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A real polynomial is called Hurwitz (stable) if all its zeros have negative real parts. For a given n ∈ N we find the smallest possible constant d n > 0 such that if the coefficients of F (z) = a 0 + a 1 z + · · · + a n z n are positive and satisfy the inequalities a k a k+1 > d n a k−1 a k+2 for k = 1, 2, . . . ,n − 2, then F (z) is Hurwitz.

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The following theorem is proved. Theorem. Suppose M = (a i,j ) be a k × k matrix with positive entries and a i,j a i+1,j+1 > 4 cos 2 π k+1 a i,j+1 a i+1,j (1 ≤ i ≤ k − 1, 1 ≤ j ≤ k − 1). Then det M > 0.The constant 4 cos 2 π k+1 in this Theorem is sharp. A few other results concerning totally positive and multiply positive matrices are obtained.

Abstract. In this note we prove a new result about (finite) multiplier sequences, i.e. linear operators acting diagonally in the standard monomial basis of R[x] and sending polynomials with all real roots to polynomials with all real roots. Namely, we show that any such operator does not decrease the logarithmic mesh when acting on an arbitrary polynomial having all roots real and of the same sign. The logarithmic mesh of such a polynomial is defined as the minimal quotient of its consecutive roots taken in the nondecreasing order of their absolute values. Multiplicateurs et mailles logarithmiquesRésumé. Les multiplicateurs considérés dans cette note sont les opérateurs linéaires qui agissent diagonalement sur R[x] muni de sa base standard (les monômes) et qui transforment les polynômesà racines réelles en polynômesà racines réelles. Nous montrons qu'un tel opérateur, appliquéà un polynôme dont toutes les racines sont réelles et de même signe, ne diminue pas la maille logarithmique, c'est-à-dire le minimum du quotient de deux racines consécutives dans l'ordre croissant des valeurs absolues.