On interrelation of Nikolskii Constants for trigonometric polynomials and entire functions of exponential type

For $0<p<\infty$, we investigate the interrelation between the Nikolskii constant for trigonometric polynomials of order at most $n$
$$ \mathcal{C}(n,p)=\sup_{T_{n}\ne 0}\frac{\|T_{n}\|_{\infty}}{\|T_{n}\|_{p}} $$
and the Nikolskii constant for entire functions of exponential type at most $1$
$$ \mathcal{L}(p)=\sup_{f\ne 0}\frac{\|f\|_{\infty}}{\|f\|_{p}}. $$

Recently E. Levin and D. Lubinsky have proved that
$$ \mathcal{C}(n,p)=\mathcal{L}(p)n^{1/p}(1+o(1)),\quad n\to \infty. $$
M. Ganzburg and S. Tikhonov have extend this result on the case of Nikolskii–Bernstein constants.
We prove inequalities
$$ n^{1/p}\mathcal{L}(p)\le \mathcal{C}(n,p)\le (n+\lceil p^{-1}\rceil)^{1/p}\mathcal{L}(p),\quad n\in \mathbb{Z}_{+},\quad 0<p<\infty, $$
which improve the result of Levin and Lubinsky. The proof follows our old approach based on properties of the integral Fejer kernel. Using this approach we proved earlier estimates for $p=1$
$$ n\mathcal{L}(1)\le \mathcal{C}(n,1)\le (n+1)\mathcal{L}(1). $$

Using such inequalities, we can estimate the constant $\mathcal{L}(p)$ solving approximately $\mathcal{C}(n,p)$ for large $n$. To do this we use recent results of V. Arestov and M. Deikalova, who expressed the Nikolskii constant $\mathcal{C}(n,p)$ using the algebraic polynomial $\rho_{n}$ that deviates least from zero in the space $L^{p}$ on the segment $[-1,1]$ with the weight $(1-t)v(t)$, where $v(t)=(1-t^{2})^{-1/2}$ is the Chebyshev weight. As consequence, we refine estimates of the Nikolskii constant $\mathcal{L}(1)$ and find that
$$ 1.081<2\pi \mathcal{L}(1)<1.082. $$
To compare previous estimates were $1.081<2\pi \mathcal{L}(1)<1.098$.