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1

On certain weighted Schur type inequalities

Beberok Tomasz

Science, Technology and Innovation

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2022

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Vol. 16, no. 3-4

42--49

EN

In this note we give sharp Schur type inequalities for univariate polynomials with convex weights. Our approach will rely on application of two-dimensional Markov type inequalities, and also certain properties of Jacobi polynomials in order to prove sharpness.

2

The stability of polynomials with the change of their degree

Wanat M.

Automatyka / Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie

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2000

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T. 4, z. 2

169-174

EN

In the paper we considre the real polynomials. It was proved in [1], that the Hadamard product of real stable polynomial is stable. It is not difficut to see, that if for an unstable real polynomial Q(x) = a(n)x(n) + + a(n-1)x(n-1)+ ... + a(1X) + a(0) there exist numbers a(m), a(m-1),..., a(n+1) (m>n) such that the polynomial Q(x) = a(m)x(m) + ... + a(n+1)x (n+1) + Q(x) is stable then for all stable real polynomials P such that deg (Q) ≥ deg(P) the Hadamard product P x Q is stable. Hence, for Q(x) = a(n)x(n)+a(n-1) + ... + a1x + + a(0) we construct: Q(x) = a(m)x(m) + ...+a(n+1)x(n+1) + Q(x) and formulate a necessary condition on Q for Q to be Hurwitz stable. We also consider the stable prolongality of polynomials in the other direction.

PL

W pracy rozważane są wielomiany rzeczywiste. W [1] udowodniono, że iloczyn Hadamarda rzeczywistych wielomianów stabilnych jest stabilny. Można zauważyć, że jeżeli dla stabilnego wielomianu rzeczywistego Q(x) = a(n)x(n)+ a(n-1)x (n-1)+ ... +a(1)x + a(0) istnieją liczby rzeczywiste a(m), a(m-1),..., a(n+1) (m > n) takie, że wielomian Q(x) = a(m)x(m) + ... + a(n+1) +... + Q(x) jest stabilny, to iloczyn Hadamarda wielomianu Q z każdym stabilnym wielomianem rzeczywistym P takim, że deg (Q) ≥ deg(P) będzie stabilny. Dla wielomianu Q{x) = = a(n)x(n) + a(n-1) + ... + a(1)x + a(0) tworzymy zatem wielomian Q(x) = = a(m) +...+ a(n+1)x(n+1) + Q(x) i formułujemy warunki konieczne, jakie musi spełniać Q, aby wielomian Q był stabilny w sensie Hurwitza. W pracy rozważana jest również przedłużalność stabilna w przeciwym kierunku.

3

On a characterization of the logarithm by a mean value property

Ger R.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

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1998

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Vol. 4

1-7

EN

Any real polynomial f(x) = ax2 + bx + c, x ∈ IR, has the property that f (x)-f (y) x-y for every (x, y) ∈ IR, x ꞊ y. It turns out that that particular form of the Lagrange mean value theorem characterizes polynomials of at most second degree. Much more can be proved: J. Aczél [1] has shown that, with no regularity assumptions, a triple (/, g, h) of functions mapping IR into itself satisfies the equation f(x)-g(y) x-y= h(x + y) for all (x, y) ∈ IR, x ≠ y, if and only if there exist real constants a, 6, c such that f (x) = g(x) = ax2 + b, x + c, x ∈ IR, and h(x) = ax + b, x ∈ IR. Generalizations involving weighted arithmetic means were also considered (see e.g. M. Falkowitz [3] and the references therein) and characterizations of polynomials of higher degrees (in the same spirit) were obtained (see [4] and [5], for instance). In what follows we are going to characterize the logarithm in a similar way. To this end, denote by D the open first quadrant of the real plane IR2 with the diagonal removed, i.e. D := (O, ∞)2 \ {(x, x) e IR2 : x ∈ (0, ∞) }.Applying the classical Lagrange mean value theorem to the logaritmic function we derive the existence of a function D 3 (x, y) -> £(x,y) € intcony {x, y} such that the equality log a:-log y x-y £(z,y)