An Estimate for the Sum of Legendre Symbols

For the sum $S$ of the Legendre symbols of a polynomial of odd degree $n\ge3$ modulo primes $p\ge3$, Weil's estimate $|S|\le(n-1)\sqrt p$ and Korobov's estimate
$$ |S|\le (n-1)\sqrt{p-\frac{(n-3)(n-4)}{4}}\qquad \text{for}\quad p\ge\frac{n^2+9}{2} $$
are well known. In this paper, we prove a stronger estimate, namely,
$$ |S|<(n-1)\sqrt{p-\frac{(n-3)(n+1)}{4}}. $$