A sufficient condition for a complex polynomial to have only simple zeros and an analog of Hutchinson's theorem for real polynomials

We find the constant $b_{\mathrm{\infty }}$ ($b_{\mathrm{\infty }}\approx 4.81058280$) such that if a complex polynomial or entire function $f(z)={\sum }_{k=0}^{\omega }{a}_k{z}^k,$ $\omega \in \{2,3,4,\dots \}\cup \{\mathrm{\infty }\},$ with nonzero coefficients satisfy the conditions $\mid \frac{a_k^2}{a_{k-1}{a}_{k+1}}\mid >b_{\mathrm{\infty }}$ for all $k=1,2,\dots ,\omega -1,$ then all the zeros of $f$ are simple. We show that the constant $b_{\mathrm{\infty }}$ in the statement above is the smallest possible. We also obtain an analog of Hutchinson’s theorem for polynomials or entire functions with real nonzero coefficients.