Markov's inequality for typically real polynomials

An Effective Lojasiewicz Inequality for Real Polynomials

Example 1. Set f1 = x d 1 and fi = xi−1 − x d i for i = 2, . . . , n. Then Φ(x) := maxi{|fi(x)|} > 0 for x 6= 0. Let p(t) = (t d , t n−2 , . . . , t). Then limt→0 ||p(t)||/|t| = 1 and Φ(p(t)) = t d . Thus the Lojasiewicz exponent is ≥ d. (In fact it equals d.) This works both over R and C. In the real case set F = ∑ f 2 i . Then degF = 2d, F has an isolated real zero at the origin and the Lojas...

Typically Real Harmonic Functions

We consider a class T O H of typically real harmonic functions on the unit disk that contains the class of normalized analytic and typically real functions. We also obtain some partial results about the region of univalence for this class.

An inequality for chromatic polynomials

Woodall, D.R., An inequality for chromatic polynomials, Discrete Mathematics 101 (1992) 327-331. It is proved that if P(G, t) is the chromatic polynomial of a simple graph G with II vertices, m edges, c components and b blocks, and if t S 1, then IP(G, t)/ 2 1t’(t l)hl(l + ys + ys2+ . + yF’ +spl), where y = m n + c, p = n c b and s = 1 t. Equality holds for several classes of graphs with few ci...

An inequality for Tutte polynomials

Let G be a graph without loops or bridges and a, b be positive real numbers with b ≥ a(a + 2). We show that the Tutte polynomial of G satisfies the inequality TG(b, 0)TG(0, b) ≥ TG(a, a). Our result was inspired by a conjecture of Merino and Welsh that TG(1, 1) ≤ max{TG(2, 0), TG(0, 2)}.

Turan’s Inequality for Appell Polynomials