math.CA — Classical Analysis and ODEs

Monotonicity of Turán-ratios of symmetric polynomials

Contributed by Suvrit Sra

We study the Turán-ratio $S_{\lambda,i}(x)^2/(S_{\lambda,i-1}(x)S_{\lambda,i+1}(x))$, where $S_{\lambda,i}(x):=s_\lambda(1^{[n-i]},x^{[i]})$, and $s_\lambda$ is the Schur polynomial in $n$ variables (indexed by partition $\lambda$) with $i$ arguments equal to $x$ and $n-i$ of them equal to $1$. Specifically, we prove that this Turán-ratio is strictly increasing for $x>1$ and $1\le i<n$ (unless all $n$ padded parts of $\lambda$ are equal, then it is identically one). For one-row shapes, our result answers a question of Ait-Haddou about complete homogeneous symmetric polynomials. For this case, we also offer a second proof through the zeros of Meixner polynomials that gives an explicit positive formula for the logarithmic derivative. We conclude with a Turán-ratio conjecture for Macdonald polynomials.