

In particular, I am interested in the case when $A(n) = \frac{n+2}{n+1}$. And I tried to check the above inequality by using maple by letting $P_n(x_1, ... , x_k) = x_1^n + ... + x_k^n$. then suprisingly(for me) I didn't find a counterexample until now, nor prove the inequality.

How can I prove (or find a counterexample) of this inequality? I really appriciate for your any comment and help.

Thank you in advance.

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  • $\begingroup$ P_n(x) = x - n ? Gerhard "Ask Me About System Design" Paseman, 2011.02.16 $\endgroup$ Feb 16 '11 at 8:10
  • $\begingroup$ And perhaps P_n(xbar) = Q(xbar) - n? Gerhard "The Power of Positive Generalization" Paseman, 2011.02.16 $\endgroup$ Feb 16 '11 at 8:14
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I am sorry. Ottem is right. $P_n$ is a polynomial of degree $n$. In particular, you may assume that $P_n$ is a symmetric function. Let's focus that $P_n(x_1, ... ,x_k) = x_1^n + ... x_k^n$, for any $k > 0 , n > 1$ given integers.

By rearrangement inequality, we can easily show that $ F(n) = \frac{P_n^2}{P_{n+1}P_{n-1} \leq 1.$ I want to know about the lower bound of $F(n)$ for each $n$.

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  • $\begingroup$ Since F(n) is homogeneous degree 0, F(n) always has a minimum point. If we take the gradient and set it to 0, we get a system of homogeneous polynomial equations(k equations and k variables), it seems that we should be able to solve it and thereafter find the minimum of F(n), but I've no idea of how to deal with such a system. $\endgroup$
    – Qingyun
    Apr 1 '11 at 18:14
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Let's try this in answer form. Let $Q = Q(x_1,\ldots,x_n)$ be a multivariate polynomial over some ordered ring containing the integers (with the usual ordering on the integers). Then define $P_n = Q - n$. Then

$(Q - (n+1))(Q - (n-1)) = Q^2 -2nQ + n^2 - 1 = (Q-n)^2 - 1$ .

This seems to satisfy your inequality, even with $A(n)= 1$ . Was there something else?

I would prefer to not find references nor do more work unless you can tell me more of what you know about the problem and more specifics on what you actually want.

Gerhard "Ask Me About System Design" Paseman, 2011.02.17

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  • $\begingroup$ Maybe the polynomials $P_n$ are meant to have degree $n$, otherwise the problem seems very trivial. $\endgroup$
    – J.C. Ottem
    Feb 18 '11 at 7:46
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  1. For the case of a single variable an obvious condition on polynomials is that a ratio $$ \frac{P_{n-1}(x)P_{n+1}(x)}{(P_n(x))^2} $$ is monotone. Then sharp estimates hold true with limits via $P_n(0),P_n(\infty)$ and so polynomial's coefficients.

  2. Consider also a reference http://arxiv.org/abs/1207.0936 with examples for sections of exponential series.

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